O C Kantana

G Venard

Ťla Baudoin

C Kantana

Ťevaluation Comparison

This Ph.D. work contributes to the digital predistortion linearization technique of power ampliĄers. Power AmpliĄer is one of the most critical elements of radiocommunication systems, which exhibits static nonlinearities and nonlinear memory effects. Achieving a good trade-off between the linearity of the power ampliĄer and its efficiency is becoming more crucial. Digital predistortion is a powerful linearization technique that aims to compensate for power ampliĄer distortions and provides linear ampliĄcation with good efficiency. The predistortion principle consists of implementing a nonlinear function, the so-called predistorter upstream of the power ampliĄer. The predistorter ideally has the inverse characteristics of the power ampliĄer. This operation allows us to consider the memory effects of the power ampliĄer, and in particular, the long-term memory. Several behavioral models have been used as predistorter. Most of them are classiĄed into two families: global models derived from the Volterra series and models based on the segmentation approach. The comparative and analysis study of these models is one of the focuses of this dissertation, in which three aspects are used for comparison: linearization performance, complexity, and hardware implementation properties. By focusing on models based on the segmentation approach, this dissertation proposes an approach to design an optimal model according to a trade-off between linearization performance and model complexity. This model is used to linearize a dual-input Doherty power ampliĄer. A global optimization algorithm combined with a control process is proposed to enhance efficiency while maintaining a good linearity level according to a proposed adaptive cost function.

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Résumé

Le travail de th' ese présenté par ce manuscrit sŠintéresse ' a la linéarisation des ampliĄcateurs de puissance en utilisant la prédistorsion numérique. LŠampliĄcateur de puissance est lŠun des modules les plus critiques des équipements de communication radio qui présente des non-linéarités statiques ainsi que des effets de mémoire. Il devient de plus en plus crucial de réaliser un compromis entre la linéarité et le rendement énergétique. La prédistorsion numérique est une technique de linéarisation efficace qui compense les distorsions dues ' a la non-linéarité en appliquant une déformation sur le signal dŠentrée, de mani' ere ' a ce que le syst' eme global réalise une ampliĄcation linéaire. Le principe de la prédistortion numérique consiste ' a mettre en İuvre une fonction non linéaire dite prédistorteur en amont de lŠampliĄcateur de puissance, et qui a idéalement les caractéristiques inverses de lŠampliĄcateur de puissance. Cette opération permet également de prendre en compte les effets mémoire, et en particulier la mémoire ' a long terme. Plusieurs mod' eles comportementaux sont proposés pour la prédistorsion qui peuvent être classés en deux catégories : les mod' eles globaux et les mod' eles par segmentation. LŠétude comparative de ces mod' eles est lŠun des axes de travail traité par cette th' ese, dont la comparaison repose sur trois aspects qui sont soulignés : les performances de linéarisation, la complexité et les propriétés de lŠimplémentation matérielle. En se concentrant sur les mod' eles basés sur lŠapproche par segmentation, un algorithme est proposé pour concevoir un mod' ele optimal en satisfaisant un bon compromis entre les performances de linéarisation et la complexité du mod' ele. Ce mod' ele est utilisé pour linéariser un ampliĄcateur de puissance Doherty ' a double entrée pour lequel un algorithme dŠoptimisation globale associé ' a un processus de contrôle est proposé pour améliorer le rendement tout en conservant un bon niveau de linéarité selon une fonction de coût adaptative.

List of Figures

List of Tables

List of Acronyms

National Conferences and Workshops

Introduction Motivation

Modern wireless communication systems have been developed over the past three decades to provide high-speed services such as mobile communications, broadcast TV, WiFi networks, video conferences, etc. With their increasing demands for higher data rates and capacity, the rapid evolution of wireless communications continuously increases the complexity of radiofrequency systems. To meet the demands of future wireless communication systems, extensive research is being conducted to develop an energy-efficient and recon-Ągurable radio transmitter that can support multiple access technologies and operate at variable frequency bands.

Constrained by limited radio frequency (RF) resources, transmitting modulations, such as orthogonal frequency division multiplexing (OFDM), are used to achieve high spectral efficiency. However, these non-constant envelope signals lead to modulated signals with a high peak-to-power average power ratio (PAPR), which are more sensitive to the transmission channelŠs nonlinearity and require a high degree of linearity at the transmitter. Besides, carrier aggregation allows for a very high data rate but creates new challenges for transmitters in terms of bandwidth, intermodulation, harmonics, and Ąltering.

The power ampliĄer (PA) is one of the essential components of radio transmitters. Besides its role in amplifying the signal, the PA is also the primary source of signal distortion and the major contributor to the radio transmitter chainŠs energy consumption. To maintain the reliability of the RF system and reduce energy consumption, PA is required to be linear and have high efficiency. Unfortunately, the design of high-efficiency and linear PAs is not easy.

On the other hand, the development of 5G and beyond 5G wireless communication creates new challenges to provide a wide range of frequency bands that 5G wireless communication systems can handle. With the cadence of this development, high efficiency and linearity are two crucial requirements that are not easy to fulĄll simultaneously, and achieving a good trade-off between them is becoming more crucial. For high efficiency, PAŠs operating point (OP) is usually driven towards the saturation region where high nonlinear behavior is exhibited. Besides strong nonlinearities, PAs may have memory effects.

The output of an ideal PA should be linear, but in reality, PA is neither perfectly linear nor entirely instantaneous when a memory effect happens. These nonlinearities generate distortions in the signal band and cause spectral regrowth in the adjacent channels. Unfortunately, a good efficiency is obtained at the price of poor linearity, especially with modern communication waveforms with very high PAPR and large bandwidths. This situation leads to ensures a trade-off between efficiency and linearity. The nonlinear behavior of the PA depends not only on its electronics components but also on the characteristics of the transmitted signal.

The behavioral modeling of PA and its linearization have given many research works in recent years. Digital predistortion (DPD) is one of the most efficient linearization techniques used to linearize high PAs, pushing the OP of PA towards its high-efficiency region. The principle of DPD consists of applying, upstream of PA, a pre-correction on the signal so that the cascade of DPD and PA produces a linear and memoryless ampliĄcation with good efficiency. The purpose of studying the distortions of PA is to design an appropriate DPD function, called predistorter, that will compensate for these distortions and be implemented in a real digital circuit (FPGA, DSP ...). The hardware implementation, which is an important aspect of this dissertation, involves two aspects: identiĄcation and predistortion implementation. Many mathematical models have been proposed in the literature in order to be used to accurately model the dynamic nonlinear systems, including linearity and different types of memory effects. One of the contributions of this Ph.D. work is the comparison of main DPD models in terms of linearization performance, complexity, and hardware implementation properties.

The DPD models are mainly derived from the Volterra series. However, other models are based on the segmentation approach, such as the canonical piecewise linear (CPWL) functions. The segmentation approach is based on splitting the input baseband signalŠs amplitude range into different regions characterized by different models. The models based on the segmentation approach can represent strong nonlinearities and are less prone to numerical problems as they are divided into segments in which a smaller order of nonlinearity is required. The design of an appropriate DPD model based on the segmentation approach with low complexity and a viable implementation strategy is an essential topic of this dissertation.

On the other hand, maintaining a good linearity level using the DPD with enhanced PA efficiency becomes more complex, especially for wideband signals with high PAPR. Advanced PA architectures have been proposed in the literature to enhance efficiency and avoid wasting excessive power resources. Some of these architectures are used with two RF inputs, so having more degrees of freedom offered by the separate inputs can control the performance and improve the PA efficiency.

Objectives

The above discussion reveals signiĄcant challenges in applying existing linearization solutions to improve further the system for both current LTE and future 5G wireless communication systems. This thesis aims to address these questions through the following objectives:

This dissertation mainly focuses on:

• Analysis and comparative study of the DPD models in terms of linearization performances, model complexity, and numerical hardware properties.

• Search of the optimal structure of models with segmentation approach according to a cost function representing a trade-off between modeling accuracy and model complexity. Besides, an approach is proposed to divide the model segments optimally.

• Optimization of the structure of dual-input PA with DPD in which an optimization process is proposed to enhance the efficiency while maintaining the linearity.

Main Contributions

The main contributions of this dissertation are listed as follows:

• An analysis and comparative study of the DPD models. These models have been categorized into two families: global models derived from the Volterra series and models based on the segmentation approach. The models have been evaluated according to a trade-off between:

-Linearization performances presented by the modeling accuracy measure and the adjacent channel power ratio.

-Complexity presented by the number of DPD coefficients to be identiĄed.

-Hardware implementation related to the identiĄcation process and predistorter implementation.

• A new approach is proposed to optimally set the thresholds bounding the segments of the DPD models based on the segmentation approach. The impact of the model parameters on optimal segmentation has been investigated.

• A heuristic algorithm based on hill-climbing is proposed and conĄrmed to determine the optimal structure of the DPD model based on the segmentation model with:

-A new deĄnition of the neighborhood for the hill-climbing algorithm.

-A designed cost function is representing the trade-off between modeling accuracy and model complexity.

• A proposed system approach to design an optimal DPD model in terms of modeling accuracy, model complexity, and numerical properties. This contribution is based on:

-A proposed approach to integrate with reduced system complexity the hillclimbing algorithm and the segmentation optimization process.

-A proposed strategy for the hardware implementation of the DPD model designed by the system approach.

• A proposed auto-tuning approach to optimize the parameters of the dual-input Doherty PA to enhance its power efficiency with maintaining a good level of linearity.

The designed DPD model is used to linearize the PA and conĄrms our proposed system approach.

Context

The Ph.D. work presented in this dissertation is funded by the FUI22 APOGEES research project (AmPliĄcation recOnĄGurablE multimodES). The APOGEES project aims to solve the problems related to the reconĄgurability of ampliĄcation systems to meet the different challenges generated by the new uses of the spectrum.

The APOGEES project has been labeled by Aerospace Valley, Image Réseaux, and Elopsys. The industrial partners involved in this project (ARELIS, THALES, TELERAD) aim to implement a reconĄgurable system that deals with distortions of power ampliĄers to make the transmitter systems more Ćexible for different applications: drones control, tactical communications, aeronautical communications, etc. The work carried out within this thesisŠs framework is also done in close collaboration with the other academic partners involved in this project: Centrale Supélec, IMS Bordeaux, XLim Poitiers, and IETR University of Nantes.

International mobility was carried out as an intern at Mitsubishi Electric Research Laboratories (MERL) in Cambridge, USA, under the host Dr. Rui Ma and Dr. Mouhacine

Benosman. The internship is titled Machine Learning for Digital Predistortion Linearization of High-Efficiency Dual-Input Doherty Power Amplifier.

Outline

The dissertation consists of Ąve chapters.

Chapter 1 presents general concepts and the background of PA. Figures of merit used to assess the PA distortions are cited. PAPR reduction techniques and linearization techniques are also introduced, which are at the heart of this dissertation.

The principle of DPD is described in Chapter 2. Different DPD models are cited and reviewed. An analysis and comparison of DPD models is achieved according to modeling performances, complexity, and numerical properties.

In Chapter 3, the Decomposed Vector Rotation (DVR) model is investigated. A new approach to optimize the thresholds that determine the boundary of the segments is proposed. The determination of the optimal structure of the DVR model is introduced as well.

A system approach is proposed in Chapter 4 to design an optimal DVR model with reduced complexity by combining the thresholds optimization and the DVR modelŠs optimal sizing. The performances of the designed optimal DVR model are compared with those of the conventional DVR model. An implementation strategy of the optimal model is presented.

Chapter 5 presents the architecture of dual-input Doherty PA, in which a proposed optimization process is presented. The designed optimal DVR model from Chapter 4 is used as a DPD model to linearize the dual-input Doherty PA, where the DPD is optimally integrated into the global optimization process.

Finally, we give the conclusion and perspectives.

Chapter 1

Generalities on Radio Frequency Power Amplifier

Introduction

Wireless radiofrequency (RF) transmitter is a fundamental block of modern communication systems to transmit information from a given source to a receiver. Figure 1.1 illustrates a conventional architecture of an RF transmitter by highlighting the digital and analog domains. The information to be transmitted is Ąrst processed in digital form based on digital circuits such as FPGA (Field-Programmable Gate Array) or by digital signal processing (DSP). During this step, the information is encoded to be transmitted on two channels: in-phase I and quadrature Q. The IQ data are then passed through a block, so-called digital front-end for the RF processing, which usually contains a digital-up converter (DUC) for the frequency transposition, a crest factor reduction (CFR) to improve the PA efficiency, and digital predistortion (DPD) for the PA linearization. The signal is then converted by a digital-to-analog converter (DAC). Reconstruction Ąlters are often used after this conversion to eliminate unwanted frequency components and limit bandwidth.

The analog part of the transmitter is responsible for transposing the signal from the baseband to the RF domain. Several transmitter architectures are used, such as direct conversion (Zero-IF), which transposes the signal directly to the carrier frequency, and low-IF architecture, which transposes the signal to an intermediate frequency before being transposed to the carrier frequency. Since the power of the RF signal must be of a sufficient level to be transmitted through the channel, an ampliĄcation module is required to supply the power needed for the RF signal to ensure its transmission to destination. The RF transmitters are expected to conduct modulated signals while maintaining three performance indicators: linearity, bandwidth, and power efficiency. The linearity is affected by the nonlinear behavior of the transmitter caused by its distortions. Ensuring linearity is an essential aspect to preserve the quality of the transmitted signal and to avoid any loss of information during the transmission process. Since the bandwidth of the RF signal to be transmitted is wider than that of the input signal due to the generation of new frequency components, the bandwidth is a critical aspect to be managed. Moreover, wider bandwidths are needed to accommodate higher data. Power efficiency is an important consideration that reĆects the power level of transmitting the information. It depends on the electronic topology of the transmitter, in particular the power ampliĄer, and affects the costs of deploying and operating the communication infrastructure and the environmental impact. An ideal transmitter must achieve a better compromise between good linearity and high power efficiency in handling wideband signals.

Wireless transmitters consist of several stages, including modulation, DAC, frequency up-conversion, Ąltering, and ampliĄcation. Different wireless transmitter architectures have been introduced in the RF community, and one of them is shown in Figure 1.1. The position of DAC may change in some structures.

The distortions seen in RF transmitters are usually due to hardware imperfections. They have various origins, such as IQ imbalance, oscillator phase noise, DAC imperfections, frequency response distortions, amplitude and phase distortions, etc. The most predominant distortions are those due to the nonlinearity presented in the RF front-end and mainly the RF power ampliĄer (PA), which is identiĄed as the primary source of nonlinear distortions. Thus, modeling and compensating for nonlinear transmitter distortions are often turned to modeling and compensating for PA nonlinearities.

This chapter presents some generalities related to PAs. The distortions introduced by PA and the metrics to evaluate them are discussed. Two essential aspects concerning the PA, which are linearity and efficiency, and the techniques to improve them will be discussed.

Waveform Features

To quantify the dynamic of modulated waveform with non-constant envelope, some criteria such as peak-to-average power ratio (PAPR) and effective PAPR are used.

To statistically characterize the modulated waveformŠs instantaneous power Ćuctuations, the most used parameters are the crest factor (CF) and its quadratic value, the PAPR, which is deĄned by the ratio of the peak power P peak and the average power P avg as:

PAPR dB = 10 log P peak P avg  (1.1)
The CF is deĄned as square root of the PAPR:

CF = √ PAPR (1.2)
It is important to know that the PAPR of the baseband signal and the RF signal are not the same. However, they are linked by:

PAPR RF dB ≤ PAPR baseband dB + 3 dB (1.3)
where it is true if the carrier frequency f c is much greater than the signal bandwidth, which is met almost all the time. In this dissertation, we will only talk about PAPR baseband dB which will be denoted by PAPR. Generally, the PAPR is used to deĄne and represent the dynamics of the signal. However, this parameter is not sufficient to characterize the real dynamics of the signal.

Figure 1.2 presents the magnitude of IQ data in dBm of LTE signal, where the PAPR of the LTE signal is shown. This value so-called effective PAPR, denoted by PAPR 0 , is deĄned using CCDF by

p(PAPR ≥ PAPR 0 ) = α (1.4)

Overview of Power Amplifiers Characteristics

The PA is an essential component of the transmission block of RF wireless systems. In its conventional deĄnition, a PA is an electronic device used to increase the power of a signal.

The PA is usually the Ąnal stage before the antenna, which is designed to gain the input signal and, therefore, the power required to transmit it through the radio channel to reach the receiver. In an ideal context, the ampliĄcation operation should not clip the desired signal. However, the PA being a nonlinear device, is a source of distortions that impact the transmitted signal.

The design of PA is always subject to the antagonism between linearity and power efficiency [START_REF] Lavrador | ŤThe Linearity-Efficiency Compromise[END_REF]. Figure 1.3 shows a common behavior of output power (blue curve) and power efficiency (red curve) with respect to the input power, where the maximization of efficiency is done by sacriĄcing linearity by pushing the operating point of the PA towards its saturation zone. Figure 1.3: Trade-off between the linearity and the efficiency of PA However, the operating point needs to be backed off from the saturation zone to avoid nonlinear distortions. For this to happen, an approach is required to maximize power efficiency while keeping its distortions at a reasonable level that can be compensated at the system level using linearization techniques such as Feedforward, Feedback, or Predistortion [START_REF] Katz | ŤLinearization: reducing distortion in power ampliĄers[END_REF].

The behavior and characteristics of a PA depends on its class and topology. In the following, we present some general characteristics of PA.

Gain

The gain is one of the most important characteristics of the PA, which provides a measure of the ampliĄcation level of the transmitter system.

The power gain is the ratio between the output power and the input power of the signal, which is given in dB by

G dB = P out dBm -P in dBm (1.5)
where P in dBm and P out dBm are the input and output power, respectively.

AM-AM and AM-PM Characteristics

The nonlinearities can be characterized by curves called Amplitude-to-Amplitude (AM-AM) and Amplitude-to-Phase (AM/PM) curves, as shown in Figure 1.4 that is an example for a Doherty PA. The blue curve is the AM-AM characteristic, which shows the normalized magnitude of the PAŠs output signal versus the normalized magnitude of its input signal. The signals are the complex baseband IQ signals. The orange curve is the AM-PM characteristic, which shows the PA output signalŠs phase deviation versus the normalized input signal magnitude.

y(n) = Gx(n) (1.6)
But in reality, the gain is not constant due to the inherent nonlinearities in the PA, which are represented by the characteristic AM-AM [START_REF] Katz | ŤLinearization: reducing distortion in power ampliĄers[END_REF]. On the other hand, the nonlinear effects, whatever its source (average power, bandwidth, polarization conditions, temperature, etc.), change the phase shift between the input and output signals. This is represented by the AM-PM characteristic.

1-dB Compression Point

The AM-AM and AM-PM characteristics are widely used to characterize the transfer function of PA. It can be seen that the gain is compressed by PA when the input power increases.

By deĄnition, the 1 dB compression (P1dB) is the point where the gain of the PA is compressed by 1 dB against the small-signal gain of the linear region, as shown in Figure .  1.5. The points P in C and P out C represent the input and output power levels for which 1 dB of gain compression is obtained, respectively.

Back-Off

To ensure reliable signal transmission and avoid saturation at the PAŠs output, it is necessary to keep the operating point backed off from the saturation.

This back-off can be adjusted according to the input or output power at saturation, denoted respectively by P outsat or P insat , by using two indicators: input back-off (IBO) and output back-off (OBO), which are illustrated in Figure 1.6.

Figure 1.6: 1-dB compression point on AM-AM characteristic

The IBO is deĄned as the ratio between P insat and the mean input power P in of the signal and is expressed in dB by

IBO dB = P insat (dBm) -P in (dBm) (1.7)
The OBO is deĄned as the ratio between P outsat and the mean input power P out of the signal and is expressed in dB by

OBO dB = P outsat (dBm) -P out (dBm) (1.8)

Harmonics and Intermodulation Products

The nonlinear behavior can be analyzed by exciting the PA by signals of different types: continuous wave (CW) signal, two-tone signal and modulated signal, while monitoring saturation power, gain, 1-dB compression point, and intermodulation distortions.

CW Signal Test

Stimulating the PA by a CW signal is a typical test that aims to evaluate the PA characteristics in terms of linearity and power efficiency. As shown in Figure 1.7, the CW signal test consists of exiting the PA with a purely sinusoidal signal

x(t) = A cos(2πf 0 t)
with an amplitude A at a frequency f 0 . The behavior of the PA in the time domain shows a distortion of the signal at the PA output, which results in the frequency domain by the generation of new frequency components depending on the fundamental frequency f 0 : (2f 0 , 3f 0 , 4f 0 , ...), called harmonic components. The higher the power of these harmonic components, the stronger PAŠs nonlinearity.

The CW signal test appears to be a good approach to characterize the fundamental effects of the PA nonlinearities. However, this test remains limited for current communication systems and does not make it possible to further characterize the nonlinearity effects for modulated signals.

Two-Tone Signal Test

The two-tone signal test is widely used method for more relevant PA characterization and to have better description of the nonlinear behavior of the PA.

The two-tone signal is composed of two closely-spaced sinusoidal frequencies with amplitude A, and frequencies f 1 and f 2 :

x(t) = A cos(2πf 1 t) + A cos(2πf 2 t) = 2A cos(2πf m t) cos(2πf c t) where f m = f 2 -f 1 2 and f c = f 2 +f 1 2
. By driving the PA by the two-tone signal as shown in Figure 1.8, the output contains the frequency harmonics (2f 1 , 2f 2 , 3f 1 , 3f 2 , ...) and additional frequency components, called intermodulation (IMD) products, generated at frequencies mf 1 + nf 2 where m and n are positive or negative integer and ♣m♣+♣n♣ = k where k is the order of intermodulation. The odd-order terms of the IMD product (k = 3, 5, 7, . . . ) are the most likely to disturb the ampliĄed signal since they are close to the PAŠs useful band. In general, the 3 rd and 5 th IMD products have the highest power.

In order to quantify the spectral regrowth at the output of a PA driven by a two-tone, we use a metric called the carrier to inter-modulation distortion ratio (C/IMD), which represents the ratio in a linear scale between the power at the fundamental frequencies f 1,2 and the power generated at inter-modulation frequencies mf 1 + nf 2 , as shown in Figure 1.9. The two-tone test allows a standard evaluation of distortions by quantifying the nonlinear behavior of the PA. However, it is insufficient to represent the nonlinear behavior when the PA is driven by a real complex modulated signal used for communication systems.

Power Efficiency

An efficient PA aims to deliver a certain amount of power to the load without consuming too much power itself [START_REF] Reynaert | Rf power ampliĄers for mobile communications[END_REF].

However, the actual DC power consumption, P DC is always more extensive than the output power P out . The PA can dissipate a considerable amount of energy in the form of waste heat. Two parameters are commonly used to characterize the PA efficiency:

• Drain efficiency which is deĄned as the ratio between the output power P out and the DC power P DC :

η DC = P out P DC (1.9)
• Power added efficiency (PAE) which takes into account the PA input power P in , and is deĄned as:

η P AE = P out -P in P DC (1.10)

Memory Effect

In some instances, the PA is characterized by considering only its static characteristics. Mainly, the PA exhibits nonlinear and dynamic dispersive phenomena called memory effects, which means that the systemŠs output at any instants depends not only on the corresponding instantaneous input but also on the inputs at other previous instants. Therefore, the characteristics change with the frequency.

In narrowband wireless communication systems, the PA can be considered memoryless or quasi-memoryless. Wideband wireless communication systems need to identify whether the PA has a memory effect or not [START_REF] Raich | ŤOn the modeling of memory nonlinear effects of power ampliĄers for communication applications[END_REF].

Figure 1.10 presents the AM-AM characteristic of PA excited by a two-tone signal with a bandwidth of 1 MHz and 8 MHz. As shown in Figure 1.10b, the AM-AM characteristic for the wideband scenario exhibits strong dispersion. For PA design, substantial knowledge on the origins of memory effect and the accurate approaches to measure and quantify their impacts are required [START_REF] Boumaiza | Thermal memory effects modeling and compensation in RF power ampliĄers and predistortion linearizers[END_REF]. The origin of the memory effect can be either thermal or electrical [START_REF] Ghannouchi | Behavioral modeling and predistortion[END_REF].

The memory effect could be categorized into two types: short-term memory effect, also known as high-frequency (HF) memory effect, and long-term memory effect, also known as low-frequency (LF) memory effect.

The HF memory effects are caused by short time constants phenomena and depend on matching circuits and active components. The LF memory effects are induced by the slow variations of the signal, i.e., the same order of magnitude as the inverse of the envelope bandwidth. These undesirable effects are due to the componentsŠ self-heating, the trapping effect due to speciĄc technologies such as GaN, value of the input and/or output load of the transistor at the envelope frequency, automatic gain control, etc.

Figures of Merit

Stimulus signal may experience substantial distortions due to the transceiver chainŠs nonlinear components, such as the PA, and other imperfections such as the IQ imbalance, oscillator phase noise, and sampling jitter.

As a result of the nonlinearities, in-band and out-of-band distortions are generated. The main effects are the cloud-like shape of constellation points and the out-of-band radiation. The error vector magnitude (EVM) and the adjacent channel power ratio (ACPR) are mainly used to evaluate the distortion introduced by PA [START_REF] Isaksson | ŞA comparative analysis of behavioral models for rf power ampliĄers,Ť Microwave Theory and Techniques[END_REF]. For high PA nonlinearities with many carriers, noise power ratio (NRP) can be used to assess performance.

Adjacent Channel Power Ratio

The imperfections and nonlinearities of PA usually result in some adjacent channel spectral regrowth.

This phenomenon can be quantiĄed with ACPR deĄned as a bandwidth-limited ratio between the power in the main channel and the power in the adjacent channel as illustrated in Figure 1.11. where B represents the bandwidth of the signal and P (.) is power spectral density.

Error Vector Magnitude

The EVM is a metric that measures the in-band distortion level of the PA driven by a modulated signal. It is deĄned in the constellation domain and evaluates the deviation between the reference constellation point and the actual constellation point obtained in the presence of distortions.

Analytically, EVM is deĄned as:

EVM % = 1 N N -1 j=0 (δI 2 j + δQ 2 j ) S 2 avg × 100% (1.12)
Where δI and δQ are errors magnitude corresponding to in-phase symbol and quadrature symbol of received data compared with an ideally reconstructed constellation respectively, N is the number of symbols, S 2 avg is the average square magnitude. The offset and the rotation of the constellation can also be considered in the deĄnition of EVM.

Figure 1.12 shows an example of a 64-QAM signal constellation from Doherty PA operating at 3.1 GHz. The blue points present the constellation of the input signal. The red points present the constellation of the output signal after demodulation. In this case, the EVM is 10 %. Some measurements (IQ data) from real PA were collected in this dissertation without any demodulation information. For these waveforms, the value of EVM, denoted by EVM app , has approximately been estimated using NMSE in-band. Figure 1.12: Constellation of input-output IQ data with EVM=10%

Noise Power Ratio

Noise power ratio (NPR) is another possible measurement to characterize nonlinear distortions. As EVM it is used to evaluate in-band distortion.

In the NPR measurement procedure, multi-carrier signals with random amplitudes and phases (such as OFDM) will be simulated by an additive white Gaussian noise. A large number of tones with different frequencies with the same magnitudes and random phases with a uniform distribution can generate a Gaussian white noise [START_REF] Reveyrand | A Novel Experimental Noise Power Ratio Characterization Method for Multicarrier Microwave Power AmpliĄers[END_REF], which is Ąltered and passed through a narrowband-reject Ąlter to produce a deep notch in the noise pedestal center, then stimulate the PA and produce the IMD product.

The PA nonlinearity will result in the generation of a spurious signal in the notch. The NPR will then be used to evaluate the ratio between the average power of the signal outside the notch and the intermodulation noise in the notch. It is expressed as follows:

NRP dB = 10 log 10 2 B notch P (y)df 2 B carrier P (y)df B carrier B notch (1.13)
where B represents the bandwidth and P (.) is power spectral density.

The EVM and NPR are considered practical measurements for quantifying the inband distortion of the PA. However, NPR measurement does not require transmitted signal demodulation, unlike EVM [START_REF] Sombrin | On the formal identity of evm and npr measurement methods: Conditions for identity of error vector magnitude and noise power ratio[END_REF].

PAPR Reduction Techniques

To improve the PA efficiency, an intuitive solution is to reduce the PAPR of the input signal. PAPR reduction techniques, commonly known as Crest Factor Reduction (CFR), Ąnd their interest in improving the PAŠs efficiency.

Several PAPR reduction techniques have been proposed in the literature. As described in [START_REF] Cheaito | Analyse théorique des distorsions dans la bande et en dehors de la bande de transmission pour les signaux ' a porteuses multiples : Impact conjoint des non-linéarités de lŠampliĄcateur de puissance et effets de mémoire et prédistorsion[END_REF], PAPR reduction techniques can be classiĄed into three top categories: coding methods, probabilistic methods, and adding signal methods.

In order to evaluate the performance of every technique, the PAPR reduction is the most signiĄcant criterion. However, other metrics could be used to quantify the effectiveness of the approach used to reduce PAPR.

• Average power variation: Some PAPR reduction techniques result in a decrease or increase of the transmitted signalŠs average power. In the literature, most PAPR reduction techniques performance studies do not take into account the average power variation of the transmitted signal [START_REF] Guel | ŤAnalysis and comparison of clipping techniques for OFDM Peak-to-Average Power Ratio reduction[END_REF]. This variation has a strong impact on the quality of the transmission.

• In and out of band distortions: Some PAPR reduction techniques introduce in-band and/or out-of-band distortions because of their nonlinear operation. EVM and ACPR are used to evaluate these distortions.

• Downward compatibility: A PAPR reduction technique is said to be downward compatible if it does not involve any change on the receiving side. This is the case of tone reservation and clipping technique. The coding methods are not downward compatible as they require post-processing on the receiver side.

• Data rate loss: For some methods, the receiver needs additional information to recover useful transmitted data. These methods need an increase in the bandwidth and consequently a decrease in the spectral efficiency. If the bandwidth has to be kept constant, this information transmission involves a data rate loss. This is the case of the selective mapping technique.

Coding Methods

The coding methods consist in reducing the occurrence probability of the same phase value of signals which add up with the same phase. A simple block coding scheme was introduced in [START_REF] Jiang | An overview : Peak-to-average power ratio reduction techniques for ofdm signals[END_REF], and it consists of Ąnding out all possible codewords and then select those codewords of the lowest PAPR. It has been shown that using this technique, the PAPR of the signal can be reduced by 4 dB.

In [START_REF] Popovic | Synthesis of power efficient multi-tone signals with Ćat amplitude spectrum[END_REF] and [START_REF] Jiang | Block coding scheme for reducing papr in ofdm systems with large number of subcarriers[END_REF], the authors used the Golay complementary sequences where more than 3 dB PAPR reduction has been obtained. However, the coding methods is limited by two constraints. The Ąrst requires a brute-force approach to Ąnd the best appropriate code-word. The second is the complexity of storing in LUTs for encoding and decoding in the transmitter and receiver.

In summary, the actual beneĄts of coding for PAPR reduction for practical multicarrier systems are limited, regarding the low coding rate, the intractable required search for a good code, and the prohibitive complexity for many sub-carriers.

Probabilistic Methods

The idea behind the probabilistic methods is to perform several copies of the initial signal by modifying the phase, amplitude, and/or position of sub-carriers and then select the copy with the minimum PAPR.

These methods cannot guarantee the PAPR below a speciĄed level. Moreover, it decreases the spectral efficiency, and the computational complexity increases as the number of sub-carriers increases. The probabilistic methods include selective mapping (SLM) technique and partial transmit sequence (PTS) [START_REF] Muller | Ofdm with reduced peak-to-average power ratio by optimum combination of partial transmit sequences[END_REF]. The block diagram of SLM technique is shown in Figure 1.13.

Figure 1.13: Selective mapping technique

In the SLM technique, the input data sequences are multiplied by U different phase sequences to generate alternative input symbol sequences. Each of these alternative input data sequences are then applied to the IFFT operation, and then the one with the lowest PAPR is selected for transmission [START_REF] Bauml | Reducing the Peak to Average Power Ratio of Multicarrier Modulation by Selected Mapping[END_REF].

Therefore, its performance in reducing the PAPR depends on the number and phase factors design. The corresponding selected phase factor also needs to be transmitted to the receiver as side information to extract the original information properly.

Adding Signal Methods

This category, as its name suggests, includes all techniques of PAPR reduction that can be formulated as: PAPR(X + C papr ) < PAPR(X) (1.14) where X refers to the waveform signal and C papr refers to the peak-reduction signal.

Clipping

Clipping is one of the simplest techniques to reduce the PAPR [START_REF] May | Reducing the peak-to-average power ratio in ofdm radio transmission systems[END_REF], where the amplitude peak of a given signal x(n) is clipped according to a certain threshold A max .

x(t) = ♣x(t)♣e jϕ(x) if ♣x(t)♣ ≤ A max A max e jϕ(x) if ♣x(t)♣ > A max (1.15)
This technique generates some unwanted distortions in in-band and out-band, which degrades the system performance, including bit error rate (BER) and spectral efficiency.

Tone reservation

Tone reservation for multi-carrier signals proposed in [START_REF] Tellado-Mourelo | ŞPeak to average power reduction for multicarrier modulation[END_REF] is based on the reservation of sub-carriers that do not carry any useful information, in order to generate the PAPR reduction signal that lowers its peaks when added to the original multi-carrier signal.

Active Constellation Extension

Active constellation extension introduced in [START_REF] Jones | Peak power reduction in ofdm and dmt via active channel modiĄcation[END_REF] reduces the PAPR by appropriately extending the outer points of the signal constellations in the frequency domain. The constellation modiĄcations appropriately cancel the time domain peaks of the transmitted signal. This is achieved without any degradation in BER performance and data rate. However, this comes at the cost of a slight increase in the average power of the transmitted signal.

Linearization Techniques

Many linearization techniques compensate for PAŠs distortion, such as Feedforward, Feedback, and Predistortion.

Back-Off

The simplest method to deal with PA nonlinearity is to operate the PA with a signiĄcant power back-off from its saturation point to achieve linear ampliĄcation. This implies that the maximum input power level must be capped so that the entire signal remains within the linear region of the PA. This solution is not exactly a linearization technique. However, it is often used because of its simplicity of realization, but at the cost of poor energy efficiency.

Feedforward

The principle is to extract the error signal (distortions) at the output of PA and then subtract it from the output signal. Therefore, an input signal splits into two paths, as shown in Figure 1.14. The Ąrst path consists of the main PA and time delay with some coupling elements. The second one ampliĄes the error signal that is the difference between the output of the Ąrst PA aligned in time and amplitude with the input signal and the time-delayed input. A coupler subtracts the ampliĄed error signal from the PAŠs time-delayed distorted output at the output. Thus, the distortions in the out-band are reduced.

Feedback

The output from the feedbacked ampliĄer is subtracted from the input signal with a speciĄc attenuation. Therefore, the PA is driven by the error signal. The advantage is the relative simplicity of additional circuitry needed to provide feedback. On the contrary, the main disadvantage of feedback is caused by problems of stability and bandwidth limitation. Feedback linearization is classiĄed into two main groups: RF Feedback and Cartesian Feedback. The basic layout of RF Feedback linearization can be seen in Figure 1.15.

Predistortion

Predistortion (PD) is an efficient technique for linearizing the PAs. It introduces a module, called predistorter, upstream of the PA, which compensates for the PAŠs nonlinearities. The predistortion does not require an in-depth knowledge of the PAŠs physic circuits, and we only need to precisely measure the PAŠs actual behaviors for the predistortion procedure [START_REF] Kim | Digital predistortion of wideband signals based on power ampliĄer model with memory[END_REF].

The Ąrst PD systems for wireless communications were proposed by Nagata [START_REF] Nagata | Linear ampliĄcation technique for digital mobile communications[END_REF] in 1989, Cavers [START_REF] Cavers | AmpliĄer linearization using a digital predistorter with fast adaptation and low memory requirements[END_REF] - [START_REF] Cavers | A linearizing predistorter with fast adaptation[END_REF] in 1990 and Wright [START_REF] Wright | Experimental performance of an adaptive digital linearized power ampliĄer [for cellular telephony[END_REF] in 1992.

The implementation of PD can be done in analog or digital domain, but currently, the most common implementation is digital, applied to the baseband signal [START_REF] Ding | Memory polynomial predistorter based on the indirect learning architecture[END_REF] - [START_REF] Marsalek | Contributions to the power ampliĄer linearization using digital baseband adaptive predistortion[END_REF].

In the analog domain, the analog PD is performed at the carrier RF frequency, at the IF frequency, or at the baseband level after the DAC, using analog circuits, based on diodes or transistors [START_REF] Raab | Power ampliĄers and transmitters for RF and microwave[END_REF] - [START_REF] Mouafo | Etude dŠampliĄcaeurs faible niveau ' a haute linéarité en technologies intégrées HEMT ASGA pour applications spaciales[END_REF]. The implementation of analog PD has been proposed in [START_REF] Cripps | RF Power AmpliĄers for Wireless Communications[END_REF]. In [START_REF] Lee | An adaptive predistortion RF power ampliĄer with a spectrum monitor for multicarrier WCDMA applications[END_REF], the authors propose an approach where the modulated signal is predistorted. Furthermore, they also proposed an approach to make this analog predistortion adaptive. The analog PD circuits are generally small, facilitating their integration into a power ampliĄcation system with reduced power consumption. However, linearity is limited because the elimination of intermodulation is generally sufficient only for third-order IMD. Moreover, the drifts due to the aging of the PDŠs analog components and the PA cannot be corrected.

When speaking of analog PD, analog processing may be less power-consuming than its digital counterpart. However, it remains limited to reasonably simple predistortion functions like the memory polynomial and does not offer the same processing versatility as digital processing.

Predistortion is widely implemented in the digital domain, which offers better linearization performance. The digital predistortion (DPD) process is done by hardware, using a programmable logic device (FPGA, ASIC, DSP). Besides, the predistortion system can be adaptive by adding a feedback path to the system. This path takes a part of the PA output in order to update the predistortion block. The DPD is independent of the carrier frequency, which is a property of Ćexibility, and it remains helpful in a multi-mode or multi-band context, thanks to adaptability.

In 4G and 5G wireless communication systems, very high data rates are achieved thanks to carrier aggregation generating signals with very high bandwidths for which DPD may be challenging to implement (it would require too expensive DAC and ADC converters and high computation load).

On the other hand, the DPD is suitable for wideband applications. Furthermore, it is highly Ćexible and could be reconĄgurable. However, the variation on PA characteristics due to the variation of component tolerances and the drift of temperature should be considered. The adaptation is required.

The core of this dissertation focuses on the PA linearization using the digital predistortion, which will be detailed in the following chapter.

Conclusion

This chapter has provided an overview of the PA and its characteristics and the distortions that are introduced. The main metrics that evaluate these distortions have been presented. Two main aspects concerning PAs, which are linearity and power efficiency, are discussed. The techniques to improve each aspect are discussed, as well.

The main contributions of these dissertations are thoroughly presented in the following chapters.

Chapter 2

Analysis and Comparative Study of Digital Predistortion Models

Introduction

The concern to linearize the PA dates back to the beginning of broadcasting and the expansion of telecommunications in the last century [START_REF] Katz | The Evolution of PA Linearization: From Classic Feedforward and Feedback Through Analog and Digital Predistortion[END_REF]. As the transmissions shift to communication with higher spectral efficiency waveforms such as M-QAM, the requirement for linearity becomes more stringent and opens the way for DPD to be developed and implemented. Even if the main goal of DPD is to linearize the power ampliĄer, it also contributes, in many cases, to improve the power efficiency, which is vital since PAs are responsible for a signiĄcant part of the power consumption in base stations of wireless cellular networks [START_REF] Auer | How much energy is needed to run a wireless network[END_REF].

This chapter mainly focuses on the principle of DPD, its architectures, and the different DPD models. The comparison of these models, which presents the Ąrst contribution of this dissertation, is achieved in different scenarios and for different types of PA.

Principle of Digital Predistortion

As mentioned in Section 1.6.4, DPD is an efficient linearization technique of PAs. It consists of generating signal components of inverted proportional amplitude and opposite phase to the distortion products, in order to have a constant complex gain over the entire operating power range of the linearized ampliĄer, so that the whole system operates as a linear ampliĄcation system, as illustrated in Figure 2.1.

Figure 2.1: Principle of digital predistortion

The characteristic of the predistortion operator f PD , called predistorter, is the inverse of that of the PA. For example, for a memoryless PA represented by g PA , the linear ampliĄcation system for memoryless systems can be expressed by:

g PA (f PD (u)) = Gu (2.1)
where G is the gain of the system. The strong inverse match between the nonlinear characteristics of the predistorter and that of the PA leads to an effective DPD to cancel the distortions of the PA. Besides, it is required from DPD to achieve high linearization performances and low cost.

The DPD does not require a deep knowledge of the physical circuits of PA [START_REF] Ghannouchi | Behavioral modeling and predistortion[END_REF], but it is necessary to precisely measure the actual behavior of PA for the DPD process [START_REF] Kim | Digital predistortion of wideband signals based on power ampliĄer model with memory[END_REF].

On the other hand, the DPD needs to be made adaptive to maintain a linear ampli-Ącation system, particularly when the input signal changes characteristics (power level, frequency, and bandwidth) as well as the PA (variation of component tolerances and temperature drift).

Figure 2.2 illustrates the principle of baseband adaptive DPD. The DPD requires a reference signal for its adaption, which could come from path 1 or 2, depending on the approach used [START_REF] Baudoin | Digital predistortion[END_REF]. The feedback signal should be acquired of good quality since it is used for correction and synchronization. For this, it is advisable to use the mixers at the lowest possible power level. The DAC and ADC converters are essential components in the blog diagram of Figure 2.2. Their power consumption is not negligible (especially for wideband signals) and can degrade the efficiency, especially if the PA output power is not very high.

Digital Predistorter Models

The aim of studying PA distortions is to design an appropriate predistorter that will compensate for these distortions and ensure linear ampliĄcation of the signal to be transmitted.

Several mathematical models have been proposed in the literature to model the PA and serve as well as predistorter. Most of them are based on Volterra Series [START_REF] Zhu | ŤPruning the Volterra Series for Behavioral Modeling of Power AmpliĄers Using Physical Knowledge[END_REF].

Volterra Series

Volterra series are widely used to represent the input-output relationship of nonlinear dynamical systems. They are formed by a combination of linear convolution and a nonlinear power series so that it can be used to characterize a nonlinear dynamic system accurately [START_REF] Schetzen | The Volterra and Wiener Theories of Nonlinear Systems[END_REF]. To model the RF behavioral system, the Volterra series can be expressed as:

ỹ(t) = +∞ k=1 +∞ 0 • • • +∞ 0 hk (τ 1 , . . , τ k ) k j=1 x(t -τ j )dτ j (2.2)
where x and ỹ represent the RF input and RF output signal, respectively, hk (•) is the real-valued k-th order Volterra kernel.

In baseband, the complex envelope of the signal can be obtained by a low-pass Ąlter. Hence, the relationship between the baseband input and output in discrete time is:

y(n) = K k=0 M -1 i 1 =0 ... M -1 i 2k+1 =i 2k h k (i 1 , ..., i k ) k+1 j=1 x(n -i j ) 2k+1 j=k+2 x * (n -i j ) (2.3)
where x(n) and y(n) represent the baseband complex input and output, respectively, h p (i 1 , ...i p ) is called the Volterra series kernel, K is the nonlinearity order, and M is the memory depth.

The Volterra series model is robust and can provide greater modeling performance, but it is very complex and slowly converge. Unfortunately, the main drawback of Volterra series is that the number of coefficients increases exponentially with the order of nonlinearity and the memory depth. This drawback makes the Volterra series unattractive for real-time applications. This leads to use particular cases of the Volterra series, which retain only a few terms of these series.

Pruning of Volterra Series

In [START_REF] Zhu | ŤPruning the Volterra Series for Behavioral Modeling of Power AmpliĄers Using Physical Knowledge[END_REF], the authors proposed an efficient approach to pruning of the Volterra series according the physical properties of the PA, which signiĄcantly reduces model complexity by removing unnecessary coefficients from the general Volterra series.

The memory polynomial (MP) model and the Generalized Memory Polynomial (GMP) model are the most popular models obtained by direct pruning of the Volterra series.

The MP model proposed in [START_REF] Ding | ŤMemory polynomial predistorter based on the indirect learning architecture[END_REF] is obtained by reducing the Volterra series model to its diagonal terms. The input-output relationship for the MP can be written as:

y(n) = K k=1 M i=0 a ki x(n -i)♣x(n -i)♣ k-1 (2.4)
where a ki represent the coefficients of the model, and K and M are the nonlinearity order and memory depth, respectively. When only odd degrees are used, (2.4) can be rewritten :

y(n) = K k=0 M i=0 a 2k+1,i x(n -i)♣x(n -i)♣ 2k (2.5)
The GMP model is built by augmenting the memory polynomial model with additional basis functions, introducing cross-terms. It is proposed in [START_REF] Morgan | ŤA Generalized Memory Polynomial Model for Digital Predistortion of RF Power AmpliĄers[END_REF] to better deal with PAs with strong nonlinearities and wideband signals. The GMP model can be written as:

y(n) = Ka-1 k=0 La-1 l=0 a kl x(n -l)♣x(n -l)♣ k + K b k=1 L b -1 l=0 M b m=1 b klm x(n -l)♣x(n -l -m)♣ k + Kc k=1 Lc-1 l=0 Mc m=1 c klm x(n -l)♣x(n -l + m)♣ k (2.6)
where K a , K b , K c are the highest orders of nonlinearity. L a , L b , L c are the highest memory depths. M b , M c denote the longest lagging and leading delay tap length, respectively. a kl , b klm , c klm are the coefficients of the general memory polynomial applied on the aligned terms, the lagging and leading cross-term.

Modified or Dynamic Volterra Series

To overcome the complexity of Volterra series, several models have been proposed from modiĄed or dynamic Volterra series. In [START_REF] Ngoya | Accurate RF and microwave system level modeling of wideband nonlinear circuits[END_REF], the authors limit the number of coefficients by separating the static and dynamic parts to model the PA. In [START_REF] Zhu | ŤDynamic Deviation Reduction-Based Volterra Behavioral Modeling of RF Power AmpliĄers[END_REF], the author proposed a derived model from the dynamic Volterra series, called dynamic deviation reduction (DDR).

The DDR model can be further simpliĄed as [START_REF] Zhu | ŤSimpliĄed Volterra Series Based Behavioral Modeling of RF Power AmpliĄers Using Deviation-Reduction[END_REF] and [START_REF] Guan | ŤSimpliĄed dynamic deviation reduction-based Volterra model for Doherty power ampliĄers[END_REF]. The 1st-order truncated DDR-1 is deĄned as:

y(n) = K-1 2 k=0 M i=0 b 2k+1,1 ♣x(n)♣ 2k x(n -i) + K-1 2 k=1 M i=1 b 2k+1,2 ♣x(n)♣ 2(k-1) x 2 (n)x * (n -i) (2.7)
where b 2k+1,j is the kernel of the system. A more complex DDR model can be derived by adding some 2nd-order dynamic terms:

y(n) = K-1 2 k=0 M i=0 b 2k+1,1 ♣x(n)♣ 2k x(n -i) + K-1 2 k=1 M i=1 b 2k+1,2 ♣x(n)♣ 2(k-1) x 2 (n)x * (n -i) + K-1 2 k=1 M i=1 b 2k+1,3 ♣x(n)♣ 2(k-1) x(n)♣x(n -i)♣ 2 + K-1 2 k=1 M i=1 b 2k+1,4 ♣x(n)♣ 2(k-1) x * (n)x 2 (n -i) (2.8)
The MP model, which contains only the diagonal terms, has limited performance because some off-diagonal terms may be signiĄcant. Compared with the MP model, the DDR model includes off-diagonal terms, but has fewer diagonal terms. The ModiĄed Dynamic Deviation Reduction-based Volterra (MDDRV) proposed in [START_REF] He | ŤDigital predistortion using modiĄed dynamic deviation reduction-based volterra model[END_REF] and [START_REF] Song | ŤDynamic deviation memory polynomial model for digital predistortion[END_REF] can be viewed as the combining of the MP model and the DDR. By combining (2.4) and (2.7), the Ąrst-order MDDRV-1 model can be expressed as:

y(n) = K-1 2 k=0 M i=0 a 2k+1,1 ♣x(n)♣ 2k x(n -i) + K-1 2 k=1 M i=1 a 2k+1,2 ♣x(n)♣ 2(k-1) x 2 (n)x * (n -i) + K-1 2 k=1 M i=1 b 2k+1 ♣x(n -i)♣ 2k x(n -i) (2.9)
where a 2k+1,j and b 2k+1 are the model coefficients. The simpliĄed second order MDDRV-2 model can be expressed, by combining (2.4) and (2.8), as:

y(n) = K-1 2 k=0 M i=0 a 2k+1,1 ♣x(n)♣ 2k x(n -i) + K-1 2 k=1 M i=1 a 2k+1,2 ♣x(n)♣ 2(k-1) x 2 (n)x * (n -i) + K-1 2 k=1 M i=1 a 2k+1,3 ♣x(n)♣ 2(k-1) x(n)♣x(n -i)♣ 2 + K-1 2 k=1 M i=1 a 2k+1,4 ♣x(n)♣ 2(k-1) x * (n)x 2 (n -i) + K-1 2 k=1 M i=1 b 2k+1 ♣x(n -i)♣ 2k x(n -i) (2.10)

Models with Segmentation

Another approach to representing a nonlinear dynamic system is splitting the amplitude range into different regions or segments characterized by piecewise approaches models, such as models using B-splines approximation, vector-switched models, decomposed vector rotation, etc. These approaches are able to represent strong nonlinearities and may be less sensitive to numerical problems than global polynomial models. Furthermore, these models are based on segments of smaller orders of nonlinearity, which can be identiĄed with short buffers of training data.

In [START_REF] Chua | Section-wise piecewise-linear functions: Canonical representation, properties, and applications[END_REF], Chua proposed the Ąrst compact representation for modeling a nonlinear function by line segments, called canonical piecewise linear (CPWL) functions. The nonlinear basis functions of CPWL are constructed piecewise, making it much more Ćexible than the Volterra models. The CPWL is expressed as:

y(n) = M i=0 a i x(n -i) + K k=1 c k M i=0 a k,i x(n -i) -β k (2.11)
where K is the number of segments that are bounded by the thresholds β k . M is the memory depth and a i , c k,i and a k,i are the coefficients.

The main drawback of CPWL in (2.11) is that the model is nonlinear with respect to its coefficients, which complicates the process of identifying the coefficients c k,i and a k,i .

Zhu proposed in [START_REF] Zhu | ŤDecomposed Vector Rotation-Based Behavioral Modeling for Digital Predistortion of RF Power AmpliĄers[END_REF] an extension and modiĄed form of CPWL that it called decomposed vector rotation (DVR) model. It is based on the vector rotation technique to handle complex signals and nonlinear systems with memory.

The extension from CPWL functions consists of changing the form ♣x(ni) - (n-i) through an operation of 4 steps:

β k ♣ to ♣♣x(n -i)♣ -β k ♣e jθ
• Calculate the magnitude value of the input signal x(ni).

• Subtract away the threshold β k .

• Apply an absolute operation ♣.♣ to ♣x(ni)♣β k .

• Restore the phase θ(ni) of the signal x(ni).

The representation ♣♣x(ni)♣β k ♣e jθ(n-i) can be rewritten as:

♣♣x(n -i)♣ -β k ♣e jθ(n-i) =    x(n -i) -β k e jθ(n-i) , if ♣x(n -i)♣ ≥ β k [x(n -i) -β k e jθ(n-i) ]e j180 • , if ♣x(n -i)♣ < β k (2.12)
where the phase rotation on ♣x(ni)β k ♣ is applied if ♣x(ni)♣ is less than β k , making of it the nonlinear process introduced into the DVR model. Furthermore, the DVR model is linear with respect to its coefficients. The simplest version of the model, denoted DVR-2, is deĄned by combining the linear term and the basis function discussed below, which is expressed as:

y(n) = M i=0 a i x(n -i) + K k=1 M i=0 c ki,1 ♣♣x(n -i)♣ -β k ♣e jθ(n-i) (2.13)
The DVR-2 model refers to 2 terms: the linear term and the 1st-order basis.

The DVR model can be enriched by higher-order extension and variations of basis functions (derived from DDR, for example). We will call DVR-6 model, the DVR model that can be derived by adding some DDR terms.

y ( n) = M i=0 a i x(n -i) + K k=1 M i=0 c ki,1 ♣♣x(n -i)♣ -β k ♣e jθ(n-i) + K k=1 M i=0 c ki,21 ♣♣x(n -i)♣ -β k ♣e jθ(n-i) .♣x(n)♣ + K k=1 M i=1 c ki,22 ♣♣x(n -i)♣ -β k ♣.x(n) + K k=1 M i=1 c ki,23 ♣♣x(n -i)♣ -β k ♣.x(n -i) + K k=1 M i=1 c ki,24 ♣♣x(n)♣ -β k ♣.x(n -i) (2.14)
An important aspect concerning the DVR model is how the thresholds β k are distributed. In this chapter, The DVR model is used with a uniform distribution of the thresholds, e.g.,

β k = k/K for k = 1, 2, ..., K -1.
In the next chapter, we will propose an original approach to determine the thresholdsŠ optimal values and determine the terms to be used in the DVR model.

Predistorter Identification

An essential aspect of digital predistortion is the estimation of the digital predistorter model coefficients.

Several approaches has been proposed in the literature [START_REF] Bai | Non-Parametric Nonlinear System IdentiĄcation: A Data-Driven Orthogonal Basis Function Approach[END_REF] - [START_REF] Magesacher | Benchmarking of learning architectures for digital predistortion[END_REF]. The two main techniques are direct learning architecture (DLA) and indirect learning architecture (ILA).

In the DLA, the predistorter is determined in two steps. In the Ąrst step, the coefficients of the nonlinear model for the PA are extracted. In the second step, the identiĄed PA model is used to estimate the predistorter by minimizing a criterion based on the difference between the observed PA output and the ideal output, which is equal to the original input signal multiplied by a reference gain. Several algorithms were proposed to improve the DLA [START_REF] Zhou | Novel Adaptive Nonlinear Predistorters Based on the Direct Learning Algorithm[END_REF].

Unlike DLA, ILA consists of estimating a postdistorter that will be used as a predistorter. Besides, if the model is linear in its coefficients, the ILA requires a linear estimation, while the DLA requires a nonlinear estimation.

In [START_REF] Abi Hussein | On the system level convergence of ILA and DLA for digital predistortion[END_REF], a system-level convergence of DLA and ILA is investigated and compared. The MP model was used for both architectures to asses the robustness of the identiĄcation process, where it was shown that the DLA was found to be more robust than the ILA in the presence of noisy measurements.

Post-Distortion Identification

In this chapter, the DPD model coefficients are identiĄed using postdistortion architecture that is based on ILA, but without copying coefficients to the predistorter model, unlike the ILA as is illustrated in Figure 2 The PA output signal y(n) is divided by the ampliĄcation gain G. The signals x(n) and z(n) are used to estimate the model coefficients c using least square (LS) approach in order to minimize the LS criterion built on the difference between x(n) and z p (n), the model output that is computed using the estimated coefficient c and z(n). The instantaneous error is deĄned as:

e(n) = x(n) -z p (n) (2.15)
For the DPD models that are linear with respect to their coefficients, the relationship between its input and output can be rewritten using matrix notation. For a block of N samples: z p = Zc (2.16) where z p = [z p (1), . . . , z p (N )] T , c is a C × 1 vector containing the set of model coefficients, Z is N × C matrix of regressors containing basis functions of z. For example, in the case of a MP model, this matrix is represented as:

Z =        Φ 1,1 (z(n)) . . . Φ K,1 (z(n)) Φ 1,2 (z(n)) • • • Φ K,L (z(n)) Φ 1,1 (z(n -1)) . . . . . . . . . Φ K,L (z(n -1)) . . . . . . . . . Φ 1,1 (z(n -N + 1)) • • • Φ K,L (z(n -N + 1))       
(2.17)

where Φ k,l (z(n)) = z(n -l + 1)♣z(n -l + 1)♣ k-1 .
The least square (LS) solution will be the solution for the following equation for c:

[Z H Z]c = Z H x (2.18)
or equivalently using the pseudo-inverse:

ĉ = [Z H Z] -1 Z H x (2.19)
with x is the input signal, where (2.18) minimizes the LS criterion:

ĉ = min c (J) (2.20)
with:

J = N n=1 ♣e(n)♣ 2 = N n=1 ♣x(n) -z p (n)♣ 2
Many approaches can be used to solve (2.18) such as LU, LDL, QR decomposition, etc. It should be noticed that the matrix Z H Z is generally ill-conditioned.

Comparison of DPD models

Many comparative studies have been done to compare the linearization performance of DPD models [START_REF] Hamoud | ŤEvaluation of an optimal digital predistorter for multistandard systems[END_REF] - [START_REF] Pedro | ŤA comparative overview of microwave and wireless power-ampliĄer behavioral modeling approaches[END_REF]. In [START_REF] Pedro | ŤA comparative overview of microwave and wireless power-ampliĄer behavioral modeling approaches[END_REF], the paper presented a comparative overview of various nonlinear behavioral approaches for PAs. In [START_REF] Gotthans | ŤComparison of modeling techniques for power ampliĄers[END_REF], the study focused on the linearization performance comparison of 4 DPD models: MP, GMP, Orthogonal Polynomials with Memory (OPM), and DDR.

In [START_REF] Hadi | Assessment of digital predistortion methods for DFB-SSMF radio-over-Ąber links linearization[END_REF], a comparative study has been carried out of the MP, GMP, and DVR models for DPD to linearize the radio on Ąber laser with distributed feedback. The comparison was made in terms of NMSE, EVM, and ACPR, where the authors show through experimental results The DVR model can achieve reliable performance linearization with fewer coefficients than the MP and GMP models.

In this chapter, besides MP, GMP, and DDR, the DVR-2 and DVR-6 models are also evaluated and compared, which underlines that the main contribution of this part of our work is to compares two different categories of models: global models derived from the Volterra series and models based on segmentation approach [START_REF] Kantana | ŤLinearization Performance vs Implementation Properties of Digital Predistorter Modelling for Three-Way Doherty Power AmpliĄer[END_REF].

Metrics of Comparison

The studied DPD models are evaluated according to metrics which are divided into two categories. The Ąrst one is the metrics of accuracy, which assess the linearity performance. The second one highlights the hardware implementation complexity, which refers to the number of coefficients and numerical properties, which refers to numerical stability to extract the model and the dynamic range of the coefficients. Another complexity feature of a model comes from the construction of the matrix operator, Z in (2.18) and the auto-correlation matrix Z H Z, which is controlled by the buffer length N .

Linearization Performance

As (2.18) seeks to minimize e(n), a natural measure for the modeling accuracy is the normalized mean square error (NMSE) between the post-distortion output z p (n) and the PA input signal x(n):

NMSE dB = 10 log 10 N n=1 ♣x(n) -z p (n)♣ 2 N n=1 ♣x(n)♣ 2 (2.21)
From a system point of view, the Ągure of merit is provided by the out-band distortions which are evaluated using adjacent channel power ratio (ACPR) which is deĄned for the right and left channel as: ACPR R,dB = 10 log 10

B/2 -B/2 P (z p (t))df 3B/2 B/2 P (z p (t))df ACPR L,dB = 10 log 10 B/2 -B/2 P (z p (t))df -B/2
-3B/2 P (z p (t))df (2.22) where B represents the bandwidth of the signal and P (.) is power spectral density.

Implementation Properties

Predistorters are generally implemented in real digital circuits such as FPGA. Their iden-tiĄcation can also be implemented on FPGA or some general-purpose digital processor. For that, it is important to have Ągures of merit related to hardware implementation. Two critical aspects of the implementation are concerned: the identiĄcation and the predistorter implementation.

Identification

The complexity to solve (2.18) is related to the number of coefficients C of the DPD models, and to the buffer length N [START_REF] Wang | ŤPerformance analysis of multi-stage cascaded digital predistortion[END_REF]. Table 2.1 presents the parameters and number of coefficients of the different DPD models compared in this study.

× (M + 1) M : memory depth GMP K a , K b , K c : nonlinearity order K a × L a L a , L c , L c : memory depth +K b × L b × M b M b , M c : lagging and leading cross-terms order +K c × L c × M c DDR-1 K: nonlinearity order K+1 2 + K × M M : memory depth DDR-2 K: nonlinearity order K+1 2 × (M + 1) M : memory depth +3M K-1 2 MDDRV-1 K: nonlinearity order K+1 2 + ( 3K-1 2 ) × M M : memory depth MDDRV-2 K: nonlinearity order K+1 2 × (M + 1) M : memory depth +2 × (K -1) × M DVR-2 K: number of segments (K + 1) × (M + 1) M : memory depth DVR-6 K: number of segments 5 × K × M + 2 × K + M + 1 M : memory depth
The computation to solve (2.18) may exhibit numerical sensitivity depending on the condition number of Z H Z which will be used as an indicator of this sensitivity.

On the other hand, according to Table 2.1, we can classify the DPD models in terms of complexity. The complexity of the DPD models, derived from the Volterra series, depends on the nonlinearity order K and the memory depth M . The complexity depends on the number of segment K, the memory depth M , and the number of the basis functions for the DVR model.

Predistorter implementation

For the implementation of the predistorter, there are two aspects to consider: the overall complexity related to the number of parameters (Table 2.1) and the number of bits required to encode each of those parameters. This number of bits will depend both on the dynamic range and the resolution required for the coefficients.

The hardware implementation can be carried out by two structures: direct implementation and the implementation by LUT. A comparison study of the direct form and implementation using LUT was presented in [START_REF] Gilabert | ŤMulti-Lookup Table FPGA Implementation of an Adaptive Digital Predistorter for Linearizing RF Power AmpliĄers With Memory Effects[END_REF].

In direct implementation, complexity refers to the number of computational operations directly related to the number of coefficients. In implementation by LUT, complexity refers to the number of LUT, which is only related to the memory depth. It is worth noting that LUT implementation may not provide better implementation efficiency for all kinds of models, such as GMP.

An example implementation of the MP model with K = 3 and M = 1 is illustrated in Figure 2.5, where the direct method and LUT are respectively shown in Figure 2.5a and Figure 2.5b. In this work, the dynamic range is considered a relative indicator of the requirement for the number of bits required for each model. We thus make the assumption that every model will require the same resolution, which is certainly rough. The dynamic range of the coefficients is introduced as a comparison element, which can be calculated as:

ϵ = max[real(c), imag(c))] -min[real(c), imag(c)]
(2.23)

Comparison of MP, GMP, DDR, MDDRV, and DVR

Testbench Description

The DPD models are evaluated using the input-output signals from real PA shown in Figure 2.6. The PA line is made of a three-way Doherty PA designed for the base station (BS-PA) with three LDMOS transistors BLF7G22LS-130 from Ampleon, formerly NXP, and its associated driver. This Doherty PA can have a peak output power of 57 dBm (500 W) and has a linear gain of 16 dB. In the following, the PA in Figure 2.6 is denoted by LDMOS-TW-500W PA.

Parameters of DPD Models

The structure of the GMP model are optimally determined using a hill-climbing algorithm as proposed in [START_REF] Wang | ŤOptimal sizing of generalized memory polynomial model structure based on Hill-Climbing heuristic[END_REF] using off-line additive criterion.

For this study, we have chosen to limit the maximum nonlinearity order and memory depth to the same maximum values as the GMP model, i.e. K = 11 and M = 4, for MP, DDR-1, DDR-2, MDDRV-1, and MDDRV-2. The number of coefficients is 43.

We size the DVR model to have a model with a similar trade-off between complexity and performance compared to the other DPD models. For the DVR-2 model, the number of coefficients is 55 with K = 10 and M = 4 and 71 for the DVR-6 model with K = 3 and M = 4.

Figures of merit of every model are summarized in Tables 2.2, where it can be seen that the GMP model achieves the best performance regarding NMSE and ACPR even though DVR-6, MP, and DVR-2 also achieve good performances. Figure 2.8 shows the plot of the NMSE according to the number of coefficients for each DPD model. From this Ągure, it can be seen that the NMSE for all models, except GMP and DVR-6, converges toward an asymptote around 33 dB while the NMSE for GMP and DVR-6 is still decreasing with the number of coefficients. 2.3 shows the number of coefficients and their dynamic range for each model, and the condition number of the matrix Z H Z which is involved in the computation of the identiĄcation procedure. For the LDMOS-TW-500W PA with this waveform, DVR exhibits exceptional values for both dynamic range and condition number as there are more than ten orders of magnitude for both dynamic range and condition number between DVR models and GMP-MP models.

Even if the GMP model reaches the best linearization performance, the DVR-6 has a very similar performance but with nearly twice the number of coefficients. It is the price to pay to beneĄt from the outstanding properties of the DVR-6 model for digital hardware implementation in this scenario.

In order to validate the effectiveness of this comparative study, the GMP and DVR-6 models will be speciĄcally highlighted and compared in the following section. The comparison will be made in different scenarios with different bandwidths, and at different power levels [START_REF] Kantana | ŤComparison of GMP and DVR modelsŤ[END_REF].

Comparison of GMP and DVR

Testbench

The experiments are carried out using an LDMOS PA 50 W at 1.78 GHz center frequency. In the following, this PA is denoted by LDMOS-50W PA. The testbench used for these tests consists of software-deĄned radio (SDR) from ARELIS for generating the IQ signals and a vector signal transceiver (VST) NI for the data acquisition at 200 Ms/s.

The tests are made with a set of LTE signals of 5, 10, and 20 MHz bandwidth and a maximum PAPR of 12dB, at four output power levels: 34, 38, 42, and 44 dBm.

The nonlinearities and the memory effect of the LDMOS-50W PA can be seen from the AM-AM and AM-PM curves in Figure 2.9 for 20 MHz bandwidth case at 44 dBm.

Parameters of Models

The structure determination of the GMP model is done using the hill-climbing algorithm, which is proposed in [START_REF] Wang | ŤOptimal sizing of generalized memory polynomial model structure based on Hill-Climbing heuristic[END_REF]. With this algorithm, we size the structure of the GMP model using the criterion which makes a trade-off between modeling accuracy and complexity. For the DVR-6 model, an exhaustive search is carried out by exploring all combinations when the number of segments K ranges from 2 to 10 and the memory depth M ranges from 0 to 10.

The thresholds β k for the set of partitions are optimally selected by a proposed approach which will be the subject of the next chapter. In Figure 2.10a, the NMSE behavior versus the number of coefficients of the two DPD models has a similar form of convergence where they start from the same point and converge towards an NMSE asymptote around 40 dB. However, the DVR-6 model has faster convergence since its NMSE between 3 and 8 coefficients is better than that of the GMP model. This is clearly seen in Figure 2.10b, where the DVR-6 model has fast convergence to join the 40 dB asymptote, while the GMP model requires 15 coefficients. However, the GMP model convergence is slightly better than that of the DVR-6 model by about 0.2dB of NMSE, as shown in Ągure 2.10c.

To study the linearization performance and the numerical properties of the two DPD models, we have chosen to compare the two models with the same number of coefficients. In this case, the number of coefficients is set to 39 coefficients.

Table 2.4 summarizes the parameters of structure found by the hill-climbing algorithm for the GMP model and the structure parameters found by the exhaustive search for the DVR-6 model. By comparing the results in Figure 2.10, the values collected in Table 2.5 conĄrm that the GMP model provides slightly better accuracy than the DVR model, which can be tolerable since the results are collected using postdistortion scenario.

GMP K M K a L a K b L b M b K c L c M c 5 MHz 3 2 1 14
Figure 2.11 shows the power spectrum of the postdistorter for the GMP and DVR-6 model with 5, 10, and 20 MHz bandwidth and 44 dBm output power. As it can be seen, both DPD models have close linearization performances. 2.6 presents the dynamic ranges of the coefficients and the condition number of the matrix involved in the identiĄcation procedure. The results in Table 2.5 and Table 2.6 are given with the same complexity for both DPD models. From Table 2.6, it can be seen that the condition number for DVR is lower than for GMP, even though it is still signiĄcant. The dynamic range of the coefficients is the other Ągure of merit that assesses the numerical properties of the model. It can be seen that the DVR model exhibits a better behavior for this feature than the GMP model.

On the other hand, the performance of the GMP and DVR-6 models is investigated with the variation of output power level. For this study, we choose the signal with 20 MHz bandwidth by varying the output power at four levels: 34, 38, 42, and 44 dBm.

The parameters for the DVR-6 model are K = 4 and M = 1. The parameters of the GMP model are obtained with the hill-climbing algorithm using the additive criterion and summarized in Table 2.7, for each output power level. The number of model coefficients for each power level is 30 coefficients.

K b 4 2 2 8 L b 1 1 1 1 M b 3 6 3 1 K c 3 6 7 10 L c 2 2 2 1 M c 1 1 1 1
Table 2.8 shows the linearization performance and numerical properties for each power level. The three Ąrst rows of Table 2.8 allow making some comparison between both models on linearization performances. The two last rows present the numerical properties results. The dynamic range and condition number of the GMP model increase with the output power. On the other hand, they remain bounded for the DVR model.

The GMP model performs better than DVR when the PA exhibits lower nonlinearities, but the DVR model remains suitable for strong nonlinearity. On the other hand, the DVR-6 model conĄrms its outstanding numerical properties for different bandwidths and output power.

Conclusion

This chapter introduces the principle of digital predistortion. The Post-Distortion architecture has been presented and discussed. The DPD models are presented and described according to 2 categories: global models derived from Volterra or modiĄed Volterra series and models based on segmentation approach. These DPD models are compared according to different criterion, and their linearization performance versus implementation properties is presented.

The GMP model provides a good trade-off between accuracy performance and complexity. The DVR model may achieve similar performances but with an increase in the number of coefficients.

On the other hand, the DVR model has remarkable numerical properties that make it a good choice for hardware implementation. For that purpose, the following chapter will focus on the design and the optimization of the DVR model, determining its structure and proposing its implementation architecture.

Chapter 3 Optimization and Sizing of DVR Model

Introduction

Volterra series have good performances for the modeling of nonlinear dynamic systems. Besides, they are linear with respect to their coefficients, which simpliĄes their identi-Ącation using the least-square method. However, their complexity is very high because their number of coefficients increases exponentially with nonlinearity order and memory depth. Moreover, the Volterra series have destructive numerical properties to identify their coefficients for a high order of nonlinearity, which can be justiĄed because the model is built with a non-orthogonal basis.

Models derived from Volterra series such as MP, GMP, or DDR have proven their effectiveness to serve as a predistorter or PA model. However, PAŠs advanced architecture with good efficiencies, such as Doherty and envelope tracking, exhibits a nonlinear behavior, which is more challenging to be linearized. On the other hand, the needs for higher data rates require the use of efficient spectral modulation techniques such as orthogonal frequency division multiplexing (OFDM). As a result, the modulated signals have a high PAPR, which stimulates PA nonlinearities. Besides, 5G and MIMO systems allow a very high data rate, which is a real challenge for DPD in bandwidth, nonlinearity, and dynamic behavior. Considering all these facts, it becomes more challenging to design an efficient DPD system to achieve accurate modeling with good efficiency and low computational complexity.

This has pushed the interest of research to Ąnd a different approach for DPD. One of these approaches is the segmentation approach, where the input space is divided into several subspaces represented by small models. One of these models based on the segmentation approach is the DVR model presented in the previous chapter as a DPD model in the comparative study.

The study of the DVR model raises different questions such as:

• how to segment the input space optimally,

• how to determine the best model structure,

• how to determine the best model structure jointly with its optimal segmentation. This chapter focuses on the DVR model, mainly on how to optimize its boundary thresholds and Ąnd its optimal structure. We propose a new approach for the optimization of thresholds in which the optimization problem is decomposed into a set of unimodal sub-problems that allow using a unidirectional minimization [START_REF] Kantana | Thresholds Optimization of Decomposed Vector Rotation Model for Digital Predistortion of RF Power AmpliĄer[END_REF]. The sizing of the DVR model is based on hill-climbing (HC) algorithm to determine an optimal structure of the The authors of [START_REF] Cao | ŤA ModiĄed Decomposed Vector Rotation-Based Behavioral Model With Efficient Hardware Implementation for Digital Predistortion of RF Power AmpliĄers[END_REF] proposed a hardware implementation strategy with low complexity and reduced power dissipation, where it was achieved by modifying the DVR model into a sub-decomposed representations, which signiĄcantly reduce the computational complexity of the model extraction. The proposed model is expressed as:

y(n) = M i=0 a i x(n -i) + K k=1 M i=0 c ki,11 F (♣x(n -i)♣, k)e jθ(n-i) + K k=1 M i=0 c ki,21 F (♣x(n -i)♣, k)e jθ(n-i) .♣x(n)♣ + . . . (3.1)
where

F (♣x(n)♣, k) is a constructed segment expressed by = 0.5(β k -♣x(n)♣ + ♣β k -♣x(n)♣♣), for odd values of k 0.5(♣x(n)♣ -β k + ♣♣x(n)♣ -β k ♣), for even values of k (3.2)
In this chapter, we adopt a general presentation of the DVR model which can be expressed by:

y(n) = M lin i=0 a i x(n -i) + Tt∈S T t (3.3)
where x(n) and y(n) are the input and output of the model, M lin is the memory depth for the linear term, a i are the complex coefficients of the linear term, and S is the set of the terms T t which are used in the model with S ⊂ T and

T = [T 1,[0,...,P ] , T 2 , T 3 , T 4 , T 5 , T 6 , T 7 ]
where T 1,[0,...,P ] is the set T 1,0 , ..., T 1,P whose elements are deĄned hereafter along with T 2 , ..., T 7 .

T 1,p = K k=1 M i=0 c ki,1p ♣♣x(n -i)♣ -β k ♣e jθ(n-i) • ♣x(n)♣ p T 2 = K k=1 M i=i 2 c ki,2 ♣♣x(n -i)♣ -β k ♣ • x(n) T 3 = K k=1 M i=i 3 c ki,3 ♣♣x(n -i)♣ -β k ♣ • x(n -i) T 4 = K k=1 M i=i 4 c ki,4 ♣♣x(n)♣ -β k ♣ • x(n -i) T 5 = K k=1 M i=i 5 c ki,5 ♣♣x(n -i)♣ -β k ♣ • x 2 (n) • x * (n -i) T 6 = K k=1 M i=i 6 c ki,6 ♣♣x(n -i)♣ -β k ♣ • x(n) • ♣x(n -i)♣ 2 T 7 = K k=1 M i=i 7 c ki,7 ♣♣x(n -i)♣ -β k ♣ • x * (n) • x 2 (n -i) (3.4)
where K is the number of segments, β k are the bounds of the segments, p is the nonlinearity order, M the memory depth, and c ki are the complex coefficients of the model for each segment. The indexes i 2 , i 3 , ..., i 7 are equal to 0 or 1 depending on the set of selected T i terms. The cardinal of T is thus P + 7. The number of coefficients, denoted by C, depends on the parameters of the DVR model structure.

Experimental Testbench

Experimental acquisitions from real PA have been carried out.

The testbench is represented in Figure 3.2, which includes a PC MATLAB software, an SDR card, a Vector Signal Transceiver (VST) from National Instruments (NI), and an LDMOS PA of 50W.

Optimization of DVR Model Thresholds

State of the Art

In (3.4), the thresholds β K that deĄne the boundaries of segments can be selected using uniform or optimized segmentation. Although many works in the literature are dedicated to models with uniform segmentation, e.g., β k = k/K for k = 1; 2; ...; K -1, few of them deal with the optimal segmentation. The Ąrst work for DPD application about optimal spacing was conducted for memoryless LUT DPD [START_REF] Cavers | ŤOptimum table spacing in predistorting ampliĄer linearizers[END_REF] - [START_REF] Lin | ŤDynamically optimum lookup-table spacing for power ampliĄer predistortion linearizationŤ[END_REF], which is a challenge closely related to the optimization of segmentation for a piecewise model. In [START_REF] Cavers | ŤOptimum table spacing in predistorting ampliĄer linearizers[END_REF], the author has proposed a non-uniform systematic spacing using a companding function into table indexing, which reduces the intermodulation power. In [START_REF] Lin | ŤDynamically optimum lookup-table spacing for power ampliĄer predistortion linearizationŤ[END_REF], an iterative procedure is proposed to optimize non-uniform LUT spacing, which is based on an IMD criterion and the input signal statistics.

The beneĄt of optimal segmentation compared to uniform segmentation has already been established in [START_REF] Magesacher | ŤOptimal Segmentation for Piecewise RF Power AmpliĄer Models[END_REF] where the authors suggest reducing the complexity of the algorithm by considering the memoryless version of the actual DVR model. The authors used Gauss-Newton-based methods such as Levenberg-Marquardt [START_REF] Levenberg | ŤA method for the solution of certain non-linear problems in least squaresŤ[END_REF] to jointly optimize the thresholds and identify the model coefficients. Nevertheless, even with the reduced complexity algorithm proposed in [START_REF] Magesacher | ŤOptimal Segmentation for Piecewise RF Power AmpliĄer Models[END_REF], the problem to solve is still a nonlinear least square problem. The authors claim that the nonlinear minimization problem behaves well close to uniform segmentation.

Generally speaking, setting the thresholds optimally for piecewise models such as the DVR model requires global optimization, and in [START_REF] Mateo | ŤRadio-over-Ąber linearization with optimized genetic algorithm CPWL model[END_REF] the authors proposed an approach based on genetic algorithm (GA) to optimize the thresholds of the CPWL model for radio-over-Ąber. The authors used the ACPR as the Ątness function for GA. The GA generations used in the GA were not mentioned in the optimization process, meaning that the complexity is not a signiĄcant concern in this research.

Structure of DVR Model

In this section, we use the set

T S = [T 1,0 T 1,1 T 2 T 3 T 4 ] with P = 1,
which corresponds to the DVR-6 model presented in Chapter 2, in which it has achieved an excellent modeling performance. Therefore, the DVR model is expressed as:

y(n) = M lin i=0 a i x(n -i) + K k=1 M i=0 c ki,10 ♣♣x(n -i)♣ -β k ♣e jθ(n-i) + K k=1 M i=0 c ki,11 ♣♣x(n -i)♣ -β k ♣e jθ(n-i) .♣x(n)♣ + K k=1 M i=1 c ki,2 ♣♣x(n -i)♣ -β k ♣.x(n) + K k=1 M i=1 c ki,3 ♣♣x(n -i)♣ -β k ♣.x(n -i) + K k=1 M i=1 c ki,4 ♣♣x(n)♣ -β k ♣.x(n -i) (3.5)
where a i and c ki are the model coefficients which are identiĄed using ILA.

In the following, we set M lin to 2 and M to 1.

Motivation

The beneĄts of an optimized segmentation are highlighted in Figure 3.4 and Figure 3.5 which compares uniform and optimal segmentation for a different number of segments in terms of NMSE and ACPR for a 20 MHz bandwidth LTE signal to drive the LDMOS PA. According to Figure 3.4, the linearization performance in terms of NMSE for 8 segments using uniform segmentation could be achieved with only 3 segments using optimal segmentation. This is conĄrmed in Figure 3.5 in terms of ACPR, where the optimal segmentation requires only 4 segments to achieve the same linearization performance that the uniform segmentation with 10 segments.

The NMSE and ACPR of the DVR model with optimal segmentation converge towards an asymptote from 4 segments, while uniform segmentation requires more than 10 segments to reach the best linearization performance.

Thus, the optimal segmentation signiĄcantly reduces the complexity since the number of coefficients to be estimated is reduced. For the scenario at hand, only 43 coefficients for optimized segmentation are required to achieve the same linearization performance with 93 coefficients for uniform segmentation. This conĄrms the interest in optimizing the thresholds of the DVR model.

In the sequel, we propose an approach to optimize the thresholds β K and how the global optimization problem could be divided into a set of unimodal sub-problems requiring only a unidirectional minimization. This approach will be compared to the brute-force and GA in terms of linearization performance and complexity.

Brute-force

Brute force is an exhaustive search over the normalized input range with a given precision. It consists of testing all the possible threshold positions over the normalized input in which for each segmentation corresponding to thresholds allocation, the linearization performance metrics are computed (NMSE or ACPR or both). the threshold positions tested are located on a grid containing N positions which are uniformly spaced.

Once brute-force is Ąnished, it will be easy to Ąnd the threshold allocation with the best linearization performance. Brute-force is seen as an intuitive solution to Ąnd an optimum for an optimization problem, but it is not a realistic approach since a massive computation is required.

Regarding the optimization of DVR model thresholds, brute-force divides the normalized input range into N steps. For a given number of segments K with K -1 thresholds:

β = [β 1 β 2 . . . β K-1 ],
the search explores all possible combinations of the position of the thresholds over the normalized input range. The brute-force returns the optimal segmentation β opt with the best (minimized) NMSE among all segmentation β (q) tested in the search space V where q ∈ V.

The brute-force returns the optimal segmentation with the best (minimized) NMSE among all segmentation tested in the search space V with Q.

For a given combination, each threshold β i in the threshold vector β must be located on a unique position. The number of combinations is N K-1 , which corresponds to NMSE (or ACPR) computations, where the search grid is divided into N values. The algorithm of brute-force is presented in Algorithm (1).

Algorithm 1: Algorithm of brute-force

Given K, M lin and M for the DVR model Divide the search grid to N steps Set loop counter q = 1 for

β 1 = [ 1 N 2 N . . . N -1 N ] do for β 2 = [ 1 N 2 N . . . N -1 N ] do if β 2 ̸ = β 1 then . . . for β K-1 = [ 1 N 2 N . . . N -1 N ] do if β K-1 ̸ = β K-2 & . . . &β K-1 ̸ = β 1 then β = [β 1 β 2 . . . β K-1 ] Evaluate NMSE(β) q = q + 1 end end end end end β opt = argmin q∈V NMSE(β q )
In our study, the results of brute-force will be taken as a reference for the comparison subject.

Genetic Algorithm

GA is an efficient heuristic algorithm that is inspired by the theory of the Darwinian principle of natural evolution.

This algorithm reĆects the process of natural selection where the individuals (thresholds) are selected for reproduction to produce offspring (optimal segmentation) of the next generation using some genetic operators such as selection, crossover, and mutation.

The GA begins by creating a random initial population of individuals characterized by a set of so-called genes. The GA then generates a sequence of new populations using the current population to create the children that make up the next generation. The GA selects the parents from individuals in the current population who have better Ątness function and bring their genes to their children.

The GA uses three main techniques at each generation to produce the next generation from the current population :

• Elite selection in which the parents are chosen for the next generation based on their Ątness function. An individual with a high Ątness function can be selected more than once as a parent, contributing its genes to more than one child.

• Crossover children are created by combining pairs of parents P in the current population. At each coordinate of the child vector, the default crossover function randomly selects a gene at the same coordinate from one of the two parents and assigns it to the child.

• Mutation children are created by randomly changing the genes of individual parents, aiming to maintain diversity within the population and prevent premature convergence.

After several generations, the GA converges to the best solution, ideally the optimal threshold segmentation.

In the problem at hand, the populationŠs size has been chosen as 50 and the maximum number of generation Q=25, but GA can stop if it has already converged towards the same solution. NMSE is used as a Ątness function.

The motivation behind the use of GA in our study is to add, along with the bruteforce, another process from different optimization approach, which will be used for the comparison subject to validate the effectiveness of the proposed approach, which will be discussed in the next section.

Proposed Approach

Principle of Unidirectional Minimization

In the proposed approach, a unidirectional minimization is achieved using the golden section (GS) search.

The GS search is an optimization technique used to Ąnd the optimum for a strictly unimodal function of a single variable over a search interval without using derivatives [START_REF] Kiefer | ŤSequential minimax search for a maximumŤ[END_REF].

For a given unimodal function f (x), the principle of GS search consists of Ąnding the optimum x opt which corresponds to the minimum value of f (x), by iteratively narrowing the range of the search interval [a b] containing that optimum until a speciĄed accuracy is reached.

The GS search derives its name from the fact that the points determining the search interval are computed using the golden ratio τ , which is equal to

1+ √ 5 2
and veriĄes τ -1 = 1 τ . The golden ratio τ is intimately involved in the Fibonacci sequence, in which the ratios of successive terms of the Fibonacci sequence possess an impressive proportion, e.g., 1.618, or its inverse 0.618.

Therefore, the length of the search interval is iteratively reduced by a factor τ . At q th iteration the search interval is noted [a q b q ]. Denoting φ = 1 τ = 0.618, two intermediate points are used to update the range of the search interval. They are determined by:

x 1 = a q + φ 2 (b q -a q ) x 2 = a q + φ(b q -a q ) (3.6)
The functions f (x 1 ) and f (x 2 ) are computed and compared as described in Algorithm (2).

Algorithm 2: Algorithm of GS search q = 0 ; a 0 = a ; b 0 = b Compute x 1 and x 2 while ♣b q -a q ♣ > ϵ do Compute f (x 1 ) and f (x 2 ) if f (x 1 ) ≤ f (x 2 ) then a q+1 = a q b q+1 = x 2 x 2 = x 1 Compute x 1 end if f (x 1 ) > f (x 2 ) then a q+1 = x 1 b q+1 = b q x 1 = x 2 Compute x 2 end q = q + 1 end x opt = aq+bq 2
The computation process of f (x 1 ) and f (x 2 ) is iteratively executed and the search interval [a b] is narrowing until the accuracy ♣b qa q ♣ < ϵ is reached where ϵ is a speciĄed small value, then takes x opt = aq+bq 2 as an approximate optimum point and f (x opt ) is the approximate optimum value.

In this work, the GS search considers NMSE as the function f (x) and the range of the normalized magnitude of the baseband input IQ as the search interval [a b].

Another advantage of the GS search is that it requires only one computation of f (x) at each iteration, except for the Ąrst one where f (a 0 ), f (x 1 ), f (x 2 ) and f (b 0 ) have to be evaluated.

optimization of a single threshold

Let us consider the case of 2 segments. In this case, we have only a single threshold to determine.

Figure 3.6 presents the evolution of GS search in terms of NMSE over the intermediate points x 1 (q) and x 2 (q) in the two Ąrst iterations and the Ąnal iteration. The blue curve presents the NMSE of postdistortion for every position of the threshold between 0 and 1. The red curve presents the AM-AM characteristic of the PA.

Initially, the search interval In the general case for K segments, we have K -1 thresholds to optimize: β = β 1 ,β 2 ,..., β K-1 . We propose to optimize the thresholds successively one by one, starting with an initial uniform segmentation. The optimization interval

I i of a selected β i is [β i-1 β i+1 ].
In a normalized interval, β 0 = 0 and β K = 1 by deĄnition.

Each β i contributes to the overall NMSE, furthermore as β i is the bound of the optimization interval for β i-1 and β i+1 so the optimization of β i will in turn impact the optimal position of β i-1 and β i+1 that should be re-evaluated. That is why we propose a heuristic to handle this situation: after having optimized the last threshold β K-1 , the algorithm is run in reverse order back to β 1 , then again up to β K-1 and so on until the segmentation has converged.

Algorithm 3: Algorithm of the proposed approach

Given K, M lin and M for the DVR model Initialization

β i , ϵ Set loop counter q=1 while (1) do for i = 1, 2, ..., K -2, K -1 do β i(q) = argmin β i ∈[β i-1 (q),β i+1 (q)] NMSE(β i ) end q = q + 1 β K-1 (q) = β K-1 (q -1) for K -1, K -2, ..., 2, 1 do β i(q) = argmin β i ∈[β i-1 (q),β i+1 (q)] NMSE(β i ) end q = q + 1 β 1 (q) = β 1 (q -1) Evaluate NMSE(β (q) )
if NMSE(q) > NMSE(q -1)ϵ then end while loop end end Given the DVR model in (3.5) deĄned by K, M and the thresholds

β 1 , • • • , β K-1 . The function NMSE(β i ) corresponds to the different value of NMSE, obtained when varying β i from β i-1 to β i+1 .

The minimum of this function is obtained with argmin

β i ∈[β i-1 ,β i+1 ] NMSE(β i ).
The algorithm for the proposed approach is described in Algorithm (3) in which the steps β K-1 (q) = β K-1 (q -1) and β 1 (q) = β 1 (q -1) are introduced for notation consistency.

Complexity of GS search

The complexity of the GS search is an important aspect to be considered in the thresholds optimization process. In this work, the search complexity is assessed by three features:

• The number of NMSE computations, which are required to identify the model coefficients.

• The number of interval updates, which is deĄned by the number of times the search interval is updated, except at the Ąrst iteration when one interval update corresponds to one NMSE evaluation.

• The number of GS iterations, which is deĄned by the number of times the thresholds vector

[β 1 , • • • , β K-1 ] is optimized.
It is represented by the parameter q in Algorithm (3).

Implementation Strategy of GS search

The proposed approach based on GS search is seen as an offline process to determine the DPD model for the PA linearization.

In the case of the DVR model, we determine the model parameters, we optimize the thresholds β, then, the DVR model is ready to be inserted as a DPD model according to ILA, in which the model is used in 2 blocks: predistorter and postdistorter, from which both must contain the optimized thresholds β opt .

For real-time DPD adaptation, GS search is only required to be performed at the initial training when the global characteristics of the PA are determined. The joint architecture of DPD adaptation, which is based on ILA and the proposed approach to optimize the thresholds, is shown in Figure 3.8.

i = Q do z i = y i /G DeĄne regressor matrix Z i Estimate coefficient c i = [Z H i Z i ] -1 Z H i x i z p = Z i c i Compute NMSE between x i

and z p

Copy c i to DVR model Get x i and y i Evaluate linearization performance end However, it is recommended to rerun the proposed approach once the input u or the PA changes any of its characteristics, especially output power, signal bandwidth, center frequency, etc.

Comparison of Linearization Performances

In the following, we set K to 4, M lin = 2 and M = 1.

The results of the proposed approach using GS search are discussed and compared to those of a uniform segmentation and the optimal segmentations obtained from brute-force search and GA.

The results are summarized in Table 3.1 and discussed in terms of NMSE and ACPR. The algorithm complexity is also discussed and evaluated by the number of computations of the objective function. The Ąrst column corresponds to the uniform segmentation. The next column represents brute-force search which has been performed by dividing the input range into 100 steps. The third column represents the results of GA, which has been performed with a population size of up to 50 and a maximum number of generations up to 25. The last column presents the results of the proposed approach.

In the scenario at hand, the proposed approach has converged towards the same optimal segmentation returned by brute-force search and GA, i.e., β opt = [0.59 0.81 0.92]. According to Table 3.1, only 341 objective function evaluations are performed to optimize the thresholds, while GA requires 2404 evaluations and brute-force requires 161700 evaluations to Ąnd the exact optimal results. Compared to uniform segmentation, the NMSE of the proposed approach has been signiĄcantly improved by nearly 3 dB. The ACPR is improved by more than 4 dB as well. This is can be conĄrmed in Figure 3 Figure 3.9 presents the spectrum of input and output of PA and the spectrum of the output of predistorter with uniform and optimized segmentation.

Analysis and Improvement of the Proposed Approach

This section highlights the behavior of the proposed approach throughout its convergence and how the optimization problem is decomposed into a set of unimodal sub-problems. Furthermore, a stochastic process is proposed to select the initial segmentation that allows accelerating the convergence of the proposed approach.

Behavior of the Proposed Approach

The initial uniform segmentation is

[β 1(0) β 2(0) β 3(0) ] = [0.25 0.5 0.75]
The optimization interval vector is It can be seen from Figure 3.12 that NMSE(β 1 ) is not unimodal on I 1(0) . Hence this example highlights the following interesting behavior, even if each subproblem is not unimodal initially, they become unimodal as each sub-problem converges towards its optimum.

I (0) = [I 1(0) ; I 2(0) ; I 3(0) ] = [[0 0.5]; [0.25 0.75]; [0.5 1]]

Random Selection

In the previous section, it has been shown how the optimization problem has been decomposed into a set of unimodal sub-problems on each interval, starting from a non-unimodal problem given by the uniform segmentation.

This leads to the fact that the initial segmentation choice may be desirable to start with a segmentation that can give a unimodal criterion on each interval from the Ąrst iteration, speeding up the algorithmŠs convergence and decreasing the execution time.

For that, a stochastic selection process is proposed: N trials of random segmentation are performed, and the best one in terms of NMSE is retained as the initial segmentation for the proposed algorithm. To validate the effectiveness of the stochastic process, 100 experiments of threshold optimization with a random selection have been performed for N=50. As shown in Figure 3.16, this stochastic process could signiĄcantly improve NMSE, thanks to a good initial segmentation. Out of the 100 tests performed, the worst NMSE is -37.69 dB, which is still better than NMSE from the uniform segmentation. Furthermore, Table 3.2 Highlights that the GS search requires 4 GS iterations with 341 NMSE computations starting from the uniform segmentation. In comparison, it requires only 1 or 2 GS iterations with a total number of 135 or 219 NMSE computations (including the 50 NMSE computations from the random trials) for the best and worst case of the 100 experiments considered. As the number of NMSE computations becomes a random variable using stochastic initialization, it is necessary to look for the probability distribution function that best Ąts the experimental data to determine an upper bound on the expected number of NMSE computations.

We have investigated the probability p(N < X) that the number of NMSE computations N required for the GS search to be less than an upper bound X.

Several experiments have been carried out with different random trials as follows:

• In this study, we present two Ągures of merit: the average number of NMSE computations and the conĄdence bound X, where some probability distributions are used to Ąt the number of NMSE computations, such as gamma distribution [START_REF] Evans | Statistical Distributions[END_REF], Burr distribution [START_REF] Burr | Cumulative Frequency Functions[END_REF], inverse Gaussian [START_REF] Chhikara | The Inverse Gaussian Distribution: Theory, Methodology and Applications[END_REF], and log-logistic distribution [START_REF] Shoukri | ŤSampling Properties of Estimators of the Log-Logistic Distribution with Application to Canadian Precipitation DataŤ[END_REF].

Figure 3.17 As shown in Figure 3.17 With these parameters, the upper bound X on the number of NMSE computations is 451 with a probability of 0.99, and 701 with a probability of 0.999 for experiment 1, where 10 random trials are performed before GS search. For experiment 2 with 30 random trials, the upper bound X is 427 with a probability of 0.99 and 654 with a probability of 0.999.

Regarding experiment 3, it has an upper bound X of 362 with a probability of 0.99, and 559 with a probability of 0.999 through Burr distribution, while with log-logistic distribution, it has an upper bound X of 330 with a probability of 0.99, and 468 with a probability of 0.999 Therefore, we can conclude that the interval [30 50] makes a sufficiently good range for choosing the number of random trials before applying the GS search.

Besides, using a stochastic initialization with 30 trials, the average number of NMSE computations will be 222 and below 427 with a probability of 0.99 or 654 with a probability of 0.999.

After the random selection process, the behavior of each NMSE(β i ) on their interval I i(0) are likely to be unimodal before starting the GS algorithm, as we can see in Figure 3.20. Experiments have shown that some tens of random selection trials are likely to provide a starting point that signiĄcantly improves the initial NMSE in comparison with the uniform segmentation. It is worth noting that for every 100 experiments of the stochastic initialization, the proposed optimization approach has converged towards the same segmentation. This illustrates its convergence behavior.

Sensitivity of Thresholds to Number of Segments

Nevertheless, the proposed approach to optimizing β (K) is only applied when the number of segments K is not changing. However, in some particular processes, such as sizing the DVR model or when the characteristics of the input signal are modiĄed, it may be necessary to change the number of segments K along with the optimization processŠs running.

Once K changes, the proposed approach must be executed to Ąnd the new optimal solution starting from uniform or random segmentation since it has been shown that both initial conditions converge towards the exact optimal solution. Figure 3.22 compares the position of the optimal segmentation for different K from 2 to 10 with M lin = 2 and M = 1. Note that for each K, the proposed approach to optimize β is started from the uniform segmentation. As can be seen, the optimal thresholds tend to be concentrated towards the nonlinear region of the PA. This empirical method carried out in Figure 3.21 allows to observe a repeated behavior, where when we go from K to K + 1, the optimal position of the Ąrst threshold β

(K+1) 1opt is located in the interval [0 β (K)
1 ], and vice versa, when going from K to K -1, the optimal segmentation β

(K-1) opt = [β (K-1) 1opt , ..., β (K-1) K-2opt ] is located in the interval [β (K) 1 1].
On the other hand, it is essential to emphasize the complexity of the proposed approach for threshold optimization, which is presented in Table 3.4 for each K. The complexity is represented in terms of NMSE computation, interval updates, and GS iterations. The complexity consistently increases with the number of segments K. The complexity required to optimize a given β (K) k of K can be reduced by using the information of the optimal segmentation β

(K-1) opt of K -1 or β (K+1) opt of K + 1.
We run the GS search to optimize β (K) k for each K starting from 2 to 10. We propose to deĄne the initial segmentation of K in terms of the optimal segmentation of K -1, which can be deĄned as

β (K) initial = [β (K) 1 ; • • • ; β (K) K-1 ] = [β r ; β (K-1) 1opt ; • • • ; β (K-1) K-2 opt ]
where β r is a thresholds that could be located randomly in the interval [0 β

(K-1)
1opt ]. How-ever, we propose to deĄne it as

β r = β (K-1) 1opt 2
As this process starts from K = 2, its initials segmentation is the uniform segmentation, i.e, β

(2

) initial = [0.5] .
Since the GS search always converges towards the optimal solution, we will focus on the impact of this approach on the complexity. Table 3.5 summarizes the complexity results of this process. Compared to the complexity results reported in Table 3.4, where the GS search optimizes each K starting from a random initial segmentation, the complexity in Table 3.5 conĄrms the interest of this method, where the complexity is signiĄcantly reduced.

In reverse order, we run the GS search starting from K = 10 to 2. The idea here is at K, the initial segmentation β (K) initial will be deĄned in terms of the optimal segmentation β (K+1) opt of K + 1, where we remove the Ąrst thresholds β (K+1) 1opt from β (K+1) opt and we deĄne the rest as an β

(K) initial of K as β (K) initial = [β (K) 1 ; • • • ; β (K) K-1 ] = [β (K+1) 2opt ; • • • ; β (K+1) Kopt ]
Table 3.6 summarizes the results of complexity. As starting from K = 10 with random segmentation, its complexity is similar to that reported in the last row in Table 3.4.

As can be seen, GS iterations and NMSE computations are drastically reduced even though the process is started from K = 10 with signiĄcant complexity. The application of this method to deĄne the initial segmentation is conĄrmed by the results presented on complexity. Additionally, it can be a good alternative to the random selection process if the information of β

(K-1) opt or β (K+1) opt is available.
Moreover, the interest of this method can be valuable in processes where K changes regularly, such as the sizing process of the DVR model, which will be illustrated in the next section. Figure 3.22 highlights an interesting behavior, showing that the optimal segmentation is not very sensitive to memory depth M . The optimization can be performed for the DVR model without memory to reduce the algorithm complexity, making it an important aspect to design an optimal DVR model. Figure 3.22: Optimal segmentation versus memory depth M However, the impact of memory depth has also been noted in [START_REF] Magesacher | ŤOptimal Segmentation for Piecewise RF Power AmpliĄer Models[END_REF] without this being illustrated by data such as in Figure 3.22.

Sensitivity of Thresholds to Memory Depth

Sensitivity of Thresholds to Model Terms

In order to study the sensitivity of the optimal segmentation to the model terms T S , we propose to optimize the thresholds for the following set:

T S 1 = [T 10 ] T S 2 = [T 10 T 11 ] T S 3 = [T 10 T 11 T 2 ] T S 4 = [T 10 T 11 T 2 T 3 ] T S 5 = [T 10 T 11 T 2 T 3 T 4 ]
with K = 4, M lin = 2, and M = 1.

Figure 3.23 present the optimal position of each threshold for different T S i . Unlike the sensitivity of thresholds to the memory depth, the model terms T S slightly impact the optimal segmentation. This impact is seen particularly over the variation of the Ąrst threshold β 1 , the optimal position of which is situated in the interval of [0.47 0.61]. The optimal position of β 2 is less sensitive than that of β 1 . The optimal position of β 3 remains almost stable with the model terms. Except for T S 1 , NMSE does not vary signiĄcantly with the different optimal segmentation. For T S 1 , applying the optimal segmentation β opt (T S 3 ) slightly worsens the NMSE by around 0.8 dB, which can be considered negligible in some scenarios.

This analysis allows us to consider that the optimal segmentation is not Ąrmly insensitive to the model terms used in T S , even if their value remains strongly correlated.

Considering that the optimal segmentation is independent of all the model terms, a rough approximation is established to be used with caution to accelerate the convergence of an optimization process proposed in the next section, aiming to design an optimal DVR model.

Conclusion of Thresholds Optimization

In this framework, we showed that the optimal segmentation could signiĄcantly improve the linearization performances compared to the uniform segmentation for the DVR model at the same number of coefficients.

We have proposed a new approach for optimizing the thresholds based on the decomposition of the global optimization problem into a set of sub-problems where the objective function is unimodal. The GS search is used to solve each unimodal sub-problem. The proposed approach has been compared with GA and brute force. Both algorithms converge to the same optimal thresholds. However, our approach presents a much lower complexity than the GA algorithm making it an appropriate candidate to design the DVR model. Once the thresholds have been optimized for a given number of thresholds K, an important aspect arises as to whether this parameter K, as well as the other parameters of the DVR model, are optimal or not. Thus, it is necessary to determine the optimal structure of the DVR model that presents the best compromise between modeling accuracy and complexity. This will be the topic of the following section.

Sizing of DVR Model Structure

The DVR model sizing consists in Ąnding an optimal model structure that leads to a promising modeling accuracy with reduced complexity.

The DVR model structure can be determined using an exhaustive search (brute force) by exploring all possible parameters order combinations. Setting the maximum order of K and M to K max and M max , respectively, the exhaustive search requires K max • M max • L tests, where L presents the total number of possible combination of model terms, L = P +7 k=1 P +7 k = 2 P +7 -1, according to (3.3) and (3.4). Limiting the maximum number of coefficients to be tested could signiĄcantly reduce the search space. However, the exhaustive search remains an heavy computation load, and an optimization approach can signiĄcantly reduce that load.

State-of-the-Art

In the literature relating to behavioral modeling, it is reported that it is not easy to determine a given behavioral model structure which has reduced complexity and high linearization performance. This operation is known as model sizing, which can be carried out using mainly two methods: Selection of basis function or applying optimization algorithms to determine the model structure. According to a criterion or a cost function, these methods return the best model structure, which refers to a trade-off between the modeling accuracy and model complexity.

As is reported in Section 2.6.1.2, the complexity to solve (2.17) depends on the number of coefficients related to the number of basis functions. Some pruning techniques to reduce the number of basis functions while keeping the same modeling accuracy have been studied in [START_REF] Jiang | ŞDigital predistortion using stochastic conjugate gradient method,Ť Broadcasting[END_REF] and [START_REF] Yu | ŞDigital predistortion using adaptive basis functions[END_REF]. For the MP model, a pruning technique is proposed in [START_REF] Reina-Tosina | ŞBehavioral modeling and predistortion of power ampliĄers under sparsity hypothesis,Ť Microwave Theory and Techniques[END_REF], where the total number of model kernels, which is determined by M and K, is minimized according to the difference between the measured and predicted signal.

Optimization algorithms can also determine the model sizing according to a good trade-off between modeling performance and model complexity. The modeling performance is represented mainly by a quadratic measure such as NMSE or ACPR and the model complexity by the modelŠs number of coefficients. Different heuristics could be applied to determine the optimal structure of the DPD model. This optimization process is seen as a nonlinear integer optimization problem. These algorithms could be deterministic or stochastic with local or global search and aim to Ąnd an optimal solution in a reduced runtime. Among these heuristics, we can cite hill-climbing (HC), simulated annealing (SA), tabu search, evolutionary algorithms (such as GA, ant colony ...), particle swarm optimization (PSO).

The Ąrst work in Ąnding the optimal structure for DPD models has been conducted for the GMP model using an integer GA [START_REF] Gotthans | ŞOptimal order estimation for modeling and predistortion of power ampliĄers[END_REF]. In this algorithm, the Ątness function is an additive weighted combination of NMSE and the number of coefficients. Compared to the exhaustive search, a solution close to the optimal model structure can be found much faster.

In [START_REF] Wang | ŤPerformance analysis of multi-stage cascaded digital predistortion[END_REF], the authors proposed an algorithm based on HC heuristic aiming to Ąnd the optimal structure of the GMP model with a reduced number of computations. The searching criterion used in this study controls the HC heuristic to ensure a good tradeoff between complexity and modeling accuracy of the GMP model. The HC heuristic provides an efficient approach to minimize the proposed criterion.

In [START_REF] Wang | ŤComparison of hill-climbing and genetic algorithms for digital predistortion models sizing[END_REF], a comparative study of two optimization algorithms, HC algorithm, and genetic algorithm, has been done in terms of convergence speed and accuracy of the obtained solution. The two algorithms were performed using the same criterion and had very similar performances, which converge towards the global optimum quickly compared to an exhaustive search.

Hill-Climbing

The HC is an optimization algorithm that belongs to the family of iterative algorithms [START_REF] Sullivan | ŞA convergence analysis of generalized hill climbing algorithms[END_REF] - [START_REF] Rueda | ŞA hill-climbing approach for automatic gridding of cdna microarray images[END_REF].

The principle of the HC algorithm consists of starting the search process from a given initial solution, determining its neighbors in a deĄned space, and choosing the best solution among these neighbors. This solution will be the initial solution in the next iteration, and the HC algorithm continues this process until no better solution is found. The deĄnition of the neighborhood is a very important aspect of the HC algorithm.

The principle of the HC algorithm is described in Algorithm [START_REF] Boumaiza | Thermal memory effects modeling and compensation in RF power ampliĄers and predistortion linearizers[END_REF].

Algorithm 5: Algorithm of hill-climbing Choose the initial element x 0(1) Evaluate the initial cost function f (x 0(1) ) while (1) do Determine the neighbors of x 0(q) : x i(q) ∈ V (q) Evaluate the cost function f (x i(q) ) x s(q) = argmin x i(q) ∈V (q) f (x i(q) ) if f (x s(q) ) < f (x s(q-1) ) then q = q + 1 x 0(q) = x s(q-1) else end while loop end end Take the last solution x s(q) as the best element In this study, The proposed approach based on the HC algorithm in [START_REF] Wang | ŤA Novel Algorithm for Determining the Structure of Digital Predistortion Models[END_REF] is extended to the DVR model to Ąnd an optimal structure with reduced complexity and high modeling accuracy.

According to Algorithm (5), at the Ąrst iteration, the HC algorithm starts from a given initial element x 0(1) which represents a DVR model structure, determines the neighbors of x 0(1) in the neighborŠs subspace V , and evaluates the cost function of these neighbors. The best cost function f (x s( 1) ) of the solution x s( 1) is compared with f (x 0(1) ). If f (x s( 1) ) is lower than f (x 0(1) ), the solution x s(1) becomes x 0(2) at the next iteration, and the HC repeats this search procedure. Moreover, the HC algorithm has been sufficiently described in [START_REF] Wang | Study on complexity reduction of digital predistortion for power am-pliĄer linearization[END_REF] to determine the optimal structure of the GMP model. Figure 3.25 presents an example of the state of HC at q th iteration, which shows how the neighbors are evaluated according to f . Figure 3.25: Neighbors of x 0(q) at q th iteration The neighbors are denoted by x i(q) , where x i(q) ϵV (q) . They are presented in Figure 3.25 by blue asterisks. The cost function f (x i(q) ) associated to x i(q) is evaluated. The solution x s(q) is the element of the neighborhood V (q) with the minimum cost function value.

There are two deĄnitions to consider. Firstly, the cost function f that achieves a tradeoff between the modelŠs complexity and modeling accuracy. Secondly, the deĄnition of the neighborhood that is used for HC search.

Cost Function

The cost function is an essential aspect of the HC algorithm. It refers to a particular criterion that leads to a trade-off between modeling accuracy and model complexity. NMSE is used to evaluate the modeling accuracy, which is denoted by N . The model complexity is presented by the number of coefficients of the model, denoted by C.

The cost function f is deĄned by combining N and C in a single criterion. In [START_REF] Gotthans | ŞOptimal order estimation for modeling and predistortion of power ampliĄers[END_REF], the Ątness function of GA is deĄned as a weighted additive combination of accuracy and model complexity.

In [START_REF] Wang | ŤA Novel Algorithm for Determining the Structure of Digital Predistortion Models[END_REF], several approaches have been proposed to deĄne the cost function using two different methods to combine N and C in one single criterion: additive and multiplicative combinations.

In this work, the cost function is deĄned using the additive combination, which is expressed by: f

(x i ) = N (x i ) + αC(x i ) (3.7)
where α is a positive real value and considered as a weighting coefficient (or the NMSE tolerance per coefficient).

In [START_REF] Wang | ŤA Novel Algorithm for Determining the Structure of Digital Predistortion Models[END_REF], the authors proposed two methods to determine the weighting coefficient using offline computation and online computation.

Definition of Neighborhood

A new deĄnition of the neighborhood is proposed for the DVR model in this work, allowing the HC algorithm to converge to the optimal structure.

For the DVR model, there are four parameters to size: the number of segments K, the memory depth M lin for the linear term, the memory depth M for the set T S , and which model terms will be selected from the set T.

In the discrete space U , the element x i that represents a DVR model structure is characterized by 3 integers: K i , M lin,i , M i and by T S i which is a k combination of the set T with k = ¶1, ..., P + 7♢. These parameters can have their values changed independently, and they compose a 4-dimension discrete space of DVR model structures.

The neighbor of element x i is deĄned as an 4-tuple

(K i + δ 1 , M lin,i + δ 2 , M i + δ 3 , T S,i + δ 4 ), where δ 1,2,3 ∈ [0, ±1].
The process involved behind δ 4 corresponds to either add to T S i a term from its relative complement in T, to remove a term from T S i , or to neither remove or add a term.

Figure 3.26 presents the discrete space V of the neighbors of x i in terms of K, M lin , and M for a given set of model term T S i . The blue point is the node ¶K i ,M lin,i ,M i ♢ of x i . The red points are the model parameters K, M lin , and M of neighbors.

T S i =                          [T 1,6 T 2 T 5 ] [T 1,0 T 1,1 T 1,6 T 2 T 5 ] [T 1,0 T 1,2 T 1,6 T 2 T 5 ] [T 1,0 T 1,3 T 1,6 T 2 T 5 ] [T 1,0 T 1,4 T 1,6 T 2 T 5 ] [T 1,0 T 1,5 T 1,6 T 2 T 5 ] [T 1,0 T 2 T 5 ] [T 1,0 T 1,6 T 1,7 T 2 T 5 ] [T 1,0 T 1,6 T 1,8 T 2 T 5 ] [T 1,0 T 1,6 T 5 ] [T 1,0 T 1,6 T 1,9 T 2 T 5 ] [T 1,0 T 1,6 T 2 T 3 T 5 ] [T 1,0 T 1,6 T 2 ] [T 1,0 T 1,6 T 2 T 4 T 5 ] [T 1,0 T 1,6 T 2 T 5 T 6 ] [T 1,0 T 1,6 T 2 T 5 T 7 ]
so the model in blue has 17 × 3 3 -1 neighbors to be evaluated.

Complexity Reduction of Hill-Climbing Algorithm

The complexity is an essential aspect of optimization algorithms. In this study, we link the complexity of the HC algorithm with four features:

• The total number of elements (neighbors) evaluated by the HC algorithm, which corresponds automatically to the number of f evaluations. It includes:

-The number of neighbors evaluated per one HC iteration, which is an indicator of the distribution of the total number of neighbors over the HC iterations.

The average number of neighbors per HC iteration will be presented as well.

• The total number of iterations required for the HC algorithm to perform, where one HC iteration is done when the cost functions of neighbors of an initial solution are computed, and the optimal solution is determined.

• The execution time (runtime) of the HC algorithm is also taken into account as a relative indicator of the complexity of the HC algorithm.

• The number of model coefficients of each neighborŠs structure is introduced since it reĆects the computational complexity of the identiĄcation process. Besides, it refers to the model complexity, which is essential for sizing the DVR model. Two features will be highlighted and linked to the computational complexity:

-Condition number of the regressor matrix Z H Z in (2.18).

-Dynamic range ϵ of the model coefficients deĄned in (2.23).

These four criteria assess the complexity of the HC algorithm and the numerical properties of the models trained during the search.

The complexity of the HC algorithm can be reduced by pruning the neighborhoods by adding some constraints on the number of coefficients.

In the HC algorithm, and according to the proposed deĄnition of neighborhood, each element x i can have up to ((P + 1) + 7) × 3 3 -1 neighbors.

An element x 0(q) at q th iteration can have common neighbors with x 0(q-1) from the previous iteration. Therefore, common neighbors will not be re-evaluated, which partially reduces the complexity of the HC search.

The search space U can also be limited by setting a maximum number of coefficients C max of the model structure to be tested.

As well, the model parameters K, M lin , and M can be limited by setting an interval for each one:

2 ⩽ K ⩽ K max 0 ⩽ M lin ⩽ M max 0 ⩽ M ⩽ M max (3.8)
where K max and M max are the maximum number of segments and the maximum memory depth, respectively.

The parameter K must be greater or equal to 2, which is the minimum number of segments. In a DVR structure, at least two segments bounded by one threshold are necessary to always remain in the concept of the segmentation approach.

In some cases, the neighbors may have a considerable variation of many coefficients. Therefore, a variation limit d C is added to the HC algorithm to control the maximum variation of the number of coefficients of neighbors.

With these constraints on C max and d c , the original neighborhood space V (q) of the element x 0(q) at q th iteration is reduced to the elements x i(q) with C i(q) ≤ C max and ♣C i(q) -C 0(q) ♣ ≤ d C .

Stopping Condition

The sizing algorithm of the DVR model can be stopped once the cost function of a given solution is not better than that of the previous one. However, in our case, the stop condition of the HC algorithm is activated according to two levels. For the Ąrst level, if a solution x s(q) has no neighbors whose number of coefficients is less than C max at the following iterations. This condition immediately leads to the stop of the HC algorithm. Otherwise, the HC algorithm checks the second level for which a positive tolerance ∆ν is determined, making it possible to choose a solution x s(q) whose N (x s(q) ) satisĄes the following conditions:

N (x s(q) ) ≤ N (x s(q-1) ) + ∆ν C(x s(q-1) ) < C(x s(q) ) ≤ C max (3.9)
where the integration of this constraint in the HC algorithm makes it possible to have a continuity of the search up to C max to explore other neighbors who can have better cost functions.

These stopping levels are achieved to give more Ćexibility to the HC algorithm for Ąnding good solutions properly. Once the HC algorithm is Ąnished, the optimal solution returned is the one that has the best value of f among the solutions of each HC iteration.

Algorithm of DVR Model Sizing

By combining the proposed deĄnition of neighborhood and the complexity reduction techniques into the HC algorithm, the optimal structure of the DVR model can be determined by the HC algorithm in Algorithm [START_REF] Ghannouchi | Behavioral modeling and predistortion[END_REF]. x s(0) = x 0(1) Initialize q = 1 while (1) do for

K i ± δ 1 do for M lin,i ± δ 2 do for M i ± δ 3 do for T S i ± δ 4 do
Determine the neighbor x i(q) ∈ V (q) of x 0(q) under d C and C max constraints Compute the cost function f (x i(q) ) end end end end if V (q) = ‰ then end while loop else

x s(q) = argmin

x i(q) ∈V (q) f (x i(q) )
Save x s(q) in solution space S (q) if C(x s(q-1) ) < C(x s(q) ) ≤ C max then if N (x s(q) ) ≤ N (x s(q-1) ) + ∆ν then q = q + 1 x 0(q) = x s(q-1) else end while loop end else end while loop end end end Return the optimal solution x s = argmin xs∈S (f (x s ))

Experiments and Results

Parameters of HC Algorithm

According to Algorithm [START_REF] Ghannouchi | Behavioral modeling and predistortion[END_REF], the evolution of the sizing process of the DVR model towards the optimal solution is mainly controlled by:

• C max : the maximum number of model coefficients.

• K max : the maximum number of segments to be assigned to a tested DVR structure.

• M max : the maximum memory depth to be assigned to a tested DVR structure.

• P : the nonlinearity order of the basis functions in the DVR model.

• α: the weighting coefficient of the cost function.

• d C : variation limit constraints on the neighborŠs coefficients.

• ∆ν: the NMSE tolerance of the HC algorithm.

The HC algorithm aims to Ąnd the optimal structure of the DVR model by taking into account three properties: the high performance of the optimal solution, the short path search towards the optimal solution, and the reduced HC complexity given by the fewest computations of f , which in turn depends on the number of coefficients of each evaluated models.

Setting the HC algorithm parameters should be carefully done to achieve these properties and provide an efficient HC algorithm.

First, we set C max to 50 coefficients. This makes it possible to have a well-extended behavior of the HC algorithm, of which the impact of C max is relatively weak in order to be able to focus on the other HC parameters. We set ∆ν to 0.2 dB.

We set the maximum number of segments K max to 10 and the maximum memory depth M max , including the linear termŠs memory depth to 10. K max and M max considerably controls the complexity of the HC algorithm according to the number of neighbors tested over the search.

The parameter P presents the order of the nonlinearity of the basis function in T 1,p . To study its impact in the DVR model, we propose to set P up to 1, then to 9, to efficiently analyze its global impact on the optimal solution and the DVR model.

The parameters α and d C inĆuence the HC algorithmŠs convergence towards the optimal solution. The parameter d C limits the number of neighbors of a given element, which may potentially not choose neighbors with a better cost function value, while the parameter α controls directly the cost function f in (3.7).

It is important to emphasize that the DVR modelŠs segmentation is uniform throughout the DVR model sizing.

In this study, we use the PAŠs experimental measurements presented in Section 3.3 using the same 20 MHz LTE signal.

Sizing DVR Model with P = 1

In the Ąrst step, we perform the HC algorithm to determine the optimal structure of the DVR model by setting P in T S to 1 (P represents the nonlinearity order of the DVR basis function).

In fact, we seek to analyze the behavior and impact of P on the DVR model through the HC algorithm. Moreover, when the model terms

T S = [T 1,0 T 1,1 T 2 T 3 T 4 ] are selected
by the HC algorithm, this corresponds to the DVR-6 model presented in the previous section.

The DVR model with the optimal structure returned by the HC algorithm is indicated by the DVR-Q P model, where Q is the number of model terms presented in the optimal structure, including the linear term.

Exhaustive Search

As mentioned before, the exhaustive search could be the spontaneous solution to Ąnd the optimal structure of the DVR model, but this approach requires a very high computational complexity.

However, we carried out an exhaustive search in this study to get all the possible structures, and we take them as a reference. The conditions of this exhaustive search are identical to that of the HC algorithm, where K max and M max were set to 10, C max to 50, and

P = 1 with T = [T 1,0 T 1,1 T 2 T 3 T 4 T 5 T 6 T 7 ].
Inspection of the results allows us to determine the DVR modelŠs best structure for each number of coefficients from 1 to C max = 50. It is essential to accentuate that the DVR models tested during the exhaustive search were performed using uniform segmentation. The exhaustive search requires almost 19 hours, in which there are a total of 24357 structures tested. The optimal solution that the HC algorithm seeks to Ąnd is located on the Pareto envelope of the exhaustive search.

HC Algorithm with α=0.1 and d C =C max

We set α to 0.1, which represents the NMSE tolerance per coefficient. We set d C to C max coefficients, which makes it possible not to introduce the impact of d C on the behavior of the HC algorithm. The blue dots present the tested structures during the HC algorithm, which presents all tested neighbors. The red dot presents the Ąrst point of the HC algorithm where its structure is:

Optimal Solution

K = 2 β = [0.5] M lin = 2 ; M = 0 T S = [ ]
The green dot presents the optimal solution where its structure has the best (minimum) f value:

K = 4 β = [0.25 0.5 0.75] M lin = 2 ; M = 1 T S = [T 1,0 T 3 T 6 ]
with C = 27 coefficients and N = -38.02 dB. In the next iteration, the best solution at the second iteration will be taken as the initial solution. The HC algorithm iteratively continues until it converges. By investigating the stop condition of the HC algorithm, it was shown in Figure 3.33 that after the seventh HC iteration, no solution found among the 91 neighbors satisĄes the conditions of (3.9), which leads to stopping the HC algorithm and returning the optimal DVR model. Table 3.7 presents the number of coefficients C, NMSE N , and the cost function f of the best solution at each HC iteration. The optimal DVR model is the solution of the third iteration, which has the best (minimized) cost function. After the 3 th iteration, an asymptotic behavior is observed around -38 dB, which can be conĄrmed in Figure 3.29. HC stops

Evolution of HC Algorithm

M lin 0 1 2 2 2 2 2 2 M 0 1 1 1 1 2 1 2 T S T 6 T 3 T 1,0 T 1,0 T 1,0 T 1,0 T 1,0 T 6 T 3 T 2 T 2 T 2 T 2 T 6 T 3 T 6 T 6 T 7 T 6 T 7 Num. of
M lin 1 2 2 2 2 2 2 M 1 1 1 1 2 1 2 T S T 6 T 3 T 1,0 T 1,0 T 1,0 T 1,0 T 1,0 T 6 T 3 T 2 T 2 T 2 T 2 T 6 T 3 T 6 T 6 T 7 T 6 T 7
Each column presents the initial solution structure, the number of the neighbors tested by the HC algorithm, and the structure of the best solution according to its cost function f . After the 7 th HC iteration, the HC algorithm evaluates 91 neighbors of the best structure from the 7 th HC iteration, but without Ąnding any structure with the best cost function, which leads to stopping the algorithm.

The memory depth of the linear term M lin is different from that of the other terms M , which conĄrms the interest of specifying two different memory depths.

It can be seen that M lin remains stable at 2 after the second iteration. From 2 th to 5 th iteration, the model terms are changed, while at the 6 th iteration, T S and M are changed, and at the 7 th iteration, K, M , and T S are changed.

On the other hand, we see that the number of evaluated neighbors at every iteration decreases after the third iterations. This refers to the fact that there are common neighbors between iterations that are not re-evaluated.

Since the HC algorithmŠs complexity is an important aspect, we summarize in Table 3.9 the total number of DVR models tested during the HC algorithm, where the average number of coefficients estimated by LS is 25 coefficients with a standard deviation of 13. By investigating the best DVR model at 27 coefficients from the exhaustive search, it has been shown that the parameters of its structure are:

K = 4 β = [0.25 0.5 0.75] M lin = 2 ; M = 1 T S = [T 1,0 T 2 T 6 ]
with C = 27 coefficients and N = -38.09 dB.

The HC algorithm returns a very close structure. In fact, by examining the behavior of the HC algorithm illustrated in Figure 3.29, it was shown that the neighbors explored by the HC algorithm do not contain any structure with the model terms T S = [T 1,0 T 2 T 6 ], which explains why the HC algorithm did not converge towards the solution given by the exhaustive search.

Nevertheless, the HC algorithm is conĄrmed and validated through this scenario, where the proposed neighborhood deĄnition demonstrates its effectiveness in determining the optimal DVR model.

Variation of d

C with α = 0.1
The parameter d C controls the space of the neighbors in the HC algorithm. Reducing d C leads to reducing the runtime of the search, but better solutions may be skipped.

The choice of d C raises three major concerns:

• From which d C , the optimal solution returned does not change.

• The impact of the d C variation on the HC complexity, especially the number of HC iterations and neighbors per iteration.

• The impact of the d C variation on the search path of the HC algorithm.

Impact of d C on Optimal Solution

In this study, the HC algorithm was carried out by varying d C from 3 to C max . Table 3.10 summarizes the parameters of the optimal structure of the DVR model for each d C . The parameters C, N , and f are presented as well. From Table 3.10, we can see that the HC algorithm returns the same optimal DVR model when d C exceeds a distance of 15 coefficients. On the other hand, all the optimal structures have the same number of coefficients (27 coefficients) except when d C = 3 and

M lin 2 2 2 2 2 2 0 2 2 2 2 2 2 M 1 1 1 1 1 1 1 1 1 1 1 1 1 T S T 1,0 T 1,0 T 1,0 T 1,0 T 1,0 T 1,0 T 1,0 T 1,0 T 1,0 T 1,0 T 1,0 T 1,0 T 1,0 T 1,1 T 4 T 3 T 3 T 1,1 T 1,1 T 3 T 4 T 3 T 3 T 3 T 3 T 3 T 5 T
d C = 9.
Regarding the structure parameters, from d C = 4, the optimal solutions has the same K, M lin , and M , unlike T S for which the structures are different for the different values of d c .

Notwithstanding, the model term T 1,0 is present in all structures, which presents the basis function of the DVR model and corresponds to: (n-i) .

K k=1 M i=0 c ki,10 ♣♣x(n -i)♣ -β k ♣e jθ
Therefore, selecting the best solution depends on the cost function, on its parameter α. The parameter d C is involved as well, which controls the size of the neighborhood space.

Impact of d C on HC complexity

The complexity of the HC algorithm for each d C is investigated. Table 3.11 presents the total number of neighbors tested for each d C and the number of neighbors per HC iteration. The Ąrst column presents d C . The second one presents the total number of neighbors tested by the HC algorithm. The third column shows in detail the number of neighbors for each HC iterations. According to Table 3.11, the HC algorithm requires seven iterations to converge from d C = 9, except when d C = 10 in which the HC algorithm requires 11 iterations. In a general way, we recognize that the total number of neighbors evaluated by the HC algorithm increases with the increases of d C and the runtime taken by the HC algorithm under the MATLAB environment, while the HC iterations are decreasing.

The number of neighbors evaluated remains 1101 neighbors after a distance of 30 coefficients. This allows us to conclude that beyond d C = 30, the HC algorithmŠs behavior, particularly its complexity, does not change.

On the other hand, when d C = 5, the total number of neighbors is reduced, and the HC iterations and the HC algorithm runtime. It can be explained by the fact that a distance of 5 coefficients allows appropriately to the search to perform only 6 HC iterations in signiĄcantly reduced runtime, in which it returns the same optimal DVR model returned when d C = 50.

Conclusion on d C Impact

In conclusion, d C has a signiĄcant impact on the complexity of the HC algorithm, unlike the linearization performances where the optimal solutions for each d C have an NMSE with a tolerance of 1 dB around.

When d C = 5, the HC algorithm exhibits a good behavior in terms of complexity, particularly the number of neighbors, the computational complexity for identifying the coefficients, and the runtime of the search.

In the following, we set d C to 5.

Variation of α with d C = 5

The determination of weighting coefficients for the multi-objective optimization problems is an essential aspect of building a cost function. The cost function f controls the evolution of the HC algorithm to determine an optimal DVR model with the best compromise, determined by α, between the number of coefficients and modeling accuracy.

The impact of the weighting coefficient α (unit tolerance) is investigated in terms of the optimal solution and the complexity of the HC algorithm. In this study, The HC algorithm is performed for each α from the vector [0.01 ; 0.025 ; 0.05 ; 0.075 ; 0.1 ; 0.15 ; 0.2 ; 0.3 ; 0.4 ; 0.5]

Impact of α on Optimal Solution

Table 3.12 summarizes the results of the HC algorithm where the parameters of the optimal structure, C, N and f = N + αC are presented for each α.

It is essential to remind that the more signiĄcant α, the more one seeks to limit the number of coefficients to the detriment of NMSE. According to Table 3.12, the variation of α inĆuences the optimal solutions of the HC algorithm, in particular in terms of the number of segments K and memory depth of the linear term M lin .

M lin 7 4 2 2 2 2 2 2 0 1 M 1 1 1 1 1 1 1 1 1 1
T S T 1,0 T 1,0 T 1,0 T 1,0 T 1,0 T 1,0 T 1,0 T 3 T 3 T 3 T 3 T 3 T 3 T 3 T 3 T 3 T 3 T 6 T
The parameters K and M lin decrease swiftly with increasing α, while M is much less sensitive to the variation of α, which remains at 1. Regarding T S , the set [T 3 T 6 ] is present in all best structures.

The behavior of how K and M lin change with the variation of α leads to an essential link between the design of the cost function f and the complexity of the optimal solution. If α increases, K and M lin decrease. Consequently, the number of coefficients C decreases, which reduces the complexity.

On the other hand, on the interval, [0.05 0.15], the optimal solution is not very sensitive to the value of α, and the HC algorithm returns the same DVR model.

Impact of α on HC Complexity

On the other hand, it is worth investigating the complexity of the HC algorithm with the variation of α. Table 3.13 summarizes the total neighbors evaluated by the HC algorithm and the HC iteration. The number of neighbors tested at each HC iteration and the total number of model coefficients are shown. According to Table 3.13 and when α = 0.05, the HC algorithm evaluates the lowest total number of neighbors and the total number of coefficients. Prominently, the HC algorithm with α = 0.05 provides an optimal DVR model that exhibits a good trade-off between the modeling performance, the model complexity, and the HC complexity. The structure of the optimal solution is:

K = 4 β = [0.25 0.5 0.75] M lin = 2 ; M = 1 T S = [T 1,0 T 3 T 6 ]
where it is denoted by the optimal DVR-4 1 model.

In this search, there are a total of 399 elements tested in 1.18 minutes and 6 HC iterations. These results are summarized in Table 3.13 for α=0.05. The parameters of the best DVR model at each HC iteration are summarized in Table 3.14.

M lin 1 2 2 2 2 2 M 1 1 1 1 1 1 T S T 6 T 3 T 3 T 3 T 3 T 1,0 T 6 T 6 T 6 T 6 T 3 T 6
According to Table 3.14, the number of segments K has a signiĄcant impact on the evolution of the search path, while M lin and M remain at 2 and 1, respectively. On the other hand, the model term T 6 , which corresponds to

K k=1 M i=0 c ki,6 ♣♣x(n -i)♣ -β k ♣ • x(n) • ♣x(n -i)♣ 2
is present in the structure of the best neighbors over the HC iterations.

Sizing DVR Model with P = 9

Once the HC algorithm has been validated for the DVR model with P = 1, we carried out the HC algorithm to size the DVR model using the terms in (3.4) and setting P to 9.

Exhaustive Search

With T = [T 1,[0,...,9] T 2 T 3 T 4 T 5 T 6 T 7 ], and by setting K max to 10, M max to 10, and C max to 50, the exhaustive search requires 339341 tests to be performed, which could take almost seven days to achieve, making of it too heavy to implement in this study. As shown previously, setting α to 0.05 and d C to 5 exhibits a good trade-off between the modeling accuracy of the optimal solution and the algorithm complexity.

We set α to 0.05 and d C to 5 coefficients. HC stops

K 2 2 2 2 2 2 2 2 2 2 2 2 M lin 0 1 2 2 2 2 2 2 2 2 2 2 M 0 1 1 1 1 1 1 1 1 1 1 1 T S T 1,5 T 1,5 T 1,5 T 1,5 T 1,5 T 1,5 T 1,5 T 1,2 T 1,2 T 1,2 T 1,2 T 1,8 T 1,8 T 1,8 T 1,7 T 1,7 T 1,7 T 1,5 T 1,5 T 1,4 T 1,3 T 2 T 1,9 T 1,8 T 1,8 T 1,8 T 1,7 T 1,6 T 1,5 T 1,4 T 2 T 1,9 T 1,9 T 1,9 T 1,8 T 1,7 T 1,6 T 1,5 T 2 T 2 T 2 T 1,9 T 1,8 T 1,7 T 1,6 T 6 T 5 T 2 T 1,9 T 1,8 T 1,7 T 6 T 5 T 2 T 1,9 T 1,8 T 6 T 5 T 2 T 1,
M lin 1 2 2 2 2 2 2 2 2 2 2 M 1 1 1 1 1 1 1 1 1 1 1 T S T 1,5 T 1,5 T 1,5 T 1,5 T 1,5 T 1,5 T 1,5 T 1,2 T 1,2 T 1,2 T 1,2 T 1,8 T 1,8 T 1,8 T 1,7 T 1,7 T 1,7 T 1,5 T 1,5 T 1,4 T 1,3 T 2 T 1,9 T 1,8 T 1,8 T 1,8 T 1,7 T 1,6 T 1,5 T 1,4 T 2 T 1,9 T 1,9 T 1,9 T 1,8 T 1,7 T 1,6 T 1,5 T 2 T 2 T 2 T 1,9 T 1,8 T 1,7 T 1,6 T 6 T 5 T 2 T 1,9 T 1,8 T 1,7 T 6 T 5 T 2 T 1,9 T 1,8 T 6 T 5 T 2 T 1,9 T 6 T 5 T 2 T 6 T 5 T 6
As shown in Table 3.15, the parameters K, M lin , and M remain at 2, 2, and 1, respectively, while the model terms change from one optimal structure to another over the HC iterations, which means that the terms bring a signiĄcant impact on the convergence of the HC algorithm towards the optimal solution. Table 3.16 presents C and N of the best solution at each HC iterations. After the 5 th iteration, an asymptotic behavior is observed around -38.7 dB, which can be conĄrmed in Figure 3.35. The total number of coefficients of all DVR models tested during the HC algorithm is 20728, with a mean of 20 coefficients and a standard deviation of 11. In the light of the behavior of the HC algorithm for P = 1 and P = 9, in this section, we present a comparison of both cases, and we also highlight the impact of P on the DVR model through its sizing using the HC algorithm.

The PAs are inherently nonlinear systems, in which many baseband nonlinear dynamic models, which can represent nonlinear systems with memory effect, can be used (MP, GMP, ...) to ensure linear ampliĄcation.

As mentioned previously, the DVR model is based on the segmentation approach, which aims to reduce the nonlinearities piecewise, unlike the global models derived from the Volterra series, which consists of deploying a high nonlinearity order to compensate for the nonlinearity effects, which leads to unsatisfactory numerical properties for identi-Ącation process and complexity for hardware implementation.

In this study, we introduce the nonlinearity order p in (3.5) in the second term

T 1,p = K k=1 M i=0 c ki,1p ♣♣x(n -i)♣ -β k ♣e jθ(n-i) • ♣x(n)♣ p
in order to enhance the DVR model ability to represent strong nonlinearities. However, dealing with p should be carefully since it presents the same properties that the nonlinearity order for the global model derived from the Volterra series. In fact, this can be conĄrmed in Figure 3.36 and Figure 3.37, which show the impact of p on the modeling accuracy and numerical properties of the DVR model performance. Figure 3.36 presents the variation of the NMSE versus p, for p from 0 to 9. In this scenario, the NMSE is computed between the PA output and the DVR model output

y(n) = M lin i=0 a i x(n -i) + K k=1 M i=0 c ki,1p ♣♣x(n -i)♣ -β k ♣e jθ(n-i) • ♣x(n)♣ p
with K = 4, M lin = 2, and M = 2, which are selected arbitrary.

For this scenario at hand, the number of coefficients C is deĄned as

C = M lin + 1 + K × (M + 1)
which remains equal to 15 coefficients with the variation of p. As can be shown in Figure 3.36, the variation of p signiĄcantly inĆuences the ability of the DVR model to model accurately the behavior of the PA. In this scenario, an optimal structure is found in terms of NMSE when the nonlinearity order p = 6, which corresponds to

2 i=0 a i x(n -i) + 4 k=1 2 i=0 c ki,1p ♣♣x(n -i)♣ -β k ♣e jθ(n-i) • ♣x(n)♣ 6
On the other hand, Figure 3 The impact of p on the numerical properties of the DVR model is weighty through Figure 3.37. Figure 3.37a shows that the condition number of the regressor matrix evolves exponentially with the increase of p, while the dynamic range of the model coefficients evolves exponentially in Figure 3.37b. This motivates the need to take into consideration the numerical properties of the DVR model. By taking the optimal structure when p = 6, it is not an optimal choice if the condition number or the dynamic range is involved in the criterion to choose this optimal structure, driving to make a trade-off between modeling accuracy and the numerical properties.

In the light of the results reported in Figure 3.36 and Figure 3.37, the interest of having a high nonlinearity order in the DVR model is advantageous to represent strong nonlinearities but must be chosen with care in order not to worsen the numerical properties.

Given that the HC complexity depends essentially on the complexity and the numerical properties of the trained DVR models, we propose to focus on the impact of the nonlinearity order on the DVR model and the HC algorithm by comparing and analyzing the results of the HC algorithm with P = 1 and P = 9 in terms of optimal structure, DVR model complexity, DVR model numerical properties, and HC complexity.

Comparison of Optimal Solution

Table 3.17 summarizes the optimal structures of the DVR-4 1 and DVR-6 9 models.

T S T 1,0 T 1,5 T 3 T 1,7 T 6 T 1,8 T 1,9 T 2
As shown in Table 3.17, the structure of DVR-4 1 has a signiĄcant number of segments K, while the structure of the DVR-6 9 model is determined by the model terms. This is justiĄed by the limitation of the nonlinearity order p to P = 1, which leads the HC algorithm to explore new neighbors by increasing K. In contrast, for a limitation of P = 9, the HC algorithm explores several combinations of model terms, leading to an optimal structure composed of 6 model terms, including the linear term.

Replacing the parameters of the optimal structure in (3.3), the DVR-4 1 model is expressed by:

y DV R-4 1 (n) = 2 i=0 a i x(n -i) + 4 k=1 1 i=0 c ki,10 ♣♣x(n -i)♣ -β k ♣e jθ(n-i) + 4 k=1 1 i=0 c ki,3 ♣♣x(n -i)♣ -β k ♣ • x(n -i) + 4 k=1 1 i=0 c ki,6 ♣♣x(n -i)♣ -β k ♣ • x(n) • ♣x(n -i)♣ 2
The DVR-6 9 model is expressed by:

y DV R-6 9 (n) = 2 i=0 a i x(n -i) + 2 k=1 1 i=0 c ki,15 ♣♣x(n -i)♣ -β k ♣e jθ(n-i) • ♣x(n)♣ 5 + 2 k=1 1 i=0 c ki,17 ♣♣x(n -i)♣ -β k ♣e jθ(n-i) • ♣x(n)♣ 7 + 2 k=1 1 i=0 c ki,18 ♣♣x(n -i)♣ -β k ♣e jθ(n-i) • ♣x(n)♣ 8 + 2 k=1 1 i=0 c ki,19 ♣♣x(n -i)♣ -β k ♣e jθ(n-i) • ♣x(n)♣ 9 + 2 k=1 1 i=0 c ki,2 ♣♣x(n -i)♣ -β k ♣ • x(n)
Their performances are summarized in Table 3.18 in terms of the number of coefficients, NMSE, the dynamic range of the model coefficients, and the condition number. The HC algorithm returns the DVR-4 1 and DVR-6 9 models according to the same setting of α and d C .

Both models have almost similar modeling accuracy (NMSE) but with different complexity (number of coefficients). The DVR-6 9 model has achieved roughly the modeling accuracy of the DVR-4 1 model only with 23 coefficients, while the DVR-4 1 model requires 27 coefficients.

In contrast, the DVR-4 1 model exhibits much better numerical properties compared to the DVR-6 9 model. This can be seen in the dynamic range of the DVR-4 1 model coefficients, which presents only 2% of the dynamic range required by the DVR-6 9 model coefficients, making it an impressive property to be taken into consideration. Further, the condition number of the DVR-4 1 model is signiĄcantly reduced compared to the DVR-6 9 model reĆecting that there is a meaningful difference in the computational complexity to identify the coefficients of both models. Besides, the complexity also corresponds to the realization of the basis functions, and each order of nonlinearity corresponds to an additional multiplication.

Comparison of Postdistortion Results

Using the postdistortion architecture, Table 3.19 summarizes the linearization performance of both models, which are presented in terms of NMSE, ACPR, and EVM app . This conĄrms the rough similarity of the linearization performance of both models. As can be seen, the DVR-4 1 and DVR-6 9 models signiĄcantly improve ACPR.

Comparison of HC Complexity

In Table 3.20, we summarize the results of the HC algorithm reported in Section 3.5.8.2.5 and Section 3.5.8.3.2.2, where the HC complexity is highlighted in terms of the total number of DVR models evaluated during the search, the HC iterations, the execution time taken by the HC algorithm to perform, and the total number of coefficients by summing the coefficients of all the neighbors tested. Once again, when P = 1, this characteristic has a substantial impact on the HC complexity. By taking a general view of the results in Table 3.20, we can conclude that the HC complexity increases by around 300% if we change P from 1 to 9.

The distribution of the neighbors over the HC iterations is depicted in Figure 3.39. According to the HC iteration, the curves present the number of neighbors for P = 1 in blue and P = 9 in red. The two curves behave roughly in the same way since the number of neighbors evaluated at the 3 rd HC iteration is the highest number. This can be justiĄed because the HC algorithm explores many new neighbors compared to the previous HC iteration. The number of neighbors decreases from the 3 rd HC iteration, which amounts to the fact that the HC algorithm Ąnds common neighbors between previous HC iterations.

Comparison of Numerical Properties

The condition number can be a quantitative factor to represent the computational complexity as it is involved in identifying the model coefficients. Their dynamic range is also discussed. For this, we propose to present for each neighbor the condition number and the dynamic range, which are sorted in ascending order and represented in Figure 3.40. As shown in Figure 3.40, the numerical properties of the DVR models tested during the search are reduced when P is set to 1, unlike when P = 9. In fact, this can be interpreted by the shape of the curves. For example, in Figure 3.40a, the blue curve that shows the results with P = 1 has nearly 220 neighbors whose condition number does not exceed 10 10 . These neighbors have a reduced computational sensitivity to identify their model coefficients.

A particular behavior is observed in the blue curve, where a discontinuous form occurs at neighbor number 220 to the next (in a sorted way), where the condition number suddenly moves from 10 10 to 10 18 . This demonstrates that when P = 1, the HC algorithm evaluates neighbors of a given element with different numerical properties, even if they are subject to the deĄnition we proposed for the neighborhood, which consists of adding or removing one function to the model parameters. Unlike the blue curve, the red curve representing the HC algorithm with P = 9 has a roughly progressive linear shape over nearly 1000 neighbors, which causes an enormous calculation, which is highlighted in terms of execution time in Table 3.20.

Regarding the dynamic range of the model coefficients depicted in Figure 3.40b, the blue curve behaves as an exponential shape, which means that the majority of neighbors tested have a reduced dynamic range between 0.5 and 316, while the red curve (P = 9) has a progressive shape that goes up to around 10 6 .

Discussion

The maximum nonlinearity order P for the DVR model signiĄcantly impacts the HC algorithm to determine an optimal structure of the DVR model. Setting P to 1 drive the HC algorithm to determine an optimal structure with multiple segments K and remarkable numerical properties, while the choice of P equal 9 leads the HC algorithm to choose an optimal structure dominated by the model term T 1,p with higher nonlinearity order of p which causes wicked numerical properties. In contrast, the number of segments K remains at two segments. On the other hand, the computational complexity behind determining the optimal structure is exploded by 300 % when P vary from 1 to 9.

Conclusion of DVR Model Sizing

In this section, the HC algorithm proposed for the GMP model has been extended to the DVR model to Ąnd the optimal DVR model structure in a minimal number of computations.

Two important aspects are highlighted in this framework: the new deĄnition of the neighborhood proposed for the HC algorithm and the deĄnition of a cost function to achieve the best trade-off between modeling accuracy and complexity. The HC algorithm parameters such as coefficient distance and the weighting coefficient of the cost function are discussed.

The impact of the nonlinearity order present in the DVR model has been studied through the model sizing using the HC algorithm.

Conclusion

In this chapter, two processes are discussed and investigated.

The design of the optimal DVR is divided into two processes: The Ąrst process investigates how to optimally set the thresholds that deĄne the boundary of each segment for a given structure, while the second process studies the determination of the optimal structure of the DVR model for a uniform segmentation.

The Ąrst process to optimize the thresholds of the DVR model is done by proposing an approach based on GS search. The proposed approach is based on decomposing the global optimization problem into a set of sub-problems where the objective function is unimodal. GS search is used to solve each unimodal sub-problem. The results illustrate the effectiveness of the proposed approach and show that the optimization of the thresholds signiĄcantly improves the linearization performances compared to conventional DVR with uniform segmentation.

For the second process, we extend the HC algorithm proposed for the GMP model to the DVR model by introducing a new neighborhood deĄnition to determine the neighbors of the evaluated structure. The convergence of the HC algorithm is ensured by a cost function that exhibits the best trade-off between the linearization performance and the model complexity. A comparison between the HC algorithm and exhaustive search is carried out. The HC algorithm validates its efficiency by returning an optimal structure close to the best structure resulting from the exhaustive search. Besides, the HC algorithm was performed with minimal execution time compared to the exhaustive search. One of the advantages of the HC algorithm is that the searching path can be controlled by conĄguring some parameters such as the coefficients distance d C and the weighting coefficient of the cost function.

Chapter 4

A System Approach to Design and Implement Optimal DVR Model

Introduction

In the previous chapter, the optimal structure of the DVR model is determined according to a cost function representing the trade-off between modeling accuracy and complexity. The sizing process of the DVR model is based on the HC algorithm, where the neighborhood deĄnition we proposed allows us to converge towards the optimal solution.

On the other hand, the thresholds that bound the segments of the DVR model are optimized over the normalized input range. The approach to optimize the thresholds consists of decomposing the global optimization problem into sub-problems that require the application of GS search.

Designing an optimal DVR model involves determining its optimal structure with optimal segmentation according to the same criteria as those used in the HC algorithm to size the DVR model. This process presents a combination of the HC algorithm and the proposed approach to optimize the thresholds in a particular way.

In this chapter, we propose system approaches (SysApp) to perform this joint optimization to design the optimal DVR model. These SysApp are developed through different versions, discussed, and compared to the HC algorithm regarding the optimal solution, linearization performance, and complexity.

Once the optimal DVR model has been designed, it is essential to emphasize its numerical properties for hardware implementation. Indeed, the purpose of designing a predistorter is to implement it in a hardware device such as an FPGA.

In this chapter, we investigate and propose an implementation strategy by focusing mainly on the optimal DVR model designed by our system approaches.

Design of Optimal DVR Model

The optimal DVR model design consists of determining its optimal structure with the optimal segmentation according to the same criteria used in the HC algorithm to size the DVR model that was discussed in Section 3.5.

System approaches

In this study, we propose different SysApp to design the optimal DVR model, which is developed through different versions, and presented as follows:

• Direct approach, which is based on integrating the proposed approach in Section 3. [START_REF] Raich | ŤOn the modeling of memory nonlinear effects of power ampliĄers for communication applications[END_REF] into the HC algorithm developed to size the DVR model. This approach is indicated by D-SysApp.

• SimpliĄed approach where the thresholds of the HC envelope are optimized. It is indicated by S-SysApp.

• Optimal approach based on the junction of the thresholds optimization and the HC algorithm, of which the running of the proposed approach to optimize the thresholds is controlled by the number of segments and the model terms. It is indicated by O-SysApp.

These approaches are discussed and compared to the HC algorithm with uniform segmentation in terms of linearization performance of the optimal solution and implementation complexity, including computational complexity and numerical properties.

Direct Approach (D-SysApp)

The Ąrst system approach called direct approach (D-SysApp) can be seen as an intuitive approach to design the optimal model by combining the HC algorithm described in Algorithm [START_REF] Ghannouchi | Behavioral modeling and predistortion[END_REF] and the proposed approach based on GS search for optimizing the thresholds β described in Algorithm (3).

Indeed, the optimization of β k is performed for each neighbor through the GS search. The cost function f is computed according to C and N of each model with its optimized segmentation β opt .

At Ąrst sight, this approach has the highest computational complexity. Its complexity will be primarily evaluated by the total number of NMSE computations, including those required for the proposed approach to optimizing β.

The principle of D-SysApp is described in Algorithm [START_REF] Isaksson | ŞA comparative analysis of behavioral models for rf power ampliĄers,Ť Microwave Theory and Techniques[END_REF].

Algorithm 7: Algorithm of D-SysApp Choose the initial element x 0(1) Evaluate the initial cost function f (x 0(1) )

x s(0) = x 0(1) Initialize q = 1 while (1) do Determine the neighbors x i(q) ∈ V (q) of x 0(q) if V (q) = ‰ then end while loop else Optimize β for each x i(q)

x s(q) = argmin

x i(q) ∈V (q) f (x i(q) )
Save x s(q) in solution space S (q) if C(x s(q-1) ) < C(x s(q) ) ≤ C max then if N (x s(q) ) ≤ N (x s(q) ) + ∆ν then q = q + 1 x 0(q) = x s(q-1) else end while loop end else end while loop end end end Return the optimal solution x s = argmin xs∈S (f (x s ))

Simplified Approach (S-SysApp)

The direct approach can be simpliĄed in complexity by optimizing the thresholds only for speciĄc DVR model structures.

These structures are determined according to 2 approaches:

• Senv-SysApp: The structures which are on the envelope of the HC algorithm.

• Sitr-SysApp: The structures which are determined at the end of each HC iteration.

The principle of Senv-SysApp is to apply the threshold optimization process using GS search once the HC algorithm is Ąnished. The complexity of Senv-SysApp is mainly evaluated by the complexity of the HC algorithm, with the complexity of the GS search multiplied by the number of structures found on the envelope.

On the other hand, Sitr-SysApp consists of inserting the GS search in the HC algorithm, which will be activated once the HC iteration is Ąnished and applied to the best neighbor of this HC iteration. Similar to Sitr-SysApp, the complexity of Senv-SysApp is primarily rated by the HC complexity, with the complexity of the GS search multiplied by the number of HC iterations.

Since the threshold optimization of the best neighbor at a given HC iteration improves its cost function, it may likely force the HC algorithm to stop earlier, mainly if no neighbor in the next HC iteration is found with a better cost function than the previous one. However, Sitr-SysApp remains a relatively limited approach where determining the solution at each HC iteration relies on comparisons between a structure with optimal segmentation and others with uniform segmentation.

The algorithm of Senv-SysApp is described in Algorithm [START_REF] Reveyrand | A Novel Experimental Noise Power Ratio Characterization Method for Multicarrier Microwave Power AmpliĄers[END_REF].

Algorithm 8: Algorithm of Senv-SysApp Choose the initial element x 0(1) Evaluate the initial cost function f (x 0(1) )

x s(0) = x 0(1) Initialize q = 1 while (1) do Determine the neighbors x i(q) ∈ V (q) of x 0(q) if V (q) = ‰ then end while loop else

x s(q) = argmin

x i(q) ∈V (q) f (x i(q) )
Save x s(q) in solution space S (q) if C(x s(q-1) ) < C(x s(q) ) ≤ C max then if N (x s(q) ) ≤ N (x s(q) ) + ∆ν then q = q + 1 x 0(q) = x s(q-1) else end while loop end else end while loop end end end for x s ∈ S (q) do Optimize β of x s Update f (x s(i) ) with β opt in S (q) end Return the optimal solution x s = argmin xs∈S (f (x s ))

The algorithm of Sitr-SysApp is described in Algorithm [START_REF] Sombrin | On the formal identity of evm and npr measurement methods: Conditions for identity of error vector magnitude and noise power ratio[END_REF].

Algorithm 9: Algorithm of Sitr-SysApp

Choose the initial element x 0(1) Evaluate the initial cost function f (x 0(1) )

x s(0) = x 0(1) Initialize q = 1 while (1) do Determine the neighbors x i(q) ∈ V (q) of x 0(q) if V (q) = ‰ then end while loop else

x s(q) = argmin x i(q) ∈V (q) f (x i(q) ) Optimize β of x s(q) Update f (x s(i) ) with β opt in S (q) if C(x s(q-1) ) < C(x s(q) ) ≤ C max then if N (x s(q) ) ≤ N (x s(q) ) + ∆ν then q = q + 1 x 0(q) = x s(q-1) else end while loop end else end while loop end end end Return the optimal solution x s = argmin xs∈S (f (x s ))

Joint Optimal Approach (O-SysApp)

The motivation behind the joint optimal approach, indicated by O-SysApp, is to execute the HC algorithm to size the DVR model on a set of neighbors with optimal segmentation, i.e., the optimal segmentation of the trained DVR models is available and stored in LUT.

The size of the LUT depends on the number of times the optimization process on the thresholds β has been executed. The optimal thresholds are stored in the LUT along with the search for the optimal solution by the HC algorithm, which makes it a joint approach. Although the perfect case is where the optimal segmentation of all structures is available and stored, it cannot be performed because it requires an exhaustive search, which leads to D-SysApp.

As seen in Section 3.4.9, Section 3.4.10, and Section 3.4.11, the parameters K, M , and T S have different impacts on the thresholds optimization process. The property of each parameter impact could be used to built O-SysApp through two versions:

• Ored-SysApp: we consider that the optimal segmentation is not sensitive to parameters M and T S . The thresholds β are optimized only when the number of segments K changes over the HC search. The optimal segmentation corresponding to each K is saved in LUT.

• Ogen-SysApp: we consider that the optimal segmentation is not sensitive to the parameter M . The thresholds β are optimized when the number of segments K or model terms T S change through the HC algorithm.

The principle of Ored-SysApp is to optimize the thresholds of a given model structure only when K changes. The optimal segmentation of each K is stored in LUT. The size of LUT depends on the maximum number of segments explored by the HC algorithm.

An optimal segmentation β (K) opt for a given K covers all structures that have the same K, regardless of the other parameters (M , T S ).

On the other hand, the threshold optimization process is based on the methods proposed in Section 3.4.9 in order to speed up the convergence of the GS search.

The algorithm of Ored-SysApp is described in Algorithm [START_REF] Cheaito | Analyse théorique des distorsions dans la bande et en dehors de la bande de transmission pour les signaux ' a porteuses multiples : Impact conjoint des non-linéarités de lŠampliĄcateur de puissance et effets de mémoire et prédistorsion[END_REF]. Ogen-SysApp is an upgraded version of Ored-SysApp, where the algorithm is similar, except, when the model terms change, the thresholds optimization process is called. The Ogen-SysApp is more general than Ored-SysApp since the model terms T S are considered as a sensitive parameter in addition to K. The parameter M remains insensitive as indicated in Section 3.4.10.

The algorithm of Ogen-SysApp is described in Algorithm [START_REF] Guel | ŤAnalysis and comparison of clipping techniques for OFDM Peak-to-Average Power Ratio reduction[END_REF].

Algorithm 10: Algorithm of Ored-SysApp Choose the initial element x 0(1) Optimize β of x 0(1) Store β

(K 0 )
opt in LUT Evaluate the initial cost function f (x 0(1) ) x s(0) = x 0(1) Initialize q = 1 while (1) do Determine the neighbors x i(q) ∈ V (q) of x 0(q) if V (q) = ‰ then end while loop else if β

(K i ) opt / ∈ LUT then if β (K i -1) opt ∈ LUT then β init = [ β (K i -1) 1 opt 2 ; β (K i -1) 1opt ; • • • ; β (K i -1) K i -2 opt ] else if β (K i +1) opt ∈ LUT then β init = [β (K i +1) 2opt ; • • • ; β (K i +1) K iopt ] else β init = [ 1 K i , ..., K i -1 K i ] end Optimize β (K i ) opt of x i(q) with initial segmentation β init Store β (K i ) opt in LUT else Get β (K i ) opt from LUT end Update x i(q) with β (K i ,T S i ) opt
x s(q) = argmin x i(q) ∈V (q) f (x i(q) ) Save x s(q) in solution space S (q) if C(x s(q-1) ) < C(x s(q) ) ≤ C max then if N (x s(q) ) ≤ N (x s(q) ) + ∆ν then q = q + 1 x 0(q) = x s(q-1) else end while loop end else end while loop end end end Return the optimal solution x s = argmin x s(0) = x 0(1) Initialize q = 1 while (1) do Determine the neighbors x i(q) ∈ V (q) of x 0(q) if V (q) = ‰ then end while loop else if β

(K i ,T S i ) opt / ∈ LUT then if β (K i -1) opt ∈ LUT then β int = [ β (K i -1) 1 opt 2 ; β (K i -1) 1opt ; • • • ; β (K i -1) K i -2 opt ] else if β (K i +1) opt ∈ LUT then β init = [β (K i +1) 2opt ; • • • ; β (K i +1) K iopt ] else β init = [ 1 K i , ..., K i -1 K i ] end Optimize β (K i ,T S i ) opt of x i(q) with initial segmentation β init Store β (K i ,T S i ) opt in LUT else Get β (K i ,T S i ) opt from LUT end Update x i(q) with β (K i ,T S i ) opt
x s(q) = argmin x i(q) ∈V (q) f (x i(q) ) Save x s(q) in solution space S (q) if C(x s(q-1) ) < C(x s(q) ) ≤ C max then if N (x s(q) ) ≤ N (x s(q) ) + ∆ν then q = q + 1 x 0(q) = x s(q-1) else end while loop end else end while loop end end end Return the optimal solution x s = argmin xs∈S (f (x s ))

The structure of the optimal solution returned by D-SysApp is:

K = 4 β opt = [0.59 0.82 0.93] M lin = 2 ; M = 1 T S = [T 2 T 6 ]
with C = 19 coefficients and N = -38.6 dB.

Compared to the optimal solution of the HC algorithm, the number of segments and the memory depth are similar in both architectures. The model terms T S are different. On the other hand, the position of the optimal thresholds is almost similar to what has been shown in Section 3.4 when K = 4. The structure of the optimal solution returned by Senv-SysApp is:

S-SysApp

Senv-SysApp

K = 4 β opt = [0.55 0.78 0.92] M lin = 2 ; M = 1 T S = [T 1,0 T 3 T 6 ]
with C = 27 coefficients and N = -38.85 dB.

In this approach, the optimal solution is chosen based on the cost function, i.e., the solution with the best f is the optimal solution. As shown in Ągure 4.3, Sitr-SysApp is stopped at the third HC iteration, which is justiĄed by the fact that the GS search optimizes best neighbor thresholds, which sig-niĄcantly improves NMSE, hence the cost function. This impacts the HC algorithm, for which the probability of Ąnding neighbors with better cost functions in the successive iterations remains low, causing the algorithm to stop.

Sitr-SysApp

The structure of the optimal solution returned by Sitr-SysApp is:

K = 3 β opt = [0.26 0.88] M lin = 2 ; M = 1 T S = [T 3 T 6 ]
with C = 15 coefficients and N = -38.01 dB.

O-SysApp

Ored-SysApp

.4 presents the results of Ored-SysApp described in Algorithm [START_REF] Cheaito | Analyse théorique des distorsions dans la bande et en dehors de la bande de transmission pour les signaux ' a porteuses multiples : Impact conjoint des non-linéarités de lŠampliĄcateur de puissance et effets de mémoire et prédistorsion[END_REF], where the structure of the optimal solution returned is: 4.1 presents the LUT created by Ored-SysApp, where the optimal segmentation is saved according to the variation of K. In this scenario, the maximum number of segments K explored by Ored-SysApp is 7 segments.

Ogen-SysApp

As described in Algorithm [START_REF] Guel | ŤAnalysis and comparison of clipping techniques for OFDM Peak-to-Average Power Ratio reduction[END_REF], the results of Ogen-SysApp to design an optimal DVR model are depicted in Figure 4.5. During Ogen-SysApp, the thresholds optimization process is performed 106 times, where the optimal segmentation is saved in 2D according to K and T S .

Comparison of Search Behavioral

Figure 4.6 compares the search path of the envelope of each system approach, where the envelope of the HC algorithm with uniform segmentation is plotted in red. As can be seen, Ored-SysApp is sub-optimal from 10 to 30 coefficients, while Senv-SysApp, Sitr-SysApp, and Ogen-SysApp achieves nearly the same performance as D-SysApp from 15 coefficients. And as expected, the direct approach (D-SysApp) achieves the best performance.

Optimal Solution

Table 4.2 summarizes the optimal solutions returned by the different system approaches to design the DVR model. As can be shown, the optimal solutions have the same model structure in terms of memory depths (M lin and M ), while the number of segments K and model terms T S are different from an optimal solution to another.

T S T 1,0 T 2 T 1,0 T 3 T 1,0 T 3 T 3 T 6 T 3 T 6 T 1,
In contrast, the optimal segmentation when the optimal solutions share the number of segments is roughly located on the same region, regardless of the model terms.

On the other hand, the optimal solutions have the same linearization performance but with different complexity. By computing the cost function with α = 0.05, the best optimal solution is the one returned by Ogen-SysApp followed by D-SysApp, which is plausible since it is based on the full integration of the GS search within the HC algorithm. However, it is important to emphasize the complexity of each system approach to conclude on their effectiveness.

Complexity

The complexity of the proposed system approaches to design the DVR model is assessed in the same way as the complexity reduction of the HC algorithm described in Section 3.5.5. Table 4.3 presents the complexity of the system approach, which is evaluated by the total number of models evaluated over the system approach, the HC iterations, the total number of NMSE computations including those of the threshold optimization process, the execution time (runtime), and the total number of model coefficients identiĄed along with the system approach. According to Table 4.3, we can see that the behavior of each approach impacts the complexity of its convergence towards the optimal solution.

The runtime of D-SysApp is very important with a high number of NMSE computations, so it cannot be a realistic candidate to design the DVR model, especially for real-time DPD adaptation. Senv-SysApp and Sitr-SysApp remain candidates in terms of the runtime and number of NMSE computations. Ored-SysApp and Ogen-SysApp exhibit superior complexity behavior compared to Sens-SysApp and Sitr-SysApp.

Discussion and Conclusion

According to the results in Table 4.2 and Table 4.3, and since D-SysApp is excluded from our choice due to its computational complexity, we can conclude that Ogen-SysApp is an excellent candidate to design the DVR model since it presents a good trade-off between linearization performance and computational complexity.

On the other hand, Sitr-SysApp exhibits a similar linearization performance compared to Ogen-SysApp at reduced complexity (15 model coefficients) and with the same model terms, which cannot be removed from our choice. However, the stop criterion of Sitr-SysApp can be subject to improvement to make it more superimposed with Ogen-SysApp in terms of the search path. Another viewpoint that can be deepened and elaborated is to develop a system approach by combining Ored-SysApp and Sitr-SysApp. When Ored-SysApp allows having a particular slope of the searching behavior, we switch to Sitr-SysApp to reĄne the search path.

Hardware Implementation Strategy of Optimal DVR Model

The design of an appropriate predistorter based on a software environment such as MAT-LAB is suitable for laboratory usage but not for real-time DPD applications. Besides, software solutions provide difficulty in implementing the adaptive algorithm. Therefore, to meet the industrial requirements, the DPD system should be implemented in hardware such as FPGA, which has many advantages in digital signal processing, including high-speed processing, Ćexible implementation, and high reliability.

Two critical points to be considered during the translation from software to hardware are: accuracy and speed. Parallel operation mechanisms can improve these points.

In the hardware implementation of DPD, there are mainly two processes:

• The application of DPD on the input signal needs to be sampled at a sufficient sampling rate to meet the growing demand for wideband signals in the current telecommunication systems and to be able to accommodate the widening of the signal bandwidth by the highest nonlinearity order of the DPD.

• The identiĄcation and update of the DPD coefficients, which can be done using block per block or sample per sample approaches.

As is mentioned in Chapter II, there are two structures to implement the DPD model in hardware: the LUT method and the direct structure with multipliers and adders. The LUT method stores the precomputed value of the gain functions, which are indexed by the input magnitude. On the other hand, the direct structure with multipliers and adders, the so-called direct multiple and add method, usually takes advantage of the pipeline architecture of the FPGA to increase the throughput in which more logic resources are involved compared to the LUT method.

In this section, we propose a hardware strategy to implement the optimal DVR model designed by the joint approach.

State-of-the-Art

According to the literature related to this research line, many works have been reported to implement the DPD models on hardware.

In [START_REF] Ben Nasr | ŤOn the critical issues of DSP/FPGA mixed digital predistorter implementation[END_REF], the authors describe the platform for the software-hardware architecture of DPD implementation for 3G transceivers, where the authors used a curve-Ątting function to generates a smooth version of the AM-AM and AM-PM curves. In [START_REF] Gilabert | ŤMulti-Lookup Table FPGA Implementation of an Adaptive Digital Predistorter for Linearizing RF Power AmpliĄers With Memory Effects[END_REF], the authors proposed hardware implementation of DPD, which is based on a nonlinear auto-regressive moving average structure, and mapped into a set of scalable LUTs.

Another work dedicated to implementing models based on the Volterra series is reported in [START_REF] Liszewski | ŤLow-complexity FPGA implementation of Volterra predistorters for power ampliĄers[END_REF], where an FPGA design is proposed and considers the memory depth. In [START_REF] Gilabert | ŤLook-Up Table Implementation of a Slow Envelope Dependent Digital Predistorter for Envelope Tracking Power AmpliĄers[END_REF], the authors proposed a hardware methodology with high-level synthesis for FPGA devices to implement the arithmetic representations of the models based on the Volterra series. A combination between FPGA and ARM processors is proposed in [START_REF] Long | ŤThe design and implementation of baseband predistorter based on FPGA and ARM[END_REF] to implement the MP model, where the authors used the ARM processor for its advantage in Ćoating-point computations. In [START_REF] Li | ŤFPGA Implementation of Memory-Based Digital Predistorters with High-Level Synthesis[END_REF], the authors proposed a scalable LUT architecture to implement the GMP model using high-level synthesis.

Regarding the DVR model, few research works have been reported in the literature devoted to hardware implementation. In [START_REF] Cao | ŤA ModiĄed Decomposed Vector Rotation-Based Behavioral Model With Efficient Hardware Implementation for Digital Predistortion of RF Power AmpliĄers[END_REF], the authors proposed a hardware structure by dividing the DVR model into a sub-composed format to reduce the implementation complexity. However, the parameters of the DVR model have been set arbitrarily without raising any concerns about the model complexity reduction. In [START_REF] Li | ŤInstantaneous Sample Indexed Magnitude-Selective Affine Function-Based Behavioral Model for Digital Predistortion of RF Power AmpliĄers[END_REF], a model derived from the DVR model is proposed, where the hardware implementation has been emphasized.

Direct Multiply and Add Method

The direct method is based on the use of multipliers and adders to implement predistortion. For polynomials such as

y = a 0 + a 1 x + a 2 x 2 + • • • + a n x n
the complexity of the direct method to evaluate the polynomial expression can be reduced by using the HornerŠs rule by formulating y as

a 0 + x(a 1 + x(• • • + x(a n-2 + x(a n-1 + a n x))))
The HornerŠs rule allows to evaluate a polynomial of degree n with only n multiplications and additions and it can be implemented with modular architectures using pipeline.

By extending this concept to the polynomial with memory (MP model, for example), the function y consists of several stages of a + b♣x(ni)♣ in which n pipelined stages are required in the circuit. This block only deals with multiplication and summation algorithms, and it is simple to implement in FPGA.

However, the complexity of implementation depends on the structure of the model (in the case of the MP model, memory depth, and nonlinear order) since the multipliers are one of the most complex and expensive components in FPGA. In addition, it is necessary to take into account the dynamics of the model coefficients and the computation noise introduced by each elementary operation.

For the optimal DVR model designed by Ogen-SysApp and according to (3.3), it can be expressed by:

y(n) = 2 i=0 a i x(n -i) + 5 k=1 1 i=0 c ki,3 ♣♣x(n -i)♣ -β k ♣ • x(n -i) + 5 k=1 1 i=0 c ki,6 ♣♣x(n -i)♣ -β k ♣ • x(n) • ♣x(n -i)♣ 2 which gives
M i=0 x(n -i)( K k=0 a k,i ♣x(n -i)♣ k ) = M i=0 x(n -i)G i
G i = K k=0 a k,i ♣x(n -i)♣ k
and the number of LUT is controlled by the memory depth M . The LUT entries can be made with two methods of spacing: uniform and non-uniform [START_REF] Cavers | ŤOptimum indexing in predistorting ampliĄer linearizers[END_REF] [START_REF] Cavers | ŤOptimum table spacing in predistorting ampliĄer linearizers[END_REF]. The non-uniform spacing based on the companding function has been proven to provide the best modeling accuracy. However, the probability density function of the input signal is required, which is generally of high computational complexity. Therefore, uniform spacing is often chosen to space the LUT due to its relatively low complexity and sufficiently good results compared to non-uniform spacing.

For the DVR model, the value stored in LUT corresponds to the function G(♣x(ni)♣) such as

G(♣x(n -i)♣) = K k=1 c ki ♣♣x(n -i)♣ -β k ♣
It should be noted that the precomputation is performed when updating the DPD coef-Ącients.

For the optimal DVR model returned by Ogen-SysApp, its implementation using LUT is depicted in Figure 4.10.

c k1 ♣♣x(n -1)♣ -β k ♣
The accuracy of the function

K k=1 c ki ♣♣x(n -i)♣ -β k ♣
depends on the number of bits of the LUT data, while the size of LUT depends on the number of bits used in the quantization of the signal and the interpolation used for the LUT values.

The complexity of the LUT implementation can be evaluated by the number of LUT (e.g., memory depth M of the DVR model), the quantization bits, and the number of bits required for each memory slot which indicates the resolution of the value stored in LUT.

Having 2 LUTs in Figure 4.10 refers to the fact that the optimal structure of the DVR model has M = 1, which is optimized by the joint approach we proposed to design the DVR model.

The length of the LUT, denoted by L, is deĄned by the number of bits N Q used in the quantization of the signal by

L = 2 N Q (4.1)
In order to Ąnd the optimal L, we propose to evaluate the linearization performance of the optimal DVR model designed by the joint approach using the LUT implementation structure by varying N Q from 2 to 20 bits.

Figure 4.11 presents the behavior of linearization performance in terms of NMSE of the DVR model according to the variation of N Q from 2 bits to 20 bits. The NMSE from the direct approach (NMSE = -38.93dB) is presented as well.

Conclusion

In this chapter, we proposed a system approach combining the HC algorithm and GS search to design the optimal DVR model. The system approach is developed through different versions, where the concept is to determine the optimal structure of the DVR model with its optimal segmentation. These system approaches are discussed and compared in terms of the optimal solution and computational complexity.

LUT-based hardware implementation for optimal DVR model is proposed and optimally sized according to the linearization performances compared to the direct implementation, making it a solid strategy to implement the DVR model on hardware such as FPGA. On the other hand, it is essential to emphasize that some model parameters, such as the memory depth M , which impact the complexity of implementation (number of LUTs), could replace the number of coefficients in the cost function design since the number of coefficients has no impact on the implementation complexity by LUT. Nevertheless, the number of coefficients has an impact on the identiĄcation process.

The study of the optimal DVR model design is shown to be robust in this Chapter and will be applied as a DPD model to linearize a dual-input Doherty PA in the next chapter.

Chapter 5

Linearization and Efficiency Enhancement of Dual-Input Doherty Power Amplifier

Introduction

The introduction of non-constant amplitude modulated signal makes the enhancement of PA efficiency with maintaining an adequate level of linearity more challenging. Starting from wideband code division multiple access (W-CDMA) in 3G, the PAPR of signals increases with the use of OFDM in 4G LTE and 5G. Besides, the requirements for higher transmission rates should be satisĄed. Wider signal bandwidths are needed to accommodate higher data rates.

For modulated signals with high PAPR, PA should operate at large power back-off, which decreases PA efficiency. Advanced architectures of PA based on dynamic load or supply modulation have been proposed in the literature to avoid wasting excessive power resources. Some of the most popular solutions are Doherty PA [START_REF] Doherty | ŤA New High Efficiency Power AmpliĄer for Modulated Waves[END_REF], envelope elimination and restoration [START_REF] Kahn | ŤSingle-Sideband Transmission by Envelope Elimination and Restoration[END_REF], envelope tracking [START_REF] Wang | ŤDesign of wide-bandwidth envelope-tracking power ampliĄers for OFDM applications[END_REF], and outphasing PA [START_REF] Chireix | ŞHigh power outphasing modulationŤ[END_REF]. These highly efficient topologies require linearization techniques such as DPD to ensure the speciĄed level of linearity according to the communication standards. Introducing DPD to the ampliĄcation system becomes more necessary to meet the linearity requirement, especially with increased signal bandwidth.

The ampliĄcation architectures based on active load modulation, one of the most common PA efficiency enhancement techniques, rely on the nonlinear interaction between the main and auxiliary transistors for modulated signals with a signiĄcant dynamic. Although these architectures can be designed with a single RF input to be used in the transmitter, several studies have been reported in the literature to highlight the beneĄts in maintaining separate inputs [START_REF] Darraji | ŤA Dual-Input Digitally Driven Doherty AmpliĄer Architecture for Performance Enhancement of Doherty Transmitters[END_REF] [100] [START_REF] Andersson | ŤA 1Ű3-GHz Digitally Controlled Dual-RF Input Power-AmpliĄer Design Based on a Doherty-Outphasing Continuum Analysis[END_REF], and the advantages of dual-input Doherty PA compared to single-input has been studied in [START_REF] Piacibello | ŤComparison of S-band Analog and Dual-Input Digital Doherty Power AmpliĄers[END_REF]. Therefore, additional degrees of freedom, so-called free parameters offered by the separate inputs, can be used to control the performance or to enhance the PA efficiency [START_REF] Darraji | ŤDoherty Goes Digital: Digitally Enhanced Doherty Power AmpliĄers[END_REF].

Focusing on dual-input PA architecture, searching for optimal free parameters guaranteeing high performance requires experimental cross-validation or exhaustive search. These processes are usually costly and computationally signiĄcant, especially when the search space is enormous and not limited.

Setting these free parameters to their optimal values within a deĄned interval can be viewed as a global optimization problem. Several techniques have been proposed in the literature to Ąnd the optimal set of free parameters among large tunable ranges considered as search intervals [START_REF] Horst | ŤIntroduction to Global Optimization: Non-convex Optimization and Its ApplicationsŤ[END_REF]. Typically, the global optimization techniques could be catego-rized into three classes: the deterministic methods (Bayesian search algorithm, branch and bound algorithms, interval methods, etc.), the stochastic methods (simultaneous perturbation, simulated annealing, swarm algorithms, etc.), and the heuristic methods (GA, ant colony, tabu search, particle swarm optimization, etc.).

In this chapter, we propose a new architecture to enhance the power efficiency of the dual-input Doherty PA while meeting the linearity requirement. The proposed architecture is based on an auto-tuning approach consisting of optimizing the free parameters by a proposed hybrid optimization algorithm according to an adaptive cost function. This cost function adaptively indicates the trade-off between power efficiency and linearity. The free parameters to be optimized in the auto-tuning approach cover PA biasing, baseband calibration process such as the power level, and PAPR reduction. Besides, a DPD linearization is jointly optimized and integrated into our approach to meet the linearity requirements.

The remainder of this chapter is organized as follows.

Section II presents the system-level aspects, including the dual-input Doherty PA and the experimental testbench.

Section III presents the free parameters to be optimized.

In Section IV, we describe the proposed auto-tuning approach-based architecture to optimize the performance of the dual-input Doherty PA with the DPD linearization technique.

Section V presents the experimental results of the proposed approach. Finally, the conclusion is given in Section VI.

System-Level Aspects

Dual-Input Doherty Power Amplifier

The PA, which is based on active load modulation (Doherty, outphasing) with separate RF inputs, can be viewed, by generalization, as the block diagram depicted in Figure 5.1. The dual-input PA has two RF inputs, a drain bias V DC , and two gate-source voltages V GS,1 and V GS,2 to control the transistorŠs terminal independently. A typical example of PA with independent V GS is Doherty, where the main (carrier) ampliĄer is biased in class B and the auxiliary ampliĄer (typically named peaking ampliĄer) in class C. The instantaneous amplitude and phase of each input in baseband, as well as the V GS gate bias voltages, can be controlled and adjusted separately, which allows having a signiĄcant degree of freedom for these parameters to improve the performance of the PA, in particular its efficiency.

At a given operating back-off (BO), the main ampliĄer reaches its maximum output voltage and becomes maximally efficient. From this power level, the auxiliary ampliĄer turns on and injects current into the common node, increasing the output power and modulating the load seen by the main ampliĄer.

In this chapter, the dual-input Doherty PA presented in [START_REF] Komatsuzaki | Ť3.0Ű3.6 GHz wideband, over 46% average efficiency GaN Doherty power ampli-Ąer with frequency dependency compensating circuits[END_REF] is used as the device under test (DUT). In [START_REF] Komatsuzaki | Ť3.0Ű3.6 GHz wideband, over 46% average efficiency GaN Doherty power ampli-Ąer with frequency dependency compensating circuits[END_REF], the authors have presented a 3.0 -3.6 GHz wideband GaN Doherty PA with a frequency dependency compensating circuit. The architecture of the dual-input Doherty PA is presented in Figure 5.2. This dual-input Doherty PA can produce a peak output power of 40 dBm (10 W) and has a linear gain of 15 dB. The measurements reported in [START_REF] Komatsuzaki | Ť3.0Ű3.6 GHz wideband, over 46% average efficiency GaN Doherty power ampli-Ąer with frequency dependency compensating circuits[END_REF] have been performed under a modulated signal with 20 MHz channel LTE signal and showed, over 3.0 -3.6 GHz frequency range with an efficiency exceeding 40%. The linearizability of this PA has been studied as well, where the ACPR has been improved by 10 -15 dB using DPD linearization.

Nevertheless, the two RF inputs of Doherty PA in [START_REF] Komatsuzaki | Ť3.0Ű3.6 GHz wideband, over 46% average efficiency GaN Doherty power ampli-Ąer with frequency dependency compensating circuits[END_REF] were connected to the same RF input, and no optimization process has been reported for this DUT.

Testbench

The effectiveness of the proposed auto-tuning approach-based architecture is validated through experiments that have been carried out using a testbench.

The block diagram and the photo of the testbench are shown in Figure 5.4 and Figure 5.5, respectively. A reference clock source is injected into the AD9371 board using the MXG X-Series N5182A vector signal generator at 30.72 MHz with 5 dBm of power. The transceiver AD9371 up-converts the baseband signals to the carrier frequency f c at 3 GHz using a 245.76 MHz sampling frequency. Two low-pass Ąlters are used for each path (main and peaking) to remove unwanted frequencies.

Each RF signal stimulates the dual-input Doherty PA through a coupler ZUDC30-183 and driver AMP1132. The drivers are used to control DUTŠs input power level, while the two couplers are used to measure the RF input powers: the peaking power P p and main power P m . Two couplers are installed in cascade at the output of the DUT. The Ąrst coupler is used to attenuate the RF output signal for the observation path. A Keysight N9010A MXA spectrum analyzer is used to characterize the signalŠs spectrum at the output of DUT through the second coupler. The RF instruments and DC supplies used in this testbench are connected to the PC workstation by LAN over Ethernet.

During the testbench setup, a calibration process of the RF instruments has been carried out to ensure that the displayed measurements are traceable and reliable.

For the observation path, The RF output signal is down-converted to the baseband by AD9371, which provides the baseband signal to the PC workstation. Around 100000 IQ samples were recorded for the baseband process with a sampling rate of 245.76 MSPS.

After the reception of the IQ data through the observation Rx path, several synchronization processes are applied using Matlab to prepare the baseband input-output signals from the PA for the processing. This acquisition process is organized as follows:

• Synchronization: The input-output baseband signals are normalized and synchronized in the time domain using a cross-correlation operation.

• f c correction: This process removes the frequency offset between the transmitter and the receiver.

• Fractional time-shift: This method is proposed in [START_REF] Olkkonen | ŤFractional Time-Shift B-Spline Filter[END_REF], which is based on a Bspline Ąlter that produces a time-shift ∆ ∈ [0, 1], which is an arbitrary fraction of the sampling period.

• IQ imbalance correction: Due to the distortion of the down-converter at the receiver side, which introduces an image interference, that may have an error Ćoor limiting the demodulation performance. Moreover, the IQ imbalance introduced by the local oscillator may be assumed constant over the signal bandwidth and can be characterized by the phase mismatch that can occur between the I and Q components, which becomes not precisely 90 degrees, and an amplitude mismatch due to the difference in gain of the I and Q branch mixers. In this work, a circularity-based blind compensation algorithm proposed in [START_REF] Anttila | ŤBlind Compensation of Frequency-Selective I/Q Imbalances in Quadrature Radio Receivers: Circularity -Based Approach[END_REF] is used as an IQ imbalance compensator.

NMSE between the baseband input and output of DUT, before and after the application of this process, is used as an indicator to assess the effectiveness of these acquisition processes. Once NMSE has improved signiĄcantly, the baseband input-output signals are then ready to be trained for the identiĄcation algorithm.

Free-Parameters

To meet the objectives of optimizing the dual-input Doherty PA, which consists of improving efficiency while maintaining a better linearity level, we propose an architecture described in Figure 5.6.

In this work, the DPD is used to linearize the DUT by compensating for the nonlinearities of dual-input Doherty PA. Additionally, CFR is used to reduce the PAPR of the transmitted signal so that dual-input Doherty PA can operate with less BO. Both CFR and DPD are implemented in baseband.

On the other hand, the dual-input Doherty PA requires two separate input signals from the PC workstation. Therefore, the baseband signal to be sent to the PA should generate a peak signal v p (n), with

v p (n) = v(n) -v(n) µ ♣v(n)♣ if ♣v(n)♣ > µ 0 if ♣v(n)♣ ≤ µ (5.1)
The peak signal v p (n) is then Ąltered using a noise shaping Ąlter and the Ąnal signal u(n) is therefore calculated by subtracting a time-aligned weighted version of v f (n) from v(n).

In this block, a clipping threshold µ is considered as a free-parameter of CFR.

Free-Parameters of Digital Splitter

The motivation behind using two separate RF inputs is to eliminate analog input splitters, such as the Wilkinson divider, and to allow independent power control to the main and peaking ampliĄer.

As mentioned before, the baseband signal x(n), which is a complex signal represented by an amplitude X and a phase θ, i.e. x = Xe jθ , is divided into two complex signals x m and x p deĄned as:

x m = α m Xe j(θ) = α m x x p = α p Xe j(θ-ϕ) = α p e -jϕ x (5.2)
The three parameters α m , α p , and ϕ in (5.2) have to be carefully tuned to ensure the high efficiency of the PA.

We suggest using only two parameters to design the digital splitter: the power ratio between the amplitude of the two inputs, denoted by α, instead of α m and α p , and the phase shift ϕ between their phases.

The relationship between α and (α m ,α p ) is deĄned as follows. The power of x m and x p are deĄned as:

P m = α 2 m P in P p = α 2 p P in (5.3)
with P in = P m + P p (5.4) where P in is the input powers of the baseband signals x. It gives

α 2 m + α 2 p = 1 (5.5)
So, if we deĄne α as:

α 2 m = α (5.6)
We have

α 2 p = 1 -α 2 m = 1 -α (5.7)
Hence, the baseband parameters α m and α p can be expressed in terms of α by:

α m = √ α ; α p = √ 1 -α (5.8)
The phase shift ϕ between the main and peaking branches is critical. It affects the PA efficiency, the ampliĄcation gain, and the linearity. It can also independently be controlled at different frequencies f c .

We propose to assign the original phase θ of x to x m and add a phase shift ϕ to the peaking signal.

θ m = θ θ p = θ -ϕ (5.9)
From (5.8) and (5.9), the baseband input signals x m and x p can be expressed as:

x m = √ αx x p = √ 1 -αe -jϕ x (5.10)
The architecture of the digital splitter is described in Figure 5.8. The design of a digital splitter consists of optimizing the power ratio α and phase shift ϕ, which are taken as free parameters in the optimization engine.

Free-Parameters of Transceiver

In the calibration process, it has been shown that two essential operations are needed to be established from the baseband: Ąxing the DAC resolution, which is integrated into the RF transceiver, and setting the gain attenuation, which controls the power level of the transmitted signal.

For DAC resolution, it is recommended to scale the IQ data to 2 15 in the baseband to ensure high accuracy of data to be transmitted to minimize loss of information. The gain attenuation directly controls the power level of the signal in the Tx branch based on an internal step attenuator with a step of 0.05 dB.

The block diagram of the transceiver is depicted in Figure 5.9.

Figure 5.9: Block diagram of transceiver

In the transceiver block, we use two parameters A m and A p deĄned in the baseband to control the attenuation in the branch Tx1 occupied by the main ampliĄer and in Tx2 occupied by the peaking ampliĄer.

We have approximately estimated the relationship between A m (and A p ) and the average power of the main (and peaking) RF signal as:

P m,dBm (A m ) = A m + aPAPR(x m ) + b √ α + c P p,dBm (A p ) = A p + aPAPR(x p ) + b √ 1 -α + c (5.11)
where a, b, and c are parameters deĄned empirically from some preliminary tests and stored in LUT and indexed in terms of the center frequency f c , and the signal bandwidth. Indeed, the process behind estimating a, b, and c consists of performing many tests with waveforms of different bandwidths, at different frequencies, and for different power levels. These test results provide a datasets of a, b, and c that will be used subsequently according to the parameters of the scenario at hands. If A m and A p have the same baseband numerical value, and x m = x p with α = 0.5, the power P m,dBm and P p,dBm are assumed to be the same. However, in practice, we have observed that by assigning the same numerical value to A m and A p , the measured powers P m,dBm and P p,dBm through the power sensors are different. This difference between P m,dBm and P p,dBm can be adjusted and compensated in baseband by using a parameter denoted by ψ with P m,dBm (A m ) = P p,dBm (A p + ψ) (5.12)

Finding ψ that satisĄes (5.12) can be done in the calibration process. However, we propose to take ψ as a free parameter controlled from baseband, which could be viewed as a hardware parameter since it can adjust the input power distribution over the main and peaking ampliĄers.

Free-Parameters of DUT

The main and peaking input of dual-input Doherty PA controls the main and peaking ampliĄer, biased with V GS,m and V GS,p , respectively.

These biased voltages are controlled from baseband and deĄned within a range of DC voltage. The DC power supply used to manage the gate bias voltages is connected to the PC workstation through an Ethernet connection that enables real-time voltage monitoring from the baseband.

Therefore, V GS,m and V GS,p are taken as a free-parameters. Since the output impedance matching network is not changing, and since we conĄgure the input side of dual-input Doherty PA, not the output side, it is not necessary to change the drain bias V DC while handling the dual-input Doherty PA.

The free parameters of the proposed architecture to be controlled are summarized in Table 5.1.

Linearization and Efficiency Enhancement of Dual-Input Doherty Power Amplifier

Finding the optimal conĄguration of each block in Table 5.1 can be seen as an optimization problem to be solved. The brute-force search can help Ąnd the optimal free parameters by exploring all possible combinations in the searching space. However, the brute-force search is not a practical solution to be implemented in real-time applications. Consequently, an autotuning approach based on an optimization algorithm is proposed to meet this need.

State-of-the-Art

In the literature, the design and study of dual-input Doherty PA with enhanced efficiency have been reported in many research works. Few of them deal with the joint optimization of PA parameters and their linearization technique.

The Ąrst work has been reported in [START_REF] Cahuana | ŤLinearization of dual-input Doherty power ampliĄers[END_REF], in which the authors have conĄrmed the interest of using a combination of splitting the input signal and vector-switched digital predistorter, in which the linearity is improved while maintaining high efficiency. The splitterŠs design has been achieved by performing several combinations of α and ϕ in a simulation environment, which does not accurately present the behavior of dual-input Doherty PA.

In [START_REF] Singerl | ŤPredistortion-and developmentplatform for multi-input RF power ampliĄers[END_REF], the authors presented a hardware implementation of a real-time DPD and digital splitter for multi-input PA. The static splitter was achieved by a nonlinear static LUT function, where the signals x m and x p at the output of the splitter are expressed as:

x m (n) = x(n) × LUT 1 (♣x(n)♣) x p (n) = x(n) × LUT 2 (♣x(n)♣) (5.13)
where LUT 1 and LUT 2 are nonlinear complex function, which are created by performing an exhaustive search. However, the voltages V GS,m and V GS,p of the Doherty PA used in measurement tests are biased identically, which raises concerns about its efficiency. The Ąrst work related to the online learning-based optimization of dual-input Doherty PA is proposed in [START_REF] Niu | ŤStochastically approximated multiobjective optimization of dual input digital Doherty Power AmpliĄer[END_REF]. The authors proposed an adaptive technique based on a simultaneously perturbed stochastic approximation (SPSA) algorithm to tune the free parameters α, ϕ, V GS,m and V GS,p . This algorithm has been performed according to a cost function J deĄned as:

J(Θ) = ω × G(Θ) + (1 -ω) × PAE(Θ) (5.14)
where Θ = [α ϕ V GS,m V GS,p ] is the vector of free parameters to be optimized, ω is a weighting coefficient, and G is the gain of PA. In [START_REF] Niu | ŤStochastically approximated multiobjective optimization of dual input digital Doherty Power AmpliĄer[END_REF], the authors showed a signiĄcant improvement in G and PAE. However, the linearity requirement has not been met since SPSA has focused only on efficiency enhancement. In [START_REF] Ma | ŤMachine-Learning Based Digital Doherty Power AmpliĄer[END_REF], a machine learning-based optimization is proposed to optimize the freeparameters used in [START_REF] Niu | ŤStochastically approximated multiobjective optimization of dual input digital Doherty Power AmpliĄer[END_REF]. Practically, [START_REF] Ma | ŤMachine-Learning Based Digital Doherty Power AmpliĄer[END_REF] is an extension of the work reported in [START_REF] Niu | ŤStochastically approximated multiobjective optimization of dual input digital Doherty Power AmpliĄer[END_REF] where the cost function is updated from its previous version, which includes, in addition to PAE and gain, the output power P out and ACPR as metric referring to the linearity.

The optimization process used in [START_REF] Ma | ŤMachine-Learning Based Digital Doherty Power AmpliĄer[END_REF] consists of two phases:

• Phase I: A simulated annealing-based global optimization algorithm runs until the cost function achieves its optimal value or stops after a given number of iterations.

• Phase II: The aim is to Ąne-tune the optimized free parameters using an approach of learning-based control based on the extremum-seeking algorithm [START_REF] Benosman | Learning-based adaptive control : an extremum seeking approach -theory and applications[END_REF].

This auto-tuning process was validated in terms of the free-parameters convergence towards optimal values, which corresponds to a maximization of the cost function according to its design. It is important to emphasize that no DPD or CFR process was introduced in [START_REF] Ma | ŤMachine-Learning Based Digital Doherty Power AmpliĄer[END_REF].

On the other hand, an adaptive signal separation is proposed in [START_REF] Peng | ŤAdaptive Signal Separation for Dual-Input Doherty Power AmpliĄer[END_REF]. The splitting function is constructed using a dictionary (LUT) derived from the static measured results of the dual-input Doherty PA. The results in [START_REF] Peng | ŤAdaptive Signal Separation for Dual-Input Doherty Power AmpliĄer[END_REF] once again conĄrmed the interest in optimizing the separation of the input signals to be transmitted to the dual-input Doherty PA, which can improve efficiency.

In another work reported in [START_REF] Guillena | ŤPrediction of the Optimal Phase Shift between Control Signals in Dual-Input Power AmpliĄers[END_REF], the authors only focused on optimizing the phase shift between the two RF inputs using an exhaustive search for dual-input load modulated balanced PA. The phase shift is determined according to NMSE and ACPR by sweeping the phase over a determined interval [0°350°] with a resolution of 10°, which makes it a simple solution to be implemented but with difficulty in knowing whether the phase shift is optimal or not, since the PA performances are very sensitive to the phase shift between its inputs. Besides, sweeping the phase over a large interval can be critical, especially at intervals where there will be no output power, leading to heat dissipation in the device, which can damage it.

Proposed Auto-Tuning Approach

General Optimization Algorithms

In this work, the proposed auto-tuning approach to optimize the free-parameters is based on an efficient hybrid heuristic search control (HHSC) based on two types of modelfree optimization methods: simulated annealing (SA) as a global optimization search and extremum-seeking control (ESC) as an adaptive control to Ąne-tune the optimized results.

In this study, the choice of SA and ESC is in line with the state-of-the-art elaborated on the dual-input Doherty PA manufactured by MERL and mounted on their testbench. The Ąrst work related to this axis of research is reported in [START_REF] Niu | ŤStochastically approximated multiobjective optimization of dual input digital Doherty Power AmpliĄer[END_REF] and [START_REF] Ma | ŤMachine-Learning Based Digital Doherty Power AmpliĄer[END_REF], against which the main contribution of our study introduces DPD linearization and CFR technique by including the clipping threshold µ in the optimization process as a free-parameter.

The vector of free parameters to be optimized is denoted by Θ. The cost function corresponding to Θ is denoted by J(Θ) or J for simplicity.

Simulated Annealing

One of the best-known heuristic search methods for addressing the complex black-box global optimization problems is the SA algorithm proposed in [START_REF] Kirkpatrick | Optimization by Simulated Annealing[END_REF].

Physical annealing in the metallurgy domain inspires the principle of the SA algorithm. Physical annealing is the process of heating a material until it reaches an annealing temperature. Then it will be cooled down slowly to increase the size of its crystals and reduce their defects. When the material is hot, the molecular structure is weaker and is more likely to change. When the material cools down, the molecular structure is more rigid and is less responsive to change.

Following the analogy with metallurgy, the slow cooling in simulated annealing depends on the slight decrease in the probability of accepting a worse solution as the solution space is explored. The algorithm should perform an extensive search to Ąnd the global optimum solution, so accepting worse solutions is fundamental.

SA algorithm is an almost straightforward stochastic search based on the Metropolis Monte Carlo method [START_REF] Metropolis | ŤEquation of State Calculations by Fast Computing MachinesŤ[END_REF], the concept of which is to accepts not only the solutions that improve J, but also some solutions that worsen it with a probability p known as the Metropolis criterion and deĄned as:

p(∆E) = e -∆E k bolt T (5.15)
where ∆E is the change in cost function, k bolt is BoltzmannŠs constant, and T is the control parameter analogous to the temperature of the annealing process.

During the search, the temperature is gradually decreased until reaching zero value in the perfect case.

SA algorithm is massively used in real-life applications to solve a global optimization problem, mainly when the problem model is not explicitly given and cannot be evaluated by computer simulation. Furthermore, the deĄnition of the neighborhood is an essential aspect of the SA algorithm, particularly when a large neighborhood of SA leads to a greater probability of arriving at a global optimum.

In case of a global optimization problem, the standard procedure of SA is given by the following steps:

• Generate initial solution by choosing a random vector Θ 0 composed of the freeparameters to be optimized, and evaluate J(Θ 0 ).

• Initialize the temperature T . If T is too high, the computation time is important.

If T is too low, there is a risk of not sufficiently exploring the parameter space and remaining in a local minimum.

• Select a new solution in the neighborhood space Ψ of the current solution Θ i where a solution Θ ′ i is accepted as a new solution depending on T . J(Θ i ) and J(Θ ′ i ) are evaluated and compared. If J(Θ ′ i ) is better (maximized or minimized depends on the design of J) than J(Θ i ), then Θ ′ i is selected as a best solution at the current iteration. Otherwise, Θ ′ i can also be accepted depending on the new probability in (5.15). The cost function J will be deĄned in the next section.

• Decrease the temperature T by a temperature reduction rate C throughout the process search by T = C × T .

• Repeat the process while T > T f .

The free-parameters optimized using SA are denoted by Θ opt,SA . The cost function corresponding to Θ opt,SA is denoted by J opt,SA .

The algorithm of SA is described in Algorithm [START_REF] Jiang | An overview : Peak-to-average power ratio reduction techniques for ofdm signals[END_REF].

One of the most popular ESC methods is the perturbation-based ESC, which is proposed in [START_REF] Ariyur | Real time optimization by extremum seeking control[END_REF]. This method efficiently probes the system using a sinusoidal perturbation (or a dither signal) to obtain a gradient estimate.

The concept of perturbation-based ESC is fundamentally depicted by the block diagram in Figure 5.10, where it consists of a target system, the output of which is the cost function J, a perturbation signal asin(ωt), a gain K, and an integrator. 5.10, the system is perturbed by a dither signal asin(ωt), which results in an output of the cost function J(Θ). This output is then multiplied by the dither signal and denoted by ξ. After multiplying ξ by a gain K and passed through an integrator 1 s , the resulting signal Θ estimates the gradient of the cost function with respect to Θ.

The loop of ESC can be written as the following dynamical system

ξ = J(Θ) × a sin(ωt) d Θ dt = K × ξ Θ = Θ + a sin(ωt) (5.16)
The integratorŠs role is to optimally adjust the operating point towards the optimum according to the gradient information of the output. The choice of the gain K may inĆuence the whole control system. Besides, it decides the controller bandwidth, which indicates the speed of the controller. The bandwidth of the controller should generally be less than that of the gradient estimator. Otherwise, the controller cannot get the updated gradient information, which makes the system unstable.

In our approach, the perturbation-based ESC is used as an off-line process, which is placed downstream of SA to Ąne-tune Θ opt,SA , The input of ESC is Θ opt,SA . The optimized conĄguration by ESC is denoted by Θ opt,HHSC . The cost function corresponding to Θ opt,HHSC is denoted by J opt,HHSC .

In this study, the values of the ESC parameters ω, K, and a are initialized by the study reported in [START_REF] Ma | ŤMachine-Learning Based Digital Doherty Power AmpliĄer[END_REF].

Algorithm of HHSC

HHSC is the core of the proposed auto-tuning approach to optimize the free parameters of the dual-input Doherty PA. The principle of HHSC consists of combining SA and ESC in cascade, as illustrated in Figure 5.11.

The interest in using two algorithms of different types reinforces the convergence of the auto-tuning approach towards the optimal solution.

Starting from an initial solution Θ 0 , SA optimizes the free parameters according to a designed cost function J. The optimized solution Θ opt,SA returned by SA will be the initial solution for the ESC process.

The algorithm of HHSC is described in Algorithm (13).

T 0 , T f , iter = 0, K, a, ω Run Algorithm of SA described in (12) Θ 0,ESC = Θ opt,SA Θ = Θ 0,ESC J old = J opt,SA while 1 do ξ = J(Θ) × a sin(ωt) Θ = K × ξ Θ = Θ + a sin(ωt) Evaluate J(Θ) if J(Θ) > J old then J old = J(Θ) else
Θ opt,HHSC = Θ J opt,HHSC = J(Θ opt,HHSC ) end while loop end end Return Θ opt,HHSC

Principle of the Proposed Auto-Tuning Approach

The principle of the auto-tuning approach proposed in this work to optimize and linearize the dual-input Doherty PA is summarized in the Ćowchart in Figure 5.12.

The process of the proposed approach is mainly composed of Ąve sub-processes:

• Design of the cost function to control the convergence of HHSC.

• Joint optimization of CFR, digital splitter, transceiver, and DUT by HHSC using the cost function designed.

• DPD linearization based on ILA.

• Update the cost function designed in the Ąrst sub-process.

• Optimal pruning of free parameters in HHSC.

Each sub-process of the Ćowchart in Figure 5.12 will be detailed in the following sections.

Design of Cost Function

The cost function is an important aspect that deĄnes the optimality criterion. In this study, the cost function will be designed to ensure a good trade-off between linearity and efficiency. The linearity requirement is represented in terms of two Ągures of merit (FOMs): EVM and ACPR, while the efficiency and output power level requirement are represented by the power added efficiency PAE and the output power P out .

Note that in this study, P out represents the peak power of the signal at the output of DUT y(n), which is given by: P out = P avg + PAPR(y) (5.17) where P avg is the average power of the signal, given by the power sensor from testbench. These FOMs are accordingly weighted to their importance in the cost function. Additionally, some FOM thresholds can also be deĄned to penalize further not meeting the targeted speciĄcations.

The optimization problem is a multi-objective optimization (MOO) since more than one objective function are optimized simultaneously.

The MOO applications have been reported in the literature for many Ąelds such as engineering, economics, Ąnance, etc. The various types of MOO problems are tracked by several methods as well [START_REF] Gunantara | ŤA review of multi-objective optimization: Methods and its appli-cationsŤ[END_REF]. The MOO consists of Ąnding the optimal solution of more than one desired target. Making decisions in MOO is a signiĄcant problem that practically leads to a proper trade-off on some contradictory issues.

The MOO problem can be written as follows:

min or max f 1 (x), f 2 (x), . . . , f n (x) subject to x ∈ U (5.18)
where x is the optimal solution, n is the number of objective functions, U is variable space which embodies all different solution and f n (x) is the n th objective function. Generally, solving the problems of MOO can be classiĄed into two methods: the Pareto method and the scalarization method.

The Pareto method is based on separating the optimal solution of each objective function during the optimization process. Dominance is an essential concept of the Pareto method based on differentiating the dominated and non-dominated solutions. For the MOO problem, a solution is reached when one objective function cannot increase without impacting the others. This condition is called Pareto optimality or Pareto efficiency.

The scalarization method allows converting MOO problem into single-objective optimization (SOO) problem such that optimal solutions to SOO problem are Pareto optimal solutions to MOO problem [START_REF] Hwang | Multiple objective decision making, methods and applications : a state-of-the-art survey[END_REF]. In [START_REF] Murata | SpeciĄcation of Genetic Search Directions in Cellular Multi-objective Genetic Algorithms[END_REF], the scalarization method incorporates multiobjective functions into scalar Ątness function as in the following equation:

F (x) = w 1 f 1 (x) + w 2 f 2 (x) + • • • + w n f n (x) (5.19)
where w 1 , w 2 , . . . , w n are the parameters of the scalarization called weighting coefficients of the cost function F . A weighting coefficient w i is the weight of its objective function f i , which refers to the relative importance of that objective function to the cost function F . This method is known as the weighted sum method. This methodŠs challenge is to correctly set each objective functionŠs weight to obtain the Pareto optimal solution in the desired region in the objective space U.

Another famous method proposed in [START_REF]ŤOn a Bicriterion Formulation of the Problems of Integrated System IdentiĄcation and System Optimization[END_REF] is called the ϵ-constraint method, which is based on keeping just one of the objectives and restricting the others within user-speciĄc values. It can be deĄned as:

min or max f j (x) subject to x ∈ U f i (x) ≤ ϵ i for i ∈ ¶1, . . . , k♢ and i ̸ = j (5.20)
In this work, we propose to design the cost function J for HHSC based on the weighted sum method but with constraints. It is deĄned as:

J = w 1 J EVM + w 2 J ACPR + w 3 J PAE + w 4 J Pout (5.21) with                                J EVM = w 1,1 EVMt EVM C + w 1,2 EVMt EVM DD + w 1,3 EVMt EVM CDD J ACPR = ACPR ACPRt J PAE = PAE PAEt J Pout = Pout Pout,t (5.22)
where EVM t , ACPR t , PAE t , and P out,t are EVM target, ACPR target, output power target, and efficiency target, respectively, that the user attempts to reach.

Since CFR as a nonlinear process deteriorates EVM dramatically, we propose to use in J EV M , EVM of CFR denoted by EVM C , EVM of DPD and DUT denoted by EVM DD , and EVM of the whole system including CFR, DPD, and DUT denoted by EVM CDD .

There is a way to present the three EVMs in one feature, denoted by EVM ms by using the mean square of EVM CDD , EVM C , and EVM DD , which can be deĄned as:

EVM ms = EVM 2 C + EVM 2 DD + EVM 2 CDD 3 (5.23)
However, in this work, EVM CDD , EVM C , and EVM DD will be presented separately. The constraints of the cost function designed in (5.21) are deĄned as:

       w 1 = w 1,1 + w 1,2 + w 1,3 4
i=1 w i = 1 max J EVM = max J ACPR = max J PAE = max J Pout = 1 (5.24)
The computation of ACPR is deĄned in terms of ACPR L1 and ACPR U 1 as:

ACPR = 10 log 10   10 ACPR L1 10 + 10 ACPR U 1 10 2   (5.25)
In (5.21), each objective function corresponding to each FOM is normalized by its target value, which is deĄned as a user speciĄcation.

The cost function design is carried out such that HHSC attempts to maximize J to 1, indicating that the userŠs speciĄcations are met. When J=1, all FOMs have reached their targets, which presents the perfect case, i.e.,

J = w 1 + w 2 + w 3 + w 4 = 1
In this way, we give the cost function a logical interpretation since the different FOMs do not have the same meaning or same unit. LS. However, the integration of the estimation process of the DPD coefficients into the optimization process involves nonlinear estimation techniques, which will considerably increase the complexity of the search.

On the other hand, one of the critical issues that arise when using DPD is the normalization gain selection. In [START_REF] Zhu | ŤOpen-loop digital predistorter for RF power ampliĄers using dynamic deviation reduction-based Volterra series[END_REF], a normalization gain G is proposed to eases the DPD identiĄcation, which is deĄned as:

G = max(♣y(n)♣) max(♣x(n)♣) (5.27)
where G is chosen as the quotient between the maximum envelope values of the PA input x(n) and output y(n).

In this work, we use the deĄnition of G in (5.27) in ILA since this solution can simplify the DPD implementation because the original input and the predistorted signal can be normalized by the same scaling factor, which facilitates the power control. Figure 5.13 shows the integration of DPD using ILA to linearize the dual-input Doherty PA. Figure 5.13: Integration of DPD using ILA to the entire system

Update of weighting Coefficient

The DPD may apply a back-off to the operating point of dual-input Doherty PA, which dramatically reduces the efficiency requirement. Therefore, the cost function J with DPD, which is denoted by J HHSC+DPD , with updated FOMs and the same initial weighting coefficients will probably have deteriorated, and some free-parameters in Θ opt,HHSC will no longer be optimal.

To maximize the cost function again, we propose to rely on the design of the cost function J by adapting its weighting coefficients w = [w 1 w 2 w 3 w 4 ] according to the change effected by the DPD.

An intuitive approach can be used by attributing an equal weight to each FOM according to the following equation:

w i = 1 n (5. 28
)
where i = 1, 2, . . . , n and n is the number of objective functions that present FOMs. In our context, the cost function is designed by combining efficiency and linearity, as is shown in (5.21). On the other hand, DPD with an efficient optimal DPD model can signiĄcantly improve linearity, which leads to reducing the weight of linearity FOMs in (5.21). Hence, we propose to design an adaptive cost function, in which the weight coefficients w are adaptive according to the improvement of linearity and efficiency over each block from the Ćowchart in 5.12. The weighting coefficients w are updated with respect to how much DPD improves linearity FOMs, e.g., EVM and ACPR, compared to before applying it.

Starting from initial weighting coefficients w, we apply HHSC to optimize the freeparameters Θ with J opt,HHSC , then DPD to linearize the DUT under the optimized freeparameters.

It is required to re-compute the cost function J HHSC+DPD once the DPD is performed, and compare it to J opt,HHSC . if J HHSC+DPD < J opt,HHSC , we propose to update the weighting coefficients w 1,2 , w 1,3 , w 2 , w 3 , and w 4 . w 1,1 is not concerned since it depends on the CFR operation.

In this process, we are only focusing on w 1,2 , w 1,3 , and w 2 that refers to linearity FOMs improved by DPD. We calculate the ratio n i of the difference before and after DPD improvement for each linearity FOM. The ratio n i is expressed as:

n 1 = EVM HHSC,DD -EVM HHSC+DPD,DD EVM t n 2 = EVM HHSC,CDD -EVM HHSC+DPD,CDD EVM t n 3 = ACPR HHSC -ACPR HHSC+DPD ACPR t (5.29)
Next, we update the weighting coefficients of FOM linearity as:

w 1,2 = w 1,2 × (1 - 10 n 1 ) w 1,3 = w 1,3 × (1 - 10 n 2 ) w 1 = w 1,1 + w 1,2 + w 1,3 w 2 = w 2 × (1 -n 3 ) (5.30)
Once w 1,2 , w 1,3 , and w 2 are updated, we propose to assign an equal weight between w 3 and w 4 according to:

w 3 + w 4 = 1 -w 1,1 -w 1,2 -w 1,3 -w 2 w 4 = w 3 (5.31)
The algorithm for updating the weighting coefficients w is described in Algorithm [START_REF] Jiang | Block coding scheme for reducing papr in ofdm systems with large number of subcarriers[END_REF].

n 3 = ACPR HHSC -ACPR HHSC+DPD ACPRt w 1,2 = w 1,2 × (1 -10 n 1 ) w 1,3 = w 1,3 × (1 -10 n 2 ) w 1 = w 1,1 + w 1,2 + w 1,3 w 2 = w 2 × (1 -n 3 ) w 3 = 1-w 1 -w 2 2 w 4 = w 3 else Finish end

Optimal Pruning of free-parameters in HHSC

Once w is updated to the DPD contribution, the cost functionŠs design J is changed. According to the Ćowchart in 5.12, we run the HHSC again, but only on reduced free parameters in Θ. The HHSC will only be performed on one free parameter that is the most sensitive one in Θ and has the most signiĄcant impact on the behavior of J. The pruning process is an off-line procedure, as shown in the Ćowchart, which aims to reduce the complexity of the HHSC when the weighting coefficients are updated.

Pruning the free parameters in HHSC is optimally achieved using the HC algorithm. The motivation behind using the HC algorithm is that it is not a black-box optimization process. The neighborhood property in the HC algorithm makes it possible to follow the algorithmŠs evolution at each iteration.

Here, the cost function is used for the joint optimization of CFR and DUT in the second block from the Ćowchart. u(n) of the CFR block according to the variation of µ. PAPR of v(n) decreases with respect to µ. The EVM of v(n) degrades exponentially with increasing µ. As we target an EVM t around 3%, we deĄne the search interval [0 µ max ] = [0 1.2], for which when µ = 1.2, the degradation of the EVM by CFR is almost 3%. • Phase shift ϕ: The search range [-180°180°] is the standard interval to be set. However, based on some preliminary tests, it was shown that there is some critical search range to be avoided. Therefore, [-10°200°] is taken as a search range for the free parameter ϕ.

• Attenuation difference ψ: As ψ is categorized as a hardware free-parameter that refers to the physical power difference between the main and peaking ampliĄer, we set its search range to [ψ min ψ max ] = [-2 2] dB.

• Bias voltage V GS : We set the bias voltage search range for both V GSm and V GS,p to [V GS,min V GS,max ] = [-4 -1] V in order to provide a Ćexible variation between deep-class C condition that should enhance efficiency and a near-class B bias where linearity should be improved. The drain bias V DC is 28 V.

Before starting the auto-tuning approach, we propose to investigate the behavior of the free parameters over their search range. For this, we propose sweeping each of ϕ and α, which are taken as an example, over their search range, separately, while keeping the other free parameters on arbitrary values. This evaluation has been performed according to the linearity and efficiency FOMs used in the cost function in (5.21).

Sweeping ϕ

In this test, the phase shift ϕ is swept over its determined search range [-10°200°] with a resolution of 2°, while keeping the other free-parameters to µ = 0, α = 0.2, ψ = 0.1, V GS,m = -1.5V , V GS,p = -2.5V . Figure 5.15 presents the FOMs according to the variation of ϕ. As can be seen, FOMs exhibit important sensitivity to the variation of ϕ. All of them tend to have unimodal behavior, in which there is an optimal ϕ associated with each extremum of FOM. For the EVM extremum (here EVM DD =EVM CDD since µ = 0), the optimal phase is located nearly to 80°, while for the other FOMs extremum (PAE, P out , and ACPR), the optimal ϕ is located near to 140°. Here, ACPR is computed in terms of ACPR U 1 and ACPR L1 as is described in (5.25).

This example again proves the beneĄt of optimizing the free parameters since the efficiency and linearity could be worsened or improved, depending on how the free parameters are conĄgured. Figure 5.16 presents the FOMs versus the variation of α. As can be seen, the linearity FOMs exhibit a unimodal behavior where the EVM and ACPR extrema are associated with an optimal α, located near 0.25, which presents 44% of the input power to the main ampliĄer versus 56% to the peaking ampliĄer.

On the other hand, we can see that PAE and P out are decreasing with α. This could be interpreted so that the more input power is distributed at the peaking ampliĄer, the more the efficiency FOMs deteriorates. The cost function J opt,HHSC = 0.66. The evolution of free parameters over HHSC is illustrated in Figure 5.18, in which the free parameters have similar behavior to J according to the HHSC iterations. Table 5.4: FOMs according to the optimal free-parameters Θ opt,HHSC with P in =11.36 dBm It should be noted that HHSC shown in Figure 5.18 was performed on the linear region of the dual-input Doherty PA. Indeed, since SA generates random solutions, it was decided to reduce the operating point of DUT while HHSC is running in order to set up a security measure of the DUT.

EVM C [%] EVM DD [%] EVM CDD [%] ACPR U 1 [dB] ACPR L1 [dB] PAE [%] P out [dBm] J opt,HHSC 1
Once the free parameters are optimized, we raise the back-off by increasing the input power from 11.36 dBm to 21.16 dBm so that the system performs nearly at P out,t . The FOMs after increasing the input power are summarized in Table 5.5 Table 5.5: FOMs according to the optimal free-parameters Θ opt,HHSC with P in =21.16 dBm According to the results from Table 5.18, the efficiency FOMs are enhanced by sac-riĄcing the linearity FOMs. This conĄrms the choice of the initial weighting coefficients where w 3 and w 4 have more inĆuence than w 1 and w 2 .

EVM C [%] EVM DD [%] EVM CDD [%] ACPR U 1 [dB] ACPR L1 [dB] PAE [%] P out [dBm] J opt
By assigning the free-parameters to their optimal conĄguration Θ opt,HHSC in Table 5.3, the DUT, including the digital splitter and transceiver, can eventually be seen as a single-input single-output system where the input is x(n) and the output is denoted y(n).

The AM-AM and AM-PM characteristics of the dual-input Doherty PA are shown in Figure 5.19a, where we can see a saturation at high power, which leads to strong nonlinearities. Besides, the memory effects are exhibited as well. The spectra of the input-output signals of the dual-input Doherty PA is illustrated in Figure 5.19b

DPD Linearization

The DPD is carried out in two steps:

• Determination of the optimal DVR model.

• Convergence of linearity FOMs using ILA.

Determination of Optimal DVR Model

According to the HC algorithm presented in Chapter 3, the structure of the DVR model is optimally sized.

In this study, the cost function, denoted by Y , is deĄned as a search criterion to ensure a good trade-off between three features: modeling accuracy presented by NMSE, denoted by N , model complexity presented by the number of coefficients C, and computational complexity, which is given by the condition number of the regressor matrix Z H Z in (2.18), denoted by Cond.

The cost function is used to control the convergence of the HC algorithm to return an optimal DVR model with a considerable ability to be implemented in hardware. We deĄne the cost function as:

Y = 0.5N + 0.25C + 0.25 log 10 (Cond) (5.32)
As discussed in the previous chapter, the hardware implementation relies on the numerical properties of the DPD model, which in this case, is presented by C and Cond. The design of the cost function in (5.32) may deteriorate C, and N since the objective function of Cond is sized in such a way to override the inĆuence of C and N slightly.

However, since we aim to implement DPD on FPGA, which is a Zynq in our case, the deterioration of N can be overcome by the DPD convergence towards the solution that presents a better trade-off between linearization performance, complexity, and numerical properties. Replacing the parameters of the optimal structure in (3.3), the optimal DVR model used in DPD is expressed by:

x(n) = 1 i=0 a i u(n -i) + 4 k=1 3 i=0 c ki,1 ♣♣u(n -i)♣ -β k ♣e jθ(n-i) + 4 k=1 3 i=0 c ki,2 ♣♣u(n -i)♣ -β k ♣e jθ(n-i) • ♣u(n)♣ + 4 k=1 3 i=0 c ki,3 ♣♣u(n -i)♣ -β k ♣e jθ(n-i) • ♣u(n)♣ 4

DPD using ILA

ILA needs multiple iterations to converge. In this study, we set the number of DPD iterations to 10.

As NMSE being considered as a strong indicator of the ILA convergence, Figure 5.22 presents the evolution of the NMSE according to the DPD iterations. As can be seen, the convergence of NMSE is rapid, in which NMSE is improved signiĄcantly from the Ąrst DPD iteration. We can see that EVM CDD (EVM DD as well) is improved from value 8% to 2.97%. This can be conĄrmed in Figure 5.24, where the red dots present the IQ constellation of y(n), and the blue dots are the reference IQ constellation of v(n). Table 5.6 summarizes the FOMs without and with DPD. The cost function is computed using FOMs without and with DPD. From Table 5.6, the cost function is decreased after the DPD application. As DPD partially modiĄes the system conditions, the conĄguration Θ opt,HHSC may not always be the optimal solution, which stimulus the motivation to update the weighting coefficients and re-launch a new HHSC.

Update Weighting Coefficients

Following the Ćowchart in Figure 5.12, the weighting coefficients w should be updated to take into account the linearity improvement by DPD to the DUT in which the efficiency is inĆuenced.

According to Algorithm ( 14), the weighting coefficients w are updated as follows: In order to investigate the behavior of the HC algorithm, Figure 5.31 presents which free-parameter has been stored in the dictionary through the HC iterations. As can be seen, the free parameter ϕ has often been repeated, which means that the optimization behavior of HHSC is seen as most sensitive to ϕ.

            
On the other hand, the free-parameter µ has been stored three times, where the conĄguration [µ × (1 ± µ 10 ) α ϕ ψ V GS,m V GS,p ] has the best cost function J at the 6 th , 13 th , and 17 th HC iteration.

After the 21 th HC iteration, no conĄguration was found with a better J, which triggers the HC algorithm to stop.

HHSC with Pruned Free-Parameters

As discussed before, the goal of the HC algorithm here is to prune optimally the free parameters of HHSC, which aims to Ąnd the most sensitive free parameter to CFR and DUT with DPD.

Hence, Θ ′ = [ϕ] is the most sensitive free parameter, which will be used in HHSC according to the cost function J designed with the updated weighting coefficients w.

Starting from the initial solution Θ ′ 0 = [164 • ], which presents the optimal ϕ from the previous HHSC, Figure 5.32 present the evolution of the cost function J and the free-parameter ϕ according to the HHSC iterations. According to Figure 5.32, the free-parameter ϕ has been re-optimized, where its optimal value becomes Θ ′ opt = 175.9 • that corresponds to the optimal cost function J opt,upd = 0.81. By comparing J opt,upd and J HHSC+DPD , the DPD coefficients are required to be updated using ILA since the optimal conĄguration of DUT is changed.

Table 5.7 summarizes the results, where FOMs are presented before and after applying DPD. At this level, the application of DPD consists only of updating the existing DPD coefficients in the predistorter since the optimal pruning of free parameters HHSC was performed with DPD.

With the new optimal conĄguration, DPD has improved the linearity FOMs. On the other hand, the efficiency FOMs are improved. This can be conĄrmed in Table 5.7, where the PAE is improved by almost 5.5%. By comparing the cost function, it can be seen Table 5.7: FOMs according to the optimal free-parameters after the optimal pruning of free-parameters in HHSC that J HHSC+DPD = 0.85 becomes better than J opt,HHSC = 0.82, which leads to Ąnishing the auto-tuning approach.

Conclusion

In this chapter, we proposed an auto-tuning approach to exploit at best dual-input Doherty PA to maximize power efficiency while being compliant with the linearity speciĄca-tions.

The proposed auto-tuning approach relies on conducting a global optimization combined with a control process to Ąnd the optimal conĄguration of a set of the crucial circuit and system-level parameters that are appropriately merged with the DPD linearization and the CFR technique.

This proposed approach has been performed according to an adaptive designed cost function, representing the trade-off between efficiency and linearity.

In order to sharpen the optimal conĄguration, we propose a new approach based on the HC algorithm to prune the free parameters optimally when DPD linearization is applied.

The proposed approach has been validated through experimental results, in which we use a 20 MHz LTE signal scenario.

The proposed approach to optimizing the dual-input Doherty PA has been well validated by presenting a good trade-off between linearity, computational complexity, and efficiency. Besides, the DPD model used, which is optimally sized, has very good numerical properties, making it a perfect candidate for its implementation on hardware such as FPGA.

Conclusion and Perspectives

Contributions

In this dissertation, we have focused on digital predistortion, which is a linearization technique of power ampliĄers to eliminate the distortions due to the nonlinearities and memory effects. This dissertation mainly focuses on three aspects, which have made our contribution: study and comparison of DPD models, design of an optimal DPD model with its implementation strategy, and optimization and linearization of dual-input Doherty PA with efficiency enhancement.

1. A comparative study is made between DPD models, which are categorized in this study into two families: global models derived from the Volterra series and models with segmentation. The analysis comparison has been made according to three aspects: modeling accuracy presented by NMSE and ACPR, model complexity presented by the number of model coefficients, and numerical properties presented by the dynamic range of coefficients and the condition number of regressor matrix for the identiĄcation process of predistorter. The validation of this study has been performed using measurements from two different PAs (LDMOS 50 W PA and Doherty three-way 500 W PA) using different scenarios in terms of signal bandwidth and output power.

2. For models with segmentation, particularly the DVR model, two major studies were developed: optimizing the thresholds and determining the optimal structure. For the thresholds optimization, a new approach is proposed to set the thresholds optimally using an iterative approach based on the decomposition of the global optimization problem into a set of unimodal sub-problems so that a unidirectional minimization, such as the golden section search, can be used to optimize the positions of thresholds. For the DVR model sizing, the hill-climbing algorithm is proposed to search for the optimal model structure. The convergence of hill-climbing is conĄrmed in terms of modeling accuracy of the optimal solution, algorithm complexity, and execution time. Two major contributions were highlighted: search criterion representing a trade-off between modeling accuracy and model complexity. A neighborhood deĄnition proposed to control the algorithm toward the optimal solution.

3. A new proposed approach to design an optimal DVR model is based on the combination of threshold optimization and model sizing. The proposed approach is achieved through three different algorithm versions. The Ąrst algorithm version relies only on optimizing the thresholds in model structures located on the envelope of model sizing. In contrast, the second version consists in integrating the thresholds optimization process into the model sizing but assuming that the memory depth and model terms are not sensitive to the optimal thresholds. So, the thresholds are optimized only if the number of segments of a model structure changes. The third version is an update of the second version, in which the terms of the model are viewed as a sensitive parameter leading to optimizing the thresholds when the number of segments and the model terms are changed. On the other hand, in this framework, we have investigated the impact of nonlinearities order in the DVR model by which we highlight the interest of using models with segmentation with reduced orders of nonlinearities, unlike model from Volterra series, such as GMP, where the nonlinearities order is highly required to achieve the same performances.

4. An auto-tuning approach is proposed to Ąnd the optimal conĄguration of a set of essential circuits and system-level parameters of dual-input Doherty PA. This approach is achieved by combining two optimization methods: simulated annealing as a global optimization algorithm and extremum-seeking as an adaptive control. The proposed approachŠs convergence is constrained by a proposed adaptive cost function designed to purposely represent a trade-off between linearity and efficiency. CFR technique and DPD linearization are integrated optimally to the proposed architecture of dual-input Doherty PA to meet the user speciĄcations by enhancing efficiency while maintaining a conforming linearity level. An optimal pruning search based on the hill-climbing algorithm is proposed to reinforce and control the approachŠs convergence toward the speciĄed user targets.

Perspectives

To extend this dissertation, some works could be developed:

1. Regarding the comparison study of DPD models, the DVR model is the only model with segmentation used in this study. It will be interesting to investigate other models, such as B-spline models and vector switched models, in the analysis comparison.

2. Generally, the PA characteristics and system performance depend on many factors such as the average power, the probability density function of the signal, the signal bandwidth, the circuit temperature, etc. The approach to design the optimal DVR model needs to be redesigned to Ąt the PA characteristics if changed, leading to an optimal reconĄgurable DVR model. For that, it will be necessary to add the PA characteristics features (average power for example) into the design process of the DVR model. Besides, the runtime of the design algorithm should be decreased so that the real-time implementation can be viable.

3. Apply a stochastic approach based on simulated annealing algorithm to design the DVR model.

4.

Implement the optimal DVR model on digital hardware, e.g., FPGA.

5. Reduce the complexity of the proposed auto-tuning approach to optimize the dualinput Doherty PA by combining CFR, DPD, and digital splitter into one block, in which a nonlinear LS method, such as Newton-Raphson algorithm, will be involved to estimate the DPD coefficients, as well as the optimal conĄguration.

6. Since the optimization of dual-input Doherty PA is conducted under speciĄed conditions, especially in terms of center frequency, output power level, signal bandwidth, it will be exciting to perform the auto-tuning approach according to a vast data set of center frequencies, output power levels, and signal bandwidths, in order to apply some machine-learning models to Ąt this data and Ąnd an optimal global conĄguration.

Further research works are currently in progress for submission, which are listed along with the abstract:

• Chouaib Kantana, Olivier Venard and Genevieve Baudoin, ŞA System Approach to Design an Optimal Digital Predistortion for Power AmpliĄer LinearizationŤ.

-Abstract: The segmentation approach has proven to be an efficient approach to replace the Volterra series for behavioral modeling of power amplifier (PA) or serve as a digital predistortion (DPD) model. The Decomposed Vector Rotation (DVR) is one of the most common model based on the segmentation approach that represents nonlinear dynamical systems with memory effects. Crucial, the study of the DVR model involves two principal aspects: the optimization of the thresholds that bound the segments and the determination of the optimal model structure, i.e., determining the optimal number of segments, the optimal memory depth, and the optimal set of model terms. In this paper, we propose an efficient system approach to design an optimal DVR model by jointly optimizing the model structure and threshold segmentation. Convergence towards the optimal model is ensured by a cost function that provides the best trade-off between modeling accuracy and model complexity. Experimental results on different amplifiers confirmed the system approach's effectiveness in designing an optimal DVR model with good linearization performance, reduced number of coefficients, and remarkable numerical properties, making it friendly to hardware implementation, for which an implementation strategy is investigated.

• Chouaib Kantana, Rui Ma, Mouhacine Benosman, Olivier Venard and Genevieve Baudoin, ŞOn the Low-Complex Optimal model of Digital Predistortion: A Stochastic ApproachŤ.

-Abstract: This paper investigates the application of a stochastic approach based on simulated annealing (SA) algorithm to design an optimal decomposed vector rotation (DVR) model for digital predistortion (DPD). The design of DVR is based on the optimization of thresholds β that bounds the segments and the model sizing that find the optimal model structure. The combination of threshold optimization and model sizing is viewed as a nonlinear regression problem requiring nonlinear estimation techniques. The SA algorithm is converged to the optimal solution according to a cost function that presents a trade-off between linearization performance and model complexity. This study reveals that the SA algorithm can be used as a nonlinear estimation technique. Experimental results using measurements from dual-input Doherty power amplifier (DIDPA) are provided, demonstrating the efficiency of SA in designing the optimal DVR model. A comparison is made between the results obtained by the hill-climbing heuristic to size the model and the golden search (GS) technique to optimize β.

• Chouaib Kantana, Rui Ma, Mouhacine Benosmane, ŞA System Approach for the Linearization and Efficiency Enhancement of Dual-Input Doherty Power AmpliĄerŤ.

-Abstract: In this paper, we propose an efficient system approach to improve the power efficiency of dual-input Doherty power amplifier (DIDPA) by maintaining a reliable level of linearity following the communication standards. The system approach handles DIDPA according to a specific workflow. Firstly, an auto-tuning process based on a hybrid heuristic search control (HHSC) is applied to optimally define DIDPA configuration by optimizing its free-parameters, including peak-toaverage power (PAPR) reduction threshold. The HHSC is driven by a cost function designed to moderate the inherent trade-off between linearity and power efficiency. The digital predistortion (DPD) is then integrated to linearize DIDPA using an optimal reduced-complexity model based on the segmentation approach. The cost function coefficients are updated optimally based on the linearity improvement by DPD. An optimal pruning process of the free-parameters, based on hill-climbing (HC) heuristics, is proposed to reduce the HHSC complexity in order to refine the optimal DIDPA configuration with the updated cost function. The system approach has been approved by experimental results using an LTE 20-MHz signal with a PAPR of 8-dB PAPR.

In the first step where HHSC is applied, DIDPA exhibited a drain efficiency of 61%. DPD linearization improved linearity using a low-complex model with only 30 coefficients, which exhibited an error vector magnitude (EVM) of 2.5% and an adjacent channel power ratio (ACPR) of -50-dB at an averaged output power of 34-dBm.

By updating the cost function coefficients and pruning the free-parameters, DIDPA exhibited an EVM of 3%, an ACPR of -50-dB, and a drain efficiency of 47% at an average output power of 39-dBm.

Résumé détaillé de la thèse en français

Approche système d'optimisation d'un prédistorteur numérique avec une complexité réduite pour la linéarisation des amplificateurs de puissance RF

Au cours des derni' eres décennies, les syst' emes de communication sans Ąl ont connu une évolution spectaculaire dans la mani' ere de fournir des services de transmissions ' a des débits de plus en plus élevés pour des applications telles que la communication mobile, les réseaux Wi-Fi, la diffusion multimédia, etc. Ces progr' es ont conduit ' a une prolifération de nouveaux services radio. Or, les exigences croissantes des utilisateurs en termes de service, de qualité, et de connectivité mettent massivement ' a lŠépreuve les ressources énergétiques et fréquentielles. Dans les syst' emes de radiofréquence (en anglais : Radio Frequency -RF), lŠampliĄcateur de puissance (en anglais : Power AmpliĄer -PA) est un élément crucial de la chaîne dont le rôle est dŠampliĄer le signal transmis aĄn dŠélever le niveau de puissance de la transmission. Le PA demeure lŠun des composants les plus importants en termes de coût, dŠautonomie, et de rendement énergétique. Le rendement énergétique maximal du PA est atteint lorsque le point de fonctionnement est proche de la zone de saturation, malheureusement dans cette zone, les non-linéarités de lŠampliĄcateur distordent le signal transmis. Ces distorsions peuvent devenir particuli' erement fortes pour les signaux ' a large bande. Sur le plan spectral, des formats de transmission tels que lŠacc' es multiple par répartition en code (en anglais : Code Division Multiple Access -CDMA) et le multiplexage par répartition orthogonale de la fréquence (en anglais : Orthogonal Frequency-Division Multiplexing -OFDM) sont de plus en plus utilisés pour acheminer des Ćux dŠinformations élevés sur un espace fréquentiel minimal, dont lŠobjectif est de surmonter la limitation spectrale des ressources et de faire preuve dŠune efficacité spectrale élevée. Ces formats de transmission gén' erent une enveloppe de signal transmis non-constante, ce qui implique un rapport puissance crête sur la puissance moyenne (en anglais : Peak-to-Average Power Ratio -PAPR) élevé. Par conséquent, ces signaux présentent une sensibilité aux nonlinéarités du canal de transmission, et en particulier ' a celle du PA.

LŠétude de PA repose enĄn sur lŠassurance dŠun véritable compromis entre linéarité et efficacité énergétique. LŠefficacité énergétique représente une caractéristique fondamentale quantiĄée principalement par deux déĄnitions : le rendement de drain η DC et le rendement en puissance ajoutée η P AE (en anglais : Power Added Efficiency -PAE). Le rendement η DC est déĄni par le rapport entre la puissance de sortie et la puissance continue consommée. Le rendement η P AE est le rapport entre la puissance RF fournie en sortie du PA diminuée de la puissance RF ' a son entrée et la puissance consommée sur lŠalimentation. Le rendement énergétique est un param' etre crucial qui dépend entre autres du choix du point de polarisation des transistors. Le PA doit être optimisé lors de la phase de conception aĄn de maximiser son rendement énergétique.

Le comportement non-linéaire du PA est généralement analysé en lŠexcitant par des signaux de natures diverses (signal mono-porteuse, signal ' a deux tons, signal modulé), et déterminant le gain, la puissance de saturation, le point de compression ' a 1 dB et les distorsions dŠintermodulation. Dans le cas o' u le PA serait excité par un signal modulé, la non-linéarité est généralement quantiĄée par 2 param' etres :

EV M (%) = 100 N k=0 ♣s k,ideal -s k,reel ♣ 2 N k=0 ♣s k,ideal ♣ 2
Dans les standards de communication, lŠEVM et lŠACPR sont donnés de mani' ere complémentaire. LŠACPR évalue lŠeffet de la non-linéarité de lŠampliĄcateur sur les bandes adjacentes, tandis que lŠEVM quantiĄe les distorsions dans la bande utile. Sous certaines conditions initiales, le PA est considéré comme un syst' eme ' a fonctionnement statique, ce qui signiĄe que la sortie de PA ' a lŠinstant t ne dépend que de lŠentrée au même instant. En réalité, le PA présente souvent des phénom' enes dispersifs non-linéaires et dynamiques appelés effets de mémoire. Cela signiĄe que la sortie du PA ' a tout instant dépend non seulement de lŠentrée instantanée correspondante, mais aussi des entrées ' a dŠautres instants précédents, ce qui impacte ses caractéristiques fréquentielles.

La caractérisation du PA dans la Ągure 1 montre clairement le compromis entre linéarité et rendement énergétique. AĄn de répondre aux spéciĄcations de linéarité imposées par les normes de communication (3GPP par exemple), la puissance moyenne du signal dŠentrée doit être abaissée en appliquant un recul (en anglais : back-off) supérieur au PAPR du signal ' a ampliĄer, tel que montré dans la Ągure 1. Grâce au recul, le PA fonctionne dans sa zone linéaire. Mais dans cette zone le rendement est généralement tr' es faible. AĄn dŠajuster le point de fonctionnement qui répond au compromis linéaritérendement, le back-off doit être ajusté de mani' ere optimale. Les techniques de linéarisation et dŠamélioration du rendement ont pour but de faire fonctionner le PA pr' es de sa zone de saturation o' u le PAE est maximum, tout en assurant dŠexcellentes performances de linéarité. Quant ' a la linéarité, des techniques ont été développées dans le but de faire fonctionner le PA dans sa zone non-linéaire et de lŠassocier ' a un dispositif de compensation des nonlinéarités sans sacriĄer sa puissance de sortie. Plusieurs techniques de linéarisation ont été proposées telles que les techniques de contre-réaction ou de feedforward et la technique de la prédistorsion analogique ou numérique (en anglais : Digital Predistortion -DPD) qui est actuellement la plus répandue et constitue un axe de recherche fondamental de cette th' ese. Le principe du DPD consiste ' a introduire en amont du PA un module appelé prédistorteur qui applique une déformation (distorsion) au signal dŠentrée pour que le syst' eme issu de la cascade DPD + PA soit une ampliĄcation linéaire. Ce principe est illustré dans la Ągure 2. A travers la littérature, nous distinguons les mod' eles non-linéaires et ' a mémoire en deux catégories : les mod' eles dérivés de la série de Volterra et les mod' eles basés sur les réseaux de neurones. DŠautre part, les mod' eles de la DPD reposent généralement sur deux approches. LŠapproche globale qui consiste ' a utiliser des mod' eles avec un ordre de non-linéarité élevé et une profondeur de mémoire pour modéliser le comportement nonlinéaire du syst' eme. LŠapproche de segmentation qui consiste ' a diviser le signal dŠentrée en différentes régions ou segments et ' a appliquer un mod' ele dŠordre de non-linéarité tr' es faible sur chaque segment. La comparaison des mod' eles issus des deux approches constitue le premier objectif de cette th' ese qui consiste ' a analyser les performances de chaque mod' ele selon des crit' eres résultant du compromis entre les performances de linéarisation, la complexité et les propriétés numériques pour lŠimplémentation matérielle.

Le deuxi' eme objectif de cette th' ese est de concevoir un mod' ele optimal de DPD basé sur lŠapproche de segmentation. Dans cette étape, une analyse détaillée du mod' ele a été réalisée, aĄn dŠétudier ses caractéristiques de linéarisation. Deux aspects sont traités en relation avec lŠétude de mod' eles basés sur lŠapproche segmentation : lŠoptimisation des seuils qui séparent les segments et le dimensionnement optimal de sa structure. Le premier aspect consiste ' a positionner de mani' ere optimale les seuils qui séparent les segments. LŠavantage dŠoptimiser la segmentation permet dŠaméliorer les performances de linéarisation, contrairement ' a une segmentation uniforme o' u les seuils sont répartis uniformément sur la dynamique du signal dŠentrée. Le deuxi' eme aspect concerne le dimensionnement du mod' ele o' u un algorithme heuristique de type hill-climbing (HC) a été utilisé pour déterminer la structure optimale en fonction du nombre de segments, de la profondeur mémoire et des fonctions utilisées dans le mod' ele. Ces deux aspects ont été combinés dans une étude o' u une approche syst' eme a été proposée pour concevoir un mod' ele optimal global. Le mod' ele optimal est par la suite utilisé pour linéariser un PA dŠune architecture avancée de type Doherty ' a double entrée (en anglais : Dual-Input Doherty Power AmpliĄer -DIDPA), ce qui constitue notre troisi' eme objectif de la th' ese. Le DIDPA a dŠabord été optimisé par une proposition dŠapproche reconĄgurable dont le principe consiste ' a optimiser les param' etres du syst' eme selon une fonction de coût adaptative. Le CFR et la DPD sont insérés en amont du DIDPA aĄn de maximiser ses performances tout en conservant une linéarité respectant les standards de communication.

Cette th' ese sŠinscrivait dans le cadre du projet FUI22 APOGEES (AmPliĄcation re-cOnĄGurablE multimodes). LŠobjectif du projet APOGEES était de résoudre lŠensemble des probl' emes de reconĄgurabilité des syst' emes dŠampliĄcation pour répondre aux nombreux déĄs générés par les usages inédits du spectre et les évolutions de la société pour le traĄc aérien, les communications tactiques et lŠutilisation opportuniste du spectre pour la télévision.

Analyse et étude comparative des modèles de DPD

La fonction de prédistorsion est souvent représentée par des mod' eles mathématiques similaires aux mod' eles comportementaux des ampliĄcateurs de puissance. La plupart de ces mod' eles sont dérivés de la série de Volterra. A temps continu, la relation générale entre un signal dŠentrée x(t) et la sortie y(t) est donnée par :

y(t) = +∞ k=1 +∞ 0 • • • +∞ 0 h k (τ 1 , . . , τ k ) k j=1
x(tτ j )dτ j o' u k représente lŠordre de la non-linéarité et h k (τ 1 , . . , τ k ) représente le noyau de la série de Volterra ' a lŠordre k. A temps discret, la représentation de la série de Volterra doublement tronquée entre un signal dŠentrée x(n) et une sortie y(n) est donnée par :

y(n) = K k=1 M i 1 =0 ... M i k =0 h k (i 1 , ..., i k ) k j=1 x(n -i j )
o' u K représente lŠordre de non-linéarité et M est la profondeur mémoire. La série de Volterra est largement connue pour sa robustesse dans la modélisation de syst' emes dynamiques non-linéaires. Son inconvénient est le fait que les coefficients de la série augmentent de façon exponentielle avec ses param' etres. Cet inconvénient rend lŠutilisation de la série de Volterra trop limitée pour les applications de DPD en temps réel. Ceci motive ' a mettre en place des mod' eles basés sur la série de Volterra, mais ne retenant que quelques termes de la série. Ces mod' eles sont communément appelés mod' eles réduits ou dérivés de la série de Volterra. Les plus connus sont les polynômes ' a mémoire (en anglais : Memory Polynomial -MP), les polynômes ' a mémoire généralisés (en anglais : Generalized Memory Polynomial -GMP), les mod' eles dérivés de la série de Volterra dynamique, comme le DDR (en anglais : Dynamic Deviation Reduction), MDDRV (en anglais : ModiĄed Dynamic Deviation Reduction-based Volterra -MDDRV).

La modélisation peut être effectuée globalement par un mod' ele unique ou bien par morceau en sŠappuyant sur une segmentation de lŠamplitude en différentes régions ou segments. Les mod' eles basés sur lŠapproche segmentale sont capables de représenter de fortes non-linéarités et moins sensibles aux probl' emes numériques contrairement aux mod' eles globaux. En effet, les segments peuvent être modélisés par des mod' eles dŠordre de non-linéarité plus faible que pour le mod' ele global. Chua dans les années 70 a proposé un mod' ele par morceau appelé CPWL (en anglais : Canonical PieceWise Linear -CPWL) représentant les segments par un Ąltre linéaire et une valeur absolue. La relation générale dŠun mod' ele CPWL entre une entrée x(n) et une sortie y(n) en bande de base est donnée par :

y(n) = M i=0 a i x(n -i) + K k=1 c k,i M i=0 a k,i x(n -i) -β k o' u K est
le nombre de segments, β K est le vecteur des seuils qui délimitent les segments et M est la profondeur mémoire. Le mod' ele CPWL SŠapplique ' a des signaux réels. AĄn que le mod' ele CPWL soit utilisé pour des signaux complexes comme un prédistorteur dont les coefficients sont linéaires avec le mod' ele, une modiĄcation et une simpliĄcation ont été apportées par Anding Zhu, aboutissant ' a un mod' ele dérivé dŠune forme simpliĄée du CPWL, appelé DVR (en anglais : Decomposed Vector Rotation).

Notre étude comparative consiste ' a analyser et comparer les mod' eles DPD qui appartiennent ' a deux familles différentes : lŠapproche globale avec un mod' ele dérivé des séries de Volterra et lŠapproche par segmentation. Ces mod' eles sont présentés dans le tableau 1. Le nombre de coefficients en fonction des param' etres (tels que profondeur mémoire, ordre de non-linéarité) est également donné.

(n) = K k=1 M i=0 a ki x(n -i)♣x(n -i)♣ k-1 K: ordre de non-linéarité K × (M + 1) M : profondeur de mémoire GMP y(n) = Ka-1 k=0 La-1 l=0 a kl x(n -l)♣x(n -l)♣ k K a , K b , K c : ordre de non-linéarité K a × L a + K b k=1 L b -1 l=0 M b m=1 b klm x(n -l)♣x(n -l -m)♣ k L a , L c , L c : profondeur de mémoire +K b × L b × M b + Kc k=1 Lc-1 l=0 Mc m=1 c klm x(n -l)♣x(n -l + m)♣ k M b , M c : longueur de retard et dŠavance +K c × L c × M c DDR-1 y(n) = K-1 2 k=0 M i=0 b 2k+1,1 ♣x(n)♣ 2k x(n -i) K: ordre de non-linéarité K+1 2 + K × M + K-1 2 k=1 M i=1 b 2k+1,2 ♣x(n)♣ 2(k-1) x 2 (n)x * (n -i) M : profondeur de mémoire DDR-2 y(n) = K-1 2 k=0 M i=0 b 2k+1,1 ♣x(n)♣ 2k x(n -i) K: ordre de non-linéarité K+1 2 × (M + 1) + K-1 2 k=1 M i=1 b 2k+1,2 ♣x(n)♣ 2(k-1) x 2 (n)x * (n -i) M : profondeur de mémoire +3M K-1 2 + K-1 2 k=1 M i=1 b 2k+1,3 ♣x(n)♣ 2(k-1) x(n)♣x(n -i)♣ 2 + K-1 2 k=1 M i=1 b 2k+1,4 ♣x(n)♣ 2(k-1) x * (n)x 2 (n -i) MDDRV-1 y(n) = K-1 2 k=0 M i=0 a 2k+1,1 ♣x(n)♣ 2k x(n -i) K: ordre de non-linéarité K+1 2 + ( 3K-1 2 ) × M + K-1 2 k=1 M i=1 a 2k+1,2 ♣x(n)♣ 2(k-1) x 2 (n)x * (n -i) M : profondeur de mémoire + K-1 2 k=1 M i=1 b 2k+1 ♣x(n -i)♣ 2k x(n -i) MDDRV-2 y(n) = K-1 2 k=0 M i=0 a 2k+1,1 ♣x(n)♣ 2k x(n -i) K: ordre de non-linéarité K+1 2 × (M + 1) + K-1 2 k=1 M i=1 a 2k+1,2 ♣x(n)♣ 2(k-1) x 2 (n)x * (n -i) M : profondeur de mémoire +2 × (K -1) × M + K-1 2 k=1 M i=1 a 2k+1,3 ♣x(n)♣ 2(k-1) x(n)♣x(n -i)♣ 2 + K-1 2 k=1 M i=1 a 2k+1,4 ♣x(n)♣ 2(k-1) x * (n)x 2 (n -i) + K-1 2 k=1 M i=1 b 2k+1 ♣x(n -i)♣ 2k x(n -i) Approche de segmentation DVR-2 y(n) = M i=0 a i x(n -i) K: nombre de segments (K + 1) × (M + 1) + K k=1 M i=0 c ki,1 ♣♣x(n -i)♣ -β k ♣e jθ(n-i) M : profondeur de mémoire DVR- 6
y ( n) = M i=0 a i x(n -i) K: nombre de segments (1 + M ) × (1 + 5 × K) + K k=1 M i=0 c ki,1 ♣♣x(n -i)♣ -β k ♣e jθ(n-i) M : profondeur de mémoire + K k=1 M i=0 c ki,21 ♣♣x(n -i)♣ -β k ♣e jθ(n-i) .♣x(n)♣ + K k=1 M i=0 c ki,22 ♣♣x(n -i)♣ -β k ♣.x(n) + K k=1 M i=0 c ki,23 ♣♣x(n -i)♣ -β k ♣.x(n -i) + K k=1 M i=0 c ki,24 ♣♣x(n)♣ -β k ♣.x(n -i)
Les mod' eles du tableau 1 sont évalués selon des crit' eres qui se répartissent en deux catégories. La premi' ere catégorie comprend des param' etres pour évaluer les performances de linéarisation. En plus de lŠACPR et de lŠEVM, le NMSE est également utilisé comme param' etre pour reĆéter la convergence de lŠarchitecture DPD dans laquelle les coefficients de chaque mod' ele sont identiĄés. Le NMSE entre une entrée x(n) et la sortie y(n) est donné par :

NMSE (dB) = 10 log 10 N n=1 ♣x(n) -z p (n)♣ 2 N n=1 ♣x(n)♣ 2
La deuxi' eme catégorie de comparaison met en évidence la complexité de lŠimplémentation matérielle du mod' ele DPD. Il est présenté par le nombre de coefficients du mod' ele et les propriétés numériques qui se réf' erent ' a la stabilité numérique dŠidentiĄcation des coefficients ainsi quŠ' a la dynamique des coefficients.

Les mod' eles de DPD sont évalués sur un PA de type Doherty avec une puissance de sortie crête de 57 dBm (500 W) et un gain linéaire de En comparant les performances des deux mod' eles pour chaque largeur de bande, les résultats du tableau 4 conĄrment que le mod' ele GMP offre une capacité de modélisation lég' erement meilleure que le mod' ele DVR-6 pour les largeurs de 5 et 20 MHz.

Pour 10 MHz, le mod' ele DVR-6 a de meilleures performances de NMSE, mais moins bonnes en termes dŠACPR. Selon le tableau 5, le conditionnement de la matrice dŠidentiĄcation du DVR-6 est plus petit que celui du GMP. La dynamique des coefficients du mod' ele DVR-6 est bien inférieure ' a celle du GMP. Par conséquent, le mod' ele DVR-6 présente un bon compromis entre les performances de linéarisation et les propriétés numériques, ce qui en fait un bon candidat pour une implémentation matérielle de la DPD.

Le tableau 6 montre les performances de linéarisation et les propriétés numériques pour chaque niveau de puissance pour une largeur de bande de 20 MHz. Le mod' ele GMP fonctionne mieux que le DVR-6 lorsque le PA op' ere dans des zones de non-linéarités faibles. DŠautre part, le mod' ele DVR-6 conĄrme ses propriétés numériques pour différentes largeurs de bande et différents niveaux de puissances de sortie.

En conclusion de cette étude comparative, le mod' ele GMP offre un bon compromis entre les performances de précision et la complexité. Le mod' ele DVR peut atteindre des performances similaires, mais avec une augmentation du nombre de coefficients. Cependant, le mod' ele DVR poss' ede des propriétés numériques impressionnantes qui en font un bon choix pour lŠimplémentation matérielle. Pour cela, la contribution suivante de cette th' ese se concentrera autour de la conception et de lŠoptimisation du mod' ele DVR, en déterminant sa structure et en proposant une architecture dŠimplémentation.

Optimisation du modèle DVR

Les mod' eles dérivés de la série de Volterra tels que MP, GMP ou DDR ont prouvé leur ef-Ącacité comme mod' ele de prédistorsion. Cependant, pour un ordre élevé de non-linéarité, la matrice de régression intervenant dans lŠidentiĄcation des coefficients est tr' es mal conditionnée ce qui peut générer des instabilités numériques si nous nŠutilisons pas de méthodes de régularisation et la dynamique des coefficients peut être élevée.

LŠévolution des architectures de PA avec une efficacité énergétique élevée, telle que Doherty et suiveur dŠenveloppe (en anglais : Envolope Traking -ET), conduit ' a des PA souvent peu linéaires. En outre, les besoins de débits de données plus élevés conduisent ' a utiliser des signaux modulés avec un PAPR élevé, ce qui stimule les non-linéarités de PA. De plus, les syst' emes 5G et MIMO permettent un débit de données excessivement élevé, ce qui est un véritable déĄ pour la DPD en termes de largeur de bande, de non-linéarités et de comportements dynamiques. Compte tenu de tous ces faits, il devient plus difficile de concevoir un syst' eme de DPD efficace pour obtenir une caractéristique inverse précise avec une bonne efficacité et une faible complexité dŠidentiĄcation. Cela conduit ' a la recherche dŠune approche alternative pour la DPD. LŠune de ces approches est lŠapproche de segmentation, o' u le signal dŠentrée est divisé en plusieurs segments séparés par des seuils. LŠun de ces mod' eles basés sur lŠapproche de segmentation est le mod' ele DVR qui a fait le sujet de lŠétude comparative. LŠétude du mod' ele DVR repose généralement sur deux aspects : la mani' ere de positionner les seuils et la détermination de la structure optimale.

Comme indiqué précédemment, le mod' ele DVR proposé initialement par Anding Zhu est déĄni par relation entrée-sortie : LŠétude du mod' ele DVR est validée par des acquisitions expérimentales des signaux entrée-sortie sur un PA de type LDMOS 50 W, stimulé par un signal LTE de largeur de bande de 20 MHz avec un PAPR de 12 dB ' a une puissance de sortie de 44 dBm. 64000 échantillons de donnée I/Q sont utilisés pour lŠalgorithme dŠidentiĄcation de DPD.

y(n) = M lin i=0 a i x(n -i) + Tt∈S T t o' u x(n) et y(n) représentent
T 1,p = K k=1 M i=0 c ki,1p ♣♣x(n -i)♣ -β k ♣e jθ(n-i) • ♣x(n)♣ p T 2 = K k=1 M i=i 2 c ki,2 ♣♣x(n -i)♣ -β k ♣ • x(n) T 3 = K k=1 M i=i 3 c ki,3 ♣♣x(n -i)♣ -β k ♣ • x(n -i) T 4 = K k=1 M i=i 4 c ki,4 ♣♣x(n)♣ -β k ♣ • x(n -i) T 5 = K k=1 M i=i 5 c ki,5 ♣♣x(n -i)♣ -β k ♣ • x 2 (n) • x * (n -i) T 6 = K k=1 M i=i 6 c ki,6 ♣♣x(n -i)♣ -β k ♣ • x(n) • ♣x(n -i)♣ 2 T 7 = K k=1 M i=i 7 c ki,7 ♣♣x(n -i)♣ -β k ♣ • x * (n) • x 2 (n -i) o' u K
Le vecteur des seuils β K dans la représentation du mod' ele DVR déĄnit le positionnement des segments. Les seuils peuvent être sélectionnés dŠune façon uniforme β k = k/K pour k = 1; 2; ..; K -1 ou optimale. Dans la littérature, beaucoup de travaux consid' erent des mod' eles ' a segmentation uniforme, peu dŠentre eux traitent lŠoptimisation de la segmentation. Le premier travail sur lŠespacement optimal a été mené pour la DPD en LUT (en anglais : LookUp Table) sans mémoire [START_REF] Cavers | ŤOptimum table spacing in predistorting ampliĄer linearizers[END_REF]. Dans [START_REF] Lin | ŤDynamically optimum lookup-table spacing for power ampliĄer predistortion linearizationŤ[END_REF], une procédure itérative est proposée pour optimiser lŠespacement non-uniforme de la LUT. Pour les mod' eles de segmentation, lŠavantage dŠune segmentation optimale par rapport ' a une segmentation uniforme a été établi dans [START_REF] Magesacher | ŤOptimal Segmentation for Piecewise RF Power AmpliĄer Models[END_REF] o' u les auteurs sugg' erent de réduire la complexité de lŠalgorithme en considérant la version sans mémoire du mod' ele DVR. Les auteurs ont utilisé des méthodes basées sur Gauss-Newton telles que Levenberg-Marquardt [START_REF] Levenberg | ŤA method for the solution of certain non-linear problems in least squaresŤ[END_REF] pour optimiser conjointement les seuils et identiĄer les coefficients du mod' ele. Dans [START_REF] Mateo | ŤRadio-over-Ąber linearization with optimized genetic algorithm CPWL model[END_REF], les auteurs ont utilisé une approche dŠoptimisation globale basée sur lŠalgorithme génétique (en anglais : Genetic Algorithm -GA) pour optimiser les seuils du mod' ele CPWL pour la radio sur Ąbre (en anglais : Radio-over-Fiber -ROF). LŠACPR est utilisé comme facteur de mérite pour le GA.

Dans cette étude dŠoptimisation du mod' ele DVR, une nouvelle approche a été proposée pour optimiser les seuils β. Le principe de cette approche consiste ' a décomposer le probl' eme dŠoptimisation global en sous-probl' emes de minimisation unidirectionnelle basée sur la section dorée (en anglais : Golden Section -GS). Pour une fonction unimodale donnée f (x), le principe de GS consiste ' a trouver lŠoptimum x opt qui correspond ' a la valeur minimale de f (x), en rétrécissant itérativement lŠintervalle de recherche [a b] contenant cet optimum jusquŠ' a ce quŠune précision spéciĄée soit atteinte. La mise ' a jour de cet intervalle se fait itérativement selon un facteur appelé le nombre dŠor τ = 1+ Dans ce cadre, nous avons montré que la segmentation optimale améliore les performances de linéarisation par rapport ' a la segmentation uniforme pour le mod' ele DVR avec le même nombre de coefficients. Nous avons proposé une approche de détermination des seuils optimaux avec une complexité réduite par rapport ' a lŠapproche GA et ' a lŠapproche force brute. La détermination des seuils optimaux a été effectué pour un nombre segments K Ąxé ' a priori. Une autre question porte sur la détermination de la valeur optimale de K et des autres param' etres de structure du mod' ele DVR. Cette question fait lŠobjet de la section suivante.

Dimensionnement du modèle DVR

Le dimensionnement du mod' ele DVR consiste ' a trouver une structure de mod' ele optimal qui conduit au meilleur compromis entre précision de modélisation et complexité réduite. La structure du mod' ele DVR est déĄnie par le nombre de segments K, la profondeur de mémoire du terme linéaire M lin , la profondeur de la mémoire M , lŠordre maximum de la non-linéarité P , et le nombre de termes T t . La structure du mod' ele DVR peut être déterminée ' a lŠaide de recherches exhaustives en explorant toutes les combinaisons de valeurs de param' etres possibles. En Ąxant lŠordre maximum de

K et M ' a K max et M max , respectivement, la recherche exhaustive nécessite K max •M max •L tests, o' u L = P +7 k=1 P +7 k = 2 P +7
-1 présente le nombre total de combinaisons possibles de termes du mod' ele. Limiter le nombre maximum de coefficients ' a tester pourrait réduire considérablement lŠespace de recherche. Cependant, cette approche de la recherche exhaustive reste tr' es lourde en calcul.

Dans la littérature, un des premiers travaux dédié au dimensionnement de la structure optimale pour les mod' eles de DPD a été fait pour le mod' ele GMP en utilisant un algorithme de type GA [START_REF] Gotthans | ŤComparison of modeling techniques for power ampliĄers[END_REF]. Dans cet algorithme, la fonction de coût représente une combinaison linéaire du NMSE et du nombre de coefficients. Dans [START_REF] Wang | ŤPerformance analysis of multi-stage cascaded digital predistortion[END_REF], les auteurs ont proposé un algorithme heuristique HC pour trouver la structure optimale du mod' ele GMP avec une complexité de calcul réduite. Dans [START_REF] Wang | ŤComparison of hill-climbing and genetic algorithms for digital predistortion models sizing[END_REF], une étude comparative de deux algorithmes dŠoptimisation, lŠalgorithme HC et le GA, a été réalisée en termes de vitesse de convergence et de performance de linéarisation de la solution obtenue. Les deux algorithmes ont été comparés sur le même crit' ere de recherche et obtiennent des performances tr' es proches avec une complexité de calcul tr' es inférieure ' a une recherche exhaustive.

Dans cette étude, lŠapproche proposée dans [START_REF] Wang | ŤA Novel Algorithm for Determining the Structure of Digital Predistortion Models[END_REF] est étendue pour le mod' ele DVR pour trouver une structure optimale avec une complexité réduite et de meilleures performances de linéarisation. Le HC est un algorithme dŠoptimisation heuristique qui part dŠune solution initiale et essaie par la suite de trouver une meilleure solution en comparant la solution actuelle avec ses voisines. SŠil existe une meilleure solution parmi les voisins, elle est considérée comme la nouvelle solution. La procédure de recherche est répétée jusquŠ' a ce quŠaucune meilleure solution ne puisse être trouvée. La déĄnition du voisinage est un processus nécessaire pour pouvoir mettre en oeuvre lŠalgorithme HC.

La fonction de coût est un aspect essentiel de lŠalgorithme HC qui conduit ' a un compromis entre les performances de linéarisation et la complexité du mod' ele. Le

K i + δ 1 , M lin,i + δ 2 , M i + δ 3 , T U,i + δ 4 ), o' u δ 1,2,3 ∈ [0, ±1].
Le processus impliqué derri' ere δ 4 correspond soit ' a ajouter ' a T S i un terme de son complément relatif dans T, soit ' a retirer un terme de T S i , soit ' a laisser inchangé T S i .

La complexité de lŠalgorithme HC est caractérisé par quatre facteurs :

• Le nombre total dŠéléments (voisins) évalués par lŠalgorithme HC, qui correspond automatiquement au nombre dŠévaluations f .

• Le nombre total dŠitérations nécessaires pour exécuter lŠalgorithme HC, o' u une itération HC est effectuée lorsque les fonctions de coût des voisins dŠune solution initiale sont calculées et la solution optimale est déterminée.

• Le temps dŠexécution de lŠalgorithme HC est également pris en compte comme indicateur relatif de la complexité de lŠalgorithme HC.

• Le nombre de coefficients du mod' ele de la structure de chaque voisin est introduit, car il reĆ' ete la complexité calculatoire du processus dŠidentiĄcation. Cela fait également référence ' a la complexité du mod' ele, qui est un aspect critique pour le dimensionnement du mod' ele DVR.

Ces crit' eres sont des indicateurs pour évaluer la complexité de lŠalgorithme HC et les propriétés numériques des mod' eles évalués au cours de lŠalgorithme HC. La complexité de lŠalgorithme HC peut être réduite en élaguant les voisinages en ajoutant des contraintes sur le nombre de coefficients.

LŠalgorithme HC a été validé par des mesures expérimentales de PA avec un signal LTE de largeur de bande de 20 MHz. La Ągure 5 présente les résultats de lŠalgorithme HC. Les résultats de la recherche exhaustive obtenus en explorant toutes les structures possibles sont tracées en point bleu en fonction de N et C. Les points en violet représentent les structures évaluées par lŠalgorithme HC. Les points rouges représentent les meilleures structures trouvées ' a chaque itération. Le nombre maximum de coefficients que lŠalgorithme HC ne doit pas dépasser est Ąxé ' a 50 coefficients.

+ 4 k=1 1 i=0 c ki,3 ♣♣x(n -i)♣ -β k ♣ • x(n -i) + 4 k=1 1 i=0 c ki,6 ♣♣x(n -i)♣ -β k ♣ • x(n) • ♣x(n -i)♣ 2
Dans la Ągure 5, les résultats sont obtenus avec un ordre maximal de non-linéarité P conĄguré ' a 1. La présence de ce terme dans le mod' ele DVR est importante aĄn de représenter de fortes non-linéarités. Pourtant, lŠordre de non-linéarité dans le mod' ele DVR impacte ses performances, notamment en terme des propriétés numériques.

Le dimensionnement du mod' ele DVR est une des contributions majeures de cette th' ese o' u lŠalgorithme HC proposé pour le mod' ele GMP a été étendu au mod' ele DVR pour trouver la structure optimale du mod' ele DVR avec un temps dŠexécution réduit. Deux aspects essentiels sont mis en évidence dans ce cadre : la nouvelle déĄnition du voisinage que nous proposons pour lŠalgorithme HC au mod' ele DVR et lŠutilisation dŠune fonction de coût qui permet dŠobtenir le meilleur compromis entre précision de modélisation et complexité.

Approche système pour la conception d'un modèle DVR optimal

La conception dŠun mod' ele DVR optimal consiste ' a déterminer sa structure optimale avec une segmentation optimale selon les mêmes crit' eres que ceux utilisés précédemment. Fondamentalement, ce processus présente une combinaison de lŠalgorithme HC et de lŠapproche proposée pour optimiser les seuils. Dans cette étude, nous proposons des approches syst' eme (SysApp) pour effectuer cette optimisation conjointe aĄn de concevoir le mod' ele DVR optimal. Ces SysApps sont développées, discutées et comparées ' a lŠalgorithme HC en terme de solutions optimales, de performances de linéarisation et de complexité.

Les SysApp pour concevoir le mod' ele DVR optimal sont développées ' a travers les différentes versions suivantes :

• Approche directe qui peut se voir comme lŠapproche la plus intuitive pour concevoir le mod' ele DVR optimal. Cette approche est basée sur la combinaison de lŠapproche proposée basée sur GS pour optimiser les seuils β et lŠalgorithme HC pour dimensionner le mod' ele DVR. En effet, lŠoptimisation de β k est effectuée pour chaque élément (structure) évalué par lŠalgorithme HC. La fonction de coût f est calculée en fonction de C et N de chaque mod' ele en tenant compte de sa segmentation optimisée β opt . Cette approche est indiquée par D-SysApp (en anglais : Direct System Approach). ' A premi' ere vue, D-SysApp requiert une complexité de calcul importante puisquŠelle sera principalement évaluée par le nombre total de calculs de NMSE nécessaire par lŠalgorithme HC et la recherche GS.

• D-SysApp peut être simpliĄée en terme de complexité en optimisant les seuils uniquement pour des structures de mod' ele DVR spéciĄques. Ces structures sont déterminées selon 2 approches :

-Les structures qui se trouvent sur lŠenveloppe de lŠalgorithme HC. Cette approche est indiquée par Senv-SysApp (en anglais : SimpliĄed approach on Envelope results). Son principe consiste ' a appliquer le processus dŠoptimisation des seuils par le GS une fois que lŠalgorithme HC est terminé. La complexité de Senv-SysApp est principalement évaluée par la complexité de lŠalgorithme HC, avec la complexité de la recherche GS multipliée par le nombre de structures trouvées sur lŠenveloppe.

-Les structures qui sont les meilleures structures (solutions) ' a la Ąn de chaque iteration de lŠalgorithme HC. Cette approche est indiquée par Sitr-SysApp (en anglais : SimpliĄed approach on Iteration results). Sitr-SysApp consiste ' a insérer la recherche GS dans lŠalgorithme HC, qui sera activé une fois lŠitération HC terminée et appliquée au meilleur voisin de cette itération HC. Cela permet de réduire la complexité qui sera principalement évaluée par celle de lŠalgorithme de HC. Selon le tableau 8, les solutions optimales ont la même structure de mod' ele en termes de profondeurs de mémoires (M lin et M ), tandis que le nombre de segments K et les termes du mod' ele T S sont différents dŠune solution ' a lŠautre. Par ailleurs, les solutions optimales disposent du même niveau de performance de linéarisation, mais avec une complexité différente. En calculant la fonction de coût avec α = 0.05, la meilleure solution optimale est celle renvoyée par Ogen-SysApp suivi par D-SysApp, ce qui est tout ' a fait raisonnable puisquŠelle est basée sur une intégration totale de la recherche GS au sein de lŠalgorithme HC .

La complexité des SysApp proposée pour concevoir le mod' ele DVR est évaluée de la même mani' ere que la réduction de la complexité de lŠalgorithme HC décrite dans la section 3.5.5. Le tableau 9 présente la complexité de chaque SysApp, qui est évaluée par le nombre total de mod' eles évalués, les itérations HC, le nombre total de calculs de NMSE incluant ceux du processus dŠoptimisation des seuils, le temps dŠexécution et la somme des nombres de coefficients de tous les mod' eles identiĄés Tableau 9 : Comparaison de la complexité des approches syst' eme Selon le tableau 9, le comportement de chaque SysApp impacte la complexité de sa convergence vers la solution optimale. Le temps dŠexécution de D-SysApp est immense avec un nombre élevé de calculs NMSE, par conséquent, il ne peut pas être un candidat réaliste de SysApp pour concevoir le mod' ele DVR. Senv-SysApp et Sitr-SysApp restent des candidats acceptables en termes de temps dŠexécution et de nombre de calculs de NMSE. Ored-SysApp et Ogen-SysApp ont une complexité plus grande que Sens-SysApp et Sitr-SysApp.

DŠapr' es les résultats du tableau 8 et 9, et puisque D-SysApp est exclu de notre choix pour sa complexité de calcul, nous pouvons conclure que Ogen-SysApp représente un excellent candidat pour concevoir le mod' ele DVR, car il présente un bon compromis entre les performances de linéarisation et la complexité de calcul. Il faut aussi noter que Sitr-SysApp présente des performances de linéarisation similaires ' a Ogen-SysApp avec une complexité réduite (15 coefficients de mod' ele) et avec les mêmes termes de mod' ele.

Stratégie d'implémentation matérielle du modèle DVR Optimal

Une stratégie dŠimplémentation matérielle sur un FPGA est discutée et investiguée dans cette partie, o' u les propriétés numériques du mod' ele DVR optimal sont soulignées. Selon la littérature sur cet axe de recherche, de nombreux travaux ont été rapportés pour implémenter les mod' eles de DPD. Dans [START_REF] Liszewski | ŤLow-complexity FPGA implementation of Volterra predistorters for power ampliĄers[END_REF], un design FPGA est proposé pour implémenter un mod' ele de DPD avec mémoire. Les auteurs ont proposé dans [START_REF] Gilabert | ŤLook-Up Table Implementation of a Slow Envelope Dependent Digital Predistorter for Envelope Tracking Power AmpliĄers[END_REF] une méthodologie avec une synth' ese de haut niveau pour les dispositifs FPGA aĄn dŠimplémenter des mod' eles basés sur la série de Volterra. Une combinaison de processeurs FPGA et ARM est proposée dans [START_REF] Long | ŤThe design and implementation of baseband predistorter based on FPGA and ARM[END_REF] pour implémenter le mod' ele MP, o' u les auteurs ont utilisé le processeur ARM pour son avantage dans les calculs en virgule Ćottante. Concernant le mod' ele DVR, peu de travaux de recherche ont été rapportés dans la littérature consacrée ' a lŠimplémentation matérielle. Dans [START_REF] Cao | ŤA ModiĄed Decomposed Vector Rotation-Based Behavioral Model With Efficient Hardware Implementation for Digital Predistortion of RF Power AmpliĄers[END_REF], les auteurs ont proposé une structure matérielle en décomposant le mod' ele DVR en un format sous-composé pour réduire la complexité de mise en İuvre. Cependant, les param' etres du mod' ele DVR ont été Ąxés arbitrairement sans évoquer aucune préoccupation quant ' a la réduction de la complexité du mod' ele. Dans [START_REF] Li | ŤInstantaneous Sample Indexed Magnitude-Selective Affine Function-Based Behavioral Model for Digital Predistortion of RF Power AmpliĄers[END_REF], un mod' ele dérivé du mod' ele DVR est proposé, o' u lŠimplémentation matérielle a été soulignée.

Il y a deux points critiques ' a considérer lors de la conversion de logiciel en matériel : la précision et la vitesse. Dans la mise en İuvre matérielle de DPD, il existe principalement deux processus :

• LŠapplication de DPD au signal dŠentrée qui doit être échantillonnée ' a une fréquence dŠéchantillonnage suffisante pour répondre ' a la demande croissante de signaux ' a large bande dans les syst' emes de télécommunication actuels tels que la 5G.

• IdentiĄer et mettre ' a jour les coefficients de DPD, ce qui peut être fait en utilisant des approches bloc par bloc ou échantillon par échantillon.

Généralement, il existe deux structures pour implémenter le mod' ele de DPD : la méthode par LUT et la structure directe avec multiplieurs et additionneurs. La méthode par LUT stocke la valeur pré-calculée de la fonction de DPD, qui est indexée par le module du signal dŠentrée quantiĄé. DŠautre part, la structure directe avec multiplieurs et additionneurs, appelée méthode directe, nécessite des ressources logiques de FPGA pour effectuer lŠimplémentation dŠune fonction de DPD. LŠobjectif de cette étude est de proposer une stratégie matérielle pour implémenter le mod' ele DVR optimal conçu par lŠapproche conjointe.

Dans cette étude, nous adoptons le mod' ele DVR optimal retourné par Ogen-SysApp qui sŠexprime selon le tableau 8 par :

y(n) = 2 i=0 a i x(n -i) + 5 k=1 1 i=0 c ki,3 ♣♣x(n -i)♣ -β k ♣ • x(n -i) + 5 k=1 1 i=0 c ki,6 ♣♣x(n -i)♣ -β k ♣ • x(n) • ♣x(n -i)♣ 2
Selon la méthode directe, lŠimplémentation de ce mod' ele optimal nécessite 42 additionneurs et 33 multiplieurs.

Une implémentation par LUT peut être réalisée en utilisant uniquement 7 additionneurs, 7 multiplieurs, et 2 LUT de 4096 cases et de 12 bits. La Ągure 7 présente lŠimplantation du mod' ele DVR optimal par LUT.

Optimisation et linéarisation de l'amplificateur de puissance Doherty à double-entrée

Avec lŠintroduction des signaux modulés avec une amplitude non-constante et un PAPR élevé dans le contexte 4G LTE et 5G, il devient de plus en plus délicat dŠaméliorer le rendement de PA tout en conservant un niveau de linéarité adéquat. Pour cela, des architectures de PA avancées basées sur une charge dynamique ou une modulation dŠalimentation ont été proposées dans la littérature pour éviter de gaspiller des ressources énergétiques excessives, telle que lŠarchitecture Doherty, suiveur dŠenveloppe (en anglais : Envelope Tracking -ET), Outphasing. Bien que ces architectures soient conçues majoritairement avec une seule entrée RF, plusieurs études ont été menées dans la littérature pour souligner les avantages de garder des entrées séparées [START_REF] Darraji | ŤA Dual-Input Digitally Driven Doherty AmpliĄer Architecture for Performance Enhancement of Doherty Transmitters[END_REF].

Le principe de la séparation des entrées RF permet dŠobtenir des degrés de liberté supplémentaires, dont des param' etres appelés param' etres libres, sont offerts par les entrées séparées, et peuvent être réglés pour améliorer les performances de PA [START_REF] Darraji | ŤDoherty Goes Digital: Digitally Enhanced Doherty Power AmpliĄers[END_REF]. la recherche des param' etres libres optimaux peut être considérée comme un probl' eme dŠoptimisation globale. Nous nous sommes intéressés ' a lŠarchitecture de Doherty, notamment, le PA Doherty ' a double-entrée (en anglais : Dual-Input Doherty PA -DIDPA), son architecture est illustrée dans la Ągure 8.

Le DIDPA se caractérise par deux entrées RF, une alimentation de drain V DC et deux tensions V GS pour polariser les ampliĄcateurs principal et de crête indépendamment. LŠamplitude et la phase instantanées de chaque signal dŠentrée, ainsi que les tensions V GS , peuvent être contrôlées et ajustées séparément en bande de base, permettant dŠaméliorer les performances du DIDPA, en particulier son rendement énergétique.

Le DIDPA est intégré dans un syst' eme comprenant en plus de la DPD comme technique de linéarisation et le CFR pour réduire le PAPR. Les param' etres libres couvrent les param' etres du DIDPA telles que les tensions de polarisation du DIDPA, le diviseur de puissance entre les 2 entrées RF, et dŠautres param' etres du syst' eme tel que le facteur de la réduction de PAPR. Peu dŠentre eux traitent lŠoptimisation des param' etres libres, la réduction du PAPR et sa linéarisation dans une approche conjointe. Le premier travail connexe est proposé dans [START_REF] Niu | ŤStochastically approximated multiobjective optimization of dual input digital Doherty Power AmpliĄer[END_REF], o' u les auteurs ont utilisé un algorithme dŠapproximation stochastique perturbée (en anglais : Simultaneous Perturbation Stochastic Approximation -SPSA) pour optimiser les param' etres libres de DIDPA. Dans [START_REF] Ma | ŤMachine-Learning Based Digital Doherty Power AmpliĄer[END_REF], les auteurs ont proposé une optimisation basée sur un recuit simulé (en anglais : Simulated Annealing -SA) pour optimiser le DIDPA avec une fonction de coût formée ' a partir des valeurs du PAE, le gain du DIDPA, la puissance de sortie P out et lŠACPR comme métrique faisant référence ' a la linéarité.

Dans cet axe de recherche, nous avons proposé une nouvelle approche de réglage automatique pour améliorer lŠefficacité énergétique du DIDPA tout en répondant ' a lŠexigence de linéarité. Cette approche consiste ' a optimiser les param' etres libres par un contrôle de recherche heuristique hybride (en anglais : Hybrid Heuristic Search Control -HHSC) selon une architecture présentée dans la Ągure 9. • CFR est utilisé pour réduire le PAPR du signal v(n).

• DPD linéarise le DIDPA pour compenser les distorsions.

• Diviseur de puissance est conçu de mani' ere optimale pour diviser le signal x(n) en deux signaux différents x m (n) et x p (n)

• DUT (en anglais : Device Under Test) présente lŠétage puissance de lŠarchitecture y compris le DIDPA avec son bloc dŠalimentation.

Chaque bloc a des param' etres libres ' a déĄnir ou ' a contrôler par lŠalgorithme dŠoptimisation HHSC :

• Le principe du CFR pour réduire le PAPR se base sur la technique dŠannulation de pics (en anglais : Peak Cancellation -PC), et se réalise ' a lŠaide dŠopérations dŠécrêtage brutal et de Ąltrage.

Le signal dŠentrée v(n) est écrêté selon un seuil µ. Un signal de pics v p (n) est construit en soustrayant le signal clippé du signal original :

v p (n) = v(n) × 1 -µ ♣v(n)♣ si ♣v(n)♣ ≥ µ 0 si ♣v(n)♣ < µ
Ce signal de pics est Ąltré avant dŠêtre soustrait du signal original. Nous utilisons le seuil dŠécrêtage µ comme param' etre libre du bloc CFR.

• La motivation derri' ere lŠutilisation de deux entrées RF séparées est dŠéliminer les séparateurs dŠentrée analogiques, tels que le diviseur de Wilkinson, et de permettre un contrôle de puissance indépendant vers lŠampliĄcateur principal et de crête de Doherty. Le diviseur de puissance divise le signal dŠentrée x = Xe jθ en deux signaux complexes x m et x p déĄnis comme :

x m = α m x ; x p = α p e -jϕ x o' u ϕ représente la différence de phase entre les deux signaux et α représente le rapport de puissance avec la condition

α m = √ α ; α p = √ 1 -α
Dans cette étude, α et ϕ sont considérés comme des param' etres libres.

• Les tensions de polarisation de lŠampliĄcateur principal et crête de DIDPA sont données respectivement par V GS,m et V GS,p , et considérées comme des param' etres libres ' a optimiser par HHSC.

• Dans cette architecture, les coefficients de la DPD sont identiĄés par lŠarchitecture ILA ' a lŠaide un processus de régression linéaire telle que la méthode des moindres carrés (en anglais : Least Square -LS). Par conséquent, la DPD ne sera pas contrôlée par HHSC.

Le principe de HHSC est basé sur la combinaison de lŠalgorithme SA comme recherche dŠoptimisation globale et de lŠapproche de commande extrémale (en anglais : Extremum Seeking Control -ESC) comme contrôle adaptatif pour affiner les résultats optimisés. Le vecteur des param' etres libres ' a optimiser est déĄni par : Θ = [µ; α; ϕ; V GS,m ; V GS,p ]. Le HHSC converge vers la solution optimale, notée par Θ opt , selon une fonction de coût J déĄnie par :

J(Θ) = i w i FOM i FOM t,i i = ¶1,

Conclusion et perspectives

Dans cette th' ese, nous nous sommes concentrés sur lŠétude de la prédistorsion numérique pour linéariser les ampliĄcateurs de puissance, et en particulier les mod' eles de segmentation. Cette th' ese sŠarticule principalement sur trois aspects : lŠétude et lŠanalyse comparative des mod' eles de la DPD, la conception dŠun mod' ele optimal basé sur lŠapproche segmentale avec une stratégie dŠimplémentation matérielle, et lŠoptimisation et la linéarisation dŠampliĄcateur de puissance Doherty ' a double-entrée avec un rendement énergétique élevé. 4. Une approche dŠauto-réglage a été proposée pour trouver la conĄguration optimale dŠun ampliĄcateur de puissance Doherty ' a double-entrée. Cette approche est obtenue en combinant deux méthodes dŠoptimisation : le recuit simulé comme étant un algorithme dŠoptimisation global et lŠapproche de commande extrémale comme étant un contrôle adaptatif. La convergence de lŠapproche proposée est guidé par une fonction de coût adaptative conçue pour conduire ' a un compromis entre linéarité et efficacité. La technique CFR et la linéarisation DPD sont intégrées de mani' ere optimale dans lŠarchitecture proposée aĄn de répondre aux spéciĄcations de lŠutilisateur en améliorant le rendement tout en maintenant un niveau satisfaisant de linéarité.

Abstract

This Ph.D. work contributes to the digital predistortion linearization technique of power ampliĄers. Power AmpliĄer is one of the most critical elements of radiocommunication systems, which exhibits static nonlinearities and nonlinear memory effects. Achieving a good trade-off between the linearity of the power ampliĄer and its efficiency is becoming more crucial. Digital predistortion is a powerful linearization technique that aims to compensate for power ampliĄer distortions and provides linear ampliĄcation with good efficiency. The predistortion principle consists of implementing a nonlinear function, the so-called predistorter upstream of the power ampliĄer. The predistorter ideally has the inverse characteristics of the power ampliĄer. This operation allows us to consider the memory effects of the power ampliĄer, and in particular, the long-term memory. Several behavioral models have been used as predistorter. Most of them are classiĄed into two families: global models derived from the Volterra series and models based on the segmentation approach. The comparative and analysis study of these models is one of the focuses of this dissertation, in which three aspects are used for comparison: linearization performance, complexity, and hardware implementation properties. By focusing on models based on the segmentation approach, this dissertation proposes an approach to design an optimal model according to a trade-off between linearization performance and model complexity. This model is used to linearize a dual-input Doherty power ampliĄer.

A global optimization algorithm combined with a control process is proposed to enhance efficiency while maintaining a good linearity level according to a proposed adaptive cost function.

Résumé

Le travail de th' ese présenté par ce manuscrit sŠintéresse ' a la linéarisation des ampliĄcateurs de puissance en utilisant la prédistorsion numérique. LŠampliĄcateur de puissance est lŠun des modules les plus critiques des équipements de communication radio qui présente des non-linéarités statiques ainsi que des effets de mémoire. Il devient de plus en plus crucial de réaliser un compromis entre la linéarité et le rendement énergétique. La prédistorsion numérique est une technique de linéarisation efficace qui compense les distorsions dues ' a la non-linéarité en appliquant une déformation sur le signal dŠentrée, de mani' ere ' a ce que le syst' eme global réalise une ampliĄcation linéaire. Le principe de la prédistortion numérique consiste ' a mettre en İuvre une fonction non linéaire dite prédistorteur en amont de lŠampliĄcateur de puissance, et qui a idéalement les caractéristiques inverses de lŠampliĄcateur de puissance. Cette opération permet également de prendre en compte les effets mémoire, et en particulier la mémoire ' a long terme. Plusieurs mod' eles comportementaux sont proposés pour la prédistorsion qui peuvent être classés en deux catégories : les mod' eles globaux et les mod' eles par segmentation. LŠétude comparative de ces mod' eles est lŠun des axes de travail traité par cette th' ese, dont la comparaison repose sur trois aspects qui sont soulignés : les performances de linéarisation, la complexité et les propriétés de lŠimplémentation matérielle. En se concentrant sur les mod' eles basés sur lŠapproche par segmentation, un algorithme est proposé pour concevoir un mod' ele optimal en satisfaisant un bon compromis entre les performances de linéarisation et la complexité du mod' ele. Ce mod' ele est utilisé pour linéariser un ampliĄcateur de puissance Doherty ' a double entrée pour lequel un algorithme dŠoptimisation globale associé ' a un processus de contrôle est proposé pour améliorer le rendement tout en conservant un bon niveau de linéarité selon une fonction de coût adaptative.

1. 1

 1 Block diagram of RF wireless transmitter . . . . . . . . . . . . . . . . . . . 1.2 IQ data of LTE signal in dBm . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Trade-off between the linearity and the efficiency of PA . . . . . . . . . . . 1.4 AM-AM and AM-PM characteristics of a Doherty PA . . . . . . . . . . . . 1.5 1-dB compression point on AM-AM characteristic . . . . . . . . . . . . . . 1.6 1-dB compression point on AM-AM characteristic . . . . . . . . . . . . . . 1.7 PA driven by CW signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8 PA driven by two-tone signal . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9 IMD products and harmonics of PA driven by two-tone signal . . . . . . . 1.10 Memory effect on AM-AM . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.11 Upper and lower adjacent channels . . . . . . . . . . . . . . . . . . . . . . 1.12 Constellation of input-output IQ data with EVM=10% . . . . . . . . . . . 1.13 Selective mapping technique . . . . . . . . . . . . . . . . . . . . . . . . . . 1.14 Principle of Feedforward . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.15 Principle of Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Principle of digital predistortion . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Baseband adaptive DPD with PA . . . . . . . . . . . . . . . . . . . . . . . 2.3 ILA and Post-Distortion architecture . . . . . . . . . . . . . . . . . . . . . 2.4 Principle of Post-Distortion architecture . . . . . . . . . . . . . . . . . . . 2.5 Implementation of an MP model with K = 3 and M = 1 . . . . . . . . . . 2.6 Three-way Doherty PA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 AM-AM & AM-PM curves of Doherty PA for an 20 MHz LTE signal . . . 2.8 Modeling accuracy vs number of coefficients . . . . . . . . . . . . . . . . . 2.9 AM-AM & AM-PM curves of LDMOS PA for a 20 MHz LTE signal . . . . 2.10 NMSE vs number of coefficients at 44 dBm output power . . . . . . . . . . 2.11 Spectra of PA input-output, GMP, and DVR-6 postdistorter at 44 dBm output power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 DVR basis function structure . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Testbench for experimental implementation . . . . . . . . . . . . . . . . . . 3.3 The PA characteristics with 20MHz LTE signal . . . . . . . . . . . . . . . 3.4 NMSE versus K with M lin = 2 and M = 1 . . . . . . . . . . . . . . . . . . 3.5 ACPR L1 and ACPR R1 versus K with M lin = 2 and M = 1 . . . . . . . . . 3.6 Evolution of GS search in terms of NMSE for K = 2 with ϵ=0.1 . . . . . . 3.7 Behavior of ACPR L1 and ACPR R1 over the normalized input . . . . . . . . 3.8 Joint architecture of ILA and thresholds optimization process . . . . . . . 3.9 Spectra of PA input-output and postdistorter output with uniform and optimized segmentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.10 AM-AM of PA and postdistorter with uniform and optimized segmentation 3.11 NMSE(β i ) on I (0) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.12 Zoom of NMSE(β 1 ) on I 1(0) . . . . . . . . . . . . . . . . . . . . . . . . . .

3. 13

 13 NMSE(β i ) on I (1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.14 NMSE(β i ) on I (2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.15 Evolution of thresholds and NMSE versus iterations . . . . . . . . . . . . . 3.16 Behavior of NMSE during random selection for 100 times . . . . . . . . . . 3.17 Fit probability distribution to data from experiment 1 . . . . . . . . . . . . 3.18 Fit probability distribution to data from experiment 2 . . . . . . . . . . . . 3.19 Fit probability distribution to data from experiment 3 . . . . . . . . . . . . 3.20 NMSE versus variation of β i over I i after random selection process where each NMSE(β i ) are unimodal . . . . . . . . . . . . . . . . . . . . . . . . . 3.21 Optimal segmentation versus number of segments K . . . . . . . . . . . . 3.22 Optimal segmentation versus memory depth M . . . . . . . . . . . . . . . 3.23 Optimal segmentation versus model terms T S . . . . . . . . . . . . . . . . 3.24 Optimal segmentation vs Terms . . . . . . . . . . . . . . . . . . . . . . . . 3.25 Neighbors of x 0(q) at q th iteration . . . . . . . . . . . . . . . . . . . . . . . 3.26 Discrete Space S of Neighbors of x i in terms of K, M lin , and M with a given T S i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.27 Exhaustive search of DVR model with P = 1 in terms of C and N . . . . . 3.28 HC algorithm results in function of C, N , and f with α=0.1 . . . . . . . . 3.29 HC algorithm and exhaustive search in terms of C, N . . . . . . . . . . . . 3.30 Evolution of HC algorithm at Ąrst iteration . . . . . . . . . . . . . . . . . 3.31 Evolution of HC algorithm at second iteration . . . . . . . . . . . . . . . . 3.32 Evolution of HC algorithm after seventh iteration . . . . . . . . . . . . . . 3.33 Evolution of HC algorithm with stopping conditions . . . . . . . . . . . . .3.34 HC algorithm with d C = 5 and α = 0.05 . . . . . . . . . . . . . . . . . . . 3.35 HC algorithm and exhaustive search in terms of C, N . . . . . . . . . . . . 3.36 Variation of p versus NMSE . . . . . . . . . . . . . . . . . . . . . . . . . . 3.37 Impact of p on numerical properties . . . . . . . . . . . . . . . . . . . . . . 3.38 Spectra of PA input-output and postdistorter output using DVR-4 1 and DVR-6 9 models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.39 Neighbors per each HC iteration . . . . . . . . . . . . . . . . . . . . . . . . 3.40 Numerical properties of all neighbors evaluated during the search sorted in ascending order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Behavior search of D-SysApp . . . . . . . . . . . . . . . . . . . . . . . . .

3. 1

 1 Comparison of linearization performances, optimal solutions and complexity 3.2 Comparison of linearization performances, optimal solutions, and complexity with random selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Probability distribution Ątting to number of NMSE computations . . . . . 3.4 Complexity of proposed approach for each K . . . . . . . . . . . . . . . . . 3.5 Complexity of GS search for increasing K from 2 to 10 . . . . . . . . . . . 3.6 Complexity of GS search for decreasing K from 10 to 2 . . . . . . . . . . . 3.7 Number of coefficients, NMSE, and cost function of the best DVR model at each HC iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 Parameters of DVR model structure (initial and best) at each HC iteration 3.9 Number of coefficients during the HC algorithm . . . . . . . . . . . . . . . 3.10 Parameters of optimal DVR model structure for d C variation . . . . . . . . 3.11 Number of neighbors versus HC iteration for each d C . . . . . . . . . . . . 3.12 Parameters of the optimal DVR model, C, N , and f with the variation α . 3.13 Complexity of HC algorithm for each α . . . . . . . . . . . . . . . . . . . . 3.14 Parameters of DVR model structure (initial and best) and the number of neighbors, at each iteration . . . . . . . . . . . . . . . . . . . . . . . . . . 3.15 Parameters of DVR model with P = 9 structure at each HC iteration . . . 3.16 C and N of the best DVR model with P = 9 at each HC iterations . . . . 3.17 Comparison of DVR-4 1 and DVR-6 9 model structures . . . . . . . . . . . 3.18 Comparison of DVR-4 1 and DVR-6 9 in terms of modeling accuracy, complexity, and numerical properties . . . . . . . . . . . . . . . . . . . . . . . 3.19 Comparison of DVR-4 1 and DVR-6 9 in terms of linearization performance111 3.20 Comparison of HC complexity for P = 1 and P = 9 . . . . . . . . . . . . . 4.1 Optimal segmentation saved through Ored-SysApp . . . . . . . . . . . . . 4.2 Comparison of optimal solutions of system approaches . . . . . . . . . . . 4.3 Complexity comparison of system approaches . . . . . . . . . . . . . . . . 5.1 Free-parameters of the proposed architecture . . . . . . . . . . . . . . . . . 5.2 Free-parameters and their searching range . . . . . . . . . . . . . . . . . . 5.3 Optimal free-parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5. 4

 4 FOMs according to the optimal free-parameters Θ opt,HHSC with P in =11.36 dBm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 FOMs according to the optimal free-parameters Θ opt,HHSC with P in =21.16 dBm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 FOMs according to the optimal free-parameters . . . . . . . . . . . . . . . 5.7 FOMs according to the optimal free-parameters after the optimal pruning of free-parameters in HHSC . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 1 . 1 :

 11 Figure 1.1: Block diagram of RF wireless transmitter

Figure 1 . 2 :

 12 Figure 1.2: IQ data of LTE signal in dBm

Figure 1 . 4 :

 14 Figure 1.4: AM-AM and AM-PM characteristics of a Doherty PA

Figure 1 .

 1 Figure 1.5: 1-dB compression point on AM-AM characteristic

Figure 1 . 7 :

 17 Figure 1.7: PA driven by CW signal

Figure 1 . 8 :

 18 Figure 1.8: PA driven by two-tone signal

Figure 1 . 9 :

 19 Figure 1.9: IMD products and harmonics of PA driven by two-tone signal

Figure 1 . 10 :

 110 Figure 1.10: Memory effect on AM-AM

Figure 1 . 11 : 2 B/ 2 P 10 B/ 2 -B/ 2 P

 111221022 Figure 1.11: Upper and lower adjacent channels

Figure 1 . 14 :

 114 Figure 1.14: Principle of Feedforward

Figure 1 . 15 :

 115 Figure 1.15: Principle of Feedback

Figure 2 . 2 :

 22 Figure 2.2: Baseband adaptive DPD with PA

  .

  3.

Figure 2 . 3 :

 23 Figure 2.3: ILA and Post-Distortion architecture

Figure 2 . 4 :

 24 Figure 2.4: Principle of Post-Distortion architecture

Figure 2 . 5 :

 25 Figure 2.5: Implementation of an MP model with K = 3 and M = 1

Figure 2 . 6 :

 26 Figure 2.6: Three-way Doherty PA

Figure 2 . 7 :

 27 Figure 2.7: AM-AM & AM-PM curves of Doherty PA for an 20 MHz LTE signal

Figure 2 . 8 :

 28 Figure 2.8: Modeling accuracy vs number of coefficients

Figure 2 . 9 :

 29 Figure 2.9: AM-AM & AM-PM curves of LDMOS PA for a 20 MHz LTE signal

Figure 2 .

 2 10 shows the plot of the modeling accuracy presented by NMSE versus the number of coefficients for 5, 10, and 20 MHz bandwidths. Theses results are achieved using postdistortion architecture. The output power is 44 dBm.

(a) 5

 5 MHz bandwidth (b) 10 MHz bandwidth (c) 20 MHz bandwidth

Figure 2 . 10 :

 210 Figure 2.10: NMSE vs number of coefficients at 44 dBm output power

Figure 2 . 11 :

 211 Figure 2.11: Spectra of PA input-output, GMP, and DVR-6 postdistorter at 44 dBm output power

Figure 3 . 3 :

 33 Figure 3.2: Testbench for experimental implementation

Figure 3 . 4 :

 34 Figure 3.4: NMSE versus K with M lin = 2 and M = 1

Figure 3 . 5 :

 35 Figure 3.5: ACPR L1 and ACPR R1 versus K with M lin = 2 and M = 1

  [a b] = [0 1]. At the Ąrst iteration, the search interval [a b] is updated with the narrower interval [a 1 1]. After 34 iterations, [a 34 b 34 ] fulĄlls the condition ♣a 34b 34 ♣ < ϵ that allows the GS search to stop and return β opt = a 34 +b 342as the optimized threshold. According to Figure3.6, the optimal threshold is located at 0.8.Thus, the NMSE from uniform segmentation to optimal segmentation is improved by nearly 4 dB, as shown in Figure3.6.

Figure 3 . 6 :

 36 Figure 3.6: Evolution of GS search in terms of NMSE for K = 2 with ϵ=0.1

Figure 3 . 7 :

 37 Figure 3.7: Behavior of ACPR L1 and ACPR R1 over the normalized input

Figure 3 . 8 :Algorithm 4 :

 384 Figure 3.8: Joint architecture of ILA and thresholds optimization process

  .4 and Figure 3.5 for K = 4. The linearization performances of the optimal segmentation β opt are presented in Figure 3.9 and Figure 3.10 where they are compared to that of the uniform segmentation β unif = [0.25 0.5 0.75].

Figure 3 . 9 :Figure 3 .

 393 Figure 3.9: Spectra of PA input-output and postdistorter output with uniform and optimized segmentation

Figure 3 .

 3 Figure 3.10: AM-AM of PA and postdistorter with uniform and optimized segmentation

Figure 3 .

 3 Figure 3.11 presents the behavior of NMSE(β i ) on each interval.

Figure 3 .

 3 Figure 3.11: NMSE(β i ) on I (0)

Figure 3 . 12 :

 312 Figure 3.12: Zoom of NMSE(β 1 ) on I 1(0)

Figure 3 .

 3 Figure 3.13: NMSE(β i ) on I (1)

Figure 3 .

 3 Figure 3.14: NMSE(β i ) on I (2)

Figure 3 .

 3 Figure 3.15 presents the evolution of thresholds β 1 , β 2 , β 3 and NMSE where the optimization of thresholds has been performed in direct order:[β 1 ; β 2 ; β 3 ; β 2 ; β 1 ; β 2 ; ...] until the algorithm has converged.

Figure 3 . 15 :

 315 Figure 3.15: Evolution of thresholds and NMSE versus iterations

Figure 3 .

 3 16 presents the 100 tests, which are plotted in colors. Each test starts with uniform segmentation, e.g., β initial = [0.25 0.5 0.75], with NMSE = -35.82 dB, and perform 50 trials with random segmentation. The NMSE is evaluated at each trial, and the best is retained as long as no better NMSE has been found.

Figure 3 . 16 :

 316 Figure 3.16: Behavior of NMSE during random selection for 100 times

  Experiment 1: (Random selection with 10 trials + GS search) × 100 times • Experiment 2: (Random selection with 30 trials + GS search) × 100 times • Experiment 3: (Random selection with 50 trials + GS search) × 100 times

  , 3.18, and 3.19 present the probability distribution Ątting to the number of NMSE computations for the experiment 1, experiment 2, and experiment 3, respectively, where probability density function (PDF) and cumulative distribution function (CDF) are shown.

Figure 3 . 17 :Figure 3 . 18 :

 317318 Figure 3.17: Fit probability distribution to data from experiment 1

  , 3.18, and 3.19, it turns out that a Burr distribution Ąts well the number of NMSE computations with a mean of 227 and 222 and a variance of 3426 and 2825 for experiments 1 and 2, while for experiment 3, Burr distribution and log-logistic distribution have similar behavior in Ątting the number of NMSE computations.

Figure 3 .

 3 Figure 3.20: NMSE versus variation of β i over I i after random selection process where each NMSE(β i ) are unimodal

Figure 3 .

 3 Figure 3.21: Optimal segmentation versus number of segments K

Figure 3 .

 3 Figure 3.22 compares the optimal position of each threshold for different memory depths M with K = 4 and M lin = 2.Figure3.22 highlights an interesting behavior, showing that the optimal segmentation is not very sensitive to memory depth M . The optimization can be performed for the DVR model without memory to reduce the algorithm complexity, making it an important aspect to design an optimal DVR model.

Figure 3 .

 3 Figure 3.23: Optimal segmentation versus model terms T S

Figure 3 . 24 :

 324 Figure 3.24: Optimal segmentation vs Terms

Figure 3 .

 3 Figure 3.26: Discrete Space S of Neighbors of x i in terms of K, M lin , and M with a given T S i

Algorithm 6 :

 6 Algorithm of DVR model sizing Initialization Get the PA input x and PA output y Set C max and d C Set ∆ν DeĄne cost function f = N + αC Set the iteration counter q = 1 Set the initial element x 0(1) with K 0 = 2, M lin,0 = M 0 = 0 and T S 0 = [] Evaluate the initial cost function f (x 0(1) )

Figure 3 .

 3 27 presents the exhaustive search results, which are plotted in blue dots, in terms of N and C.

Figure 3 . 27 :

 327 Figure 3.27: Exhaustive search of DVR model with P = 1 in terms of C and N

Figure 3 .

 3 Figure 3.28 presents the HC algorithm results for the 20 MHz LTE signal. The results are presented in 3D regarding the Ątness (value of the cost function f ), the number of coefficients C, and NMSE N .

Figure 3 .

 3 Figure 3.28: HC algorithm results in function of C, N , and f with α=0.1

Figure 3 .

 3 Figure 3.29 presents the HC algorithmŠs behavior in 2D in terms of C and N , where the exhaustive search results are plotted in the background.

Figure 3 .

 3 Figure 3.29: HC algorithm and exhaustive search in terms of C, N

Figure 3 . 30 :

 330 Figure 3.30: Evolution of HC algorithm at Ąrst iteration

Figure 3 . 31 :

 331 Figure 3.31: Evolution of HC algorithm at second iteration

At the 7

 7 th iteration, the parameters of the best solution are: K = 5 β = [0.20 0.40 0.60 0.80] M lin = 2 ; M = 2 T S = [T 1,0 T 2 T 7 ] with C = 48 coefficients and N = -38.59 dB. After the 7 th iteration, 91 neighbors of this solution, plotted by purple dots in Figure 3.32, are evaluated by the HC algorithm.

Figure 3 . 32 :

 332 Figure 3.32: Evolution of HC algorithm after seventh iteration

Figure 3 . 33 :

 333 Figure 3.33: Evolution of HC algorithm with stopping conditions

3. 5 . 8 . 2 . 5 HC

 5825 Algorithm with α = 0.05 and d C = 5 By setting α to 0.05 and d C to 5, the search path of the HC algorithm is shown in Figure 3.34, which almost follows the Pareto envelope of the exhaustive search and stops at 27 coefficients, with an NMSE of -38.02 dB.

Figure 3 .

 3 Figure 3.34: HC algorithm with d C = 5 and α = 0.05

3. 5 . 8 . 3 . 2 HC

 5832 Algorithm with α = 0.05 and d C = 5

Figure 3 .

 3 Figure 3.35 presents the DVR model sizing with P = 9 using the HC algorithm in terms of C and N . The blue dots present the neighbors evaluated over the search. The red circles present the best solution at each HC iteration.

Figure 3 . 36 :

 336 Figure 3.36: Variation of p versus NMSE

Figure 3 . 37 :

 337 Figure 3.37: Impact of p on numerical properties

Figure 3 .

 3 Figure 3.38 shows the power spectrum of the DVR-4 1 and DVR-6 9 models, which are compared to the PA output.

Figure 3 . 38 :

 338 Figure 3.38: Spectra of PA input-output and postdistorter output using DVR-4 1 and DVR-6 9 models

Figure 3 . 39 :

 339 Figure 3.39: Neighbors per each HC iteration

  (a) Condition number of regressor matrix (b) Dynamic range of model coefficients

Figure 3 . 40 :

 340 Figure 3.40: Numerical properties of all neighbors evaluated during the search sorted in ascending order

Algorithm 11 :

 11 xs∈S (f (x s )) Algorithm of Ogen-SysApp Choose the initial element x 0(1) Optimize β of x 0(1) Store β (K 0 ,T S 0 ) opt in LUT Evaluate the initial cost function f (x 0(1) )

Figure 4 .

 4 Figure 4.2 presents the results of Senv-SysApp, which consists of applying the GS search on the envelope of the HC algorithm.

Figure 4 . 2 :

 42 Figure 4.2: Behavior search of Senv-SysApp

Figure 4 .

 4 Figure 4.3 presents the results of Sitr-SysApp, which consists of optimizing the structure of the best neighbor at each HC iteration.

Figure 4 . 3 :

 43 Figure 4.3: Behavior search of Sitr-SysApp

K = 4 β

 4 opt = [0.66 0.83 0.93] M lin = 2 ; M = 1 T S = [T 1,0 T 1,1 T 4 T 5 ] with C = 35 coefficients and N = -38.80 dB.

Figure 4 . 4 :

 44 Figure 4.4: Behavior search of Ored-SysApp

Figure 4 . 5 : 5 β

 455 Figure 4.5: Behavior search of Ogen-SysApp

Figure 4 . 6 :

 46 Figure 4.6: Behavior search comparison of different SysApp

Figure 4 . 7 :

 47 Figure 4.7: Implementation of optimal DVR model by direct method

Figure 4 . 8 :

 48 Figure 4.8: Basic cell of LUT method for polynomial

Figure 4 . 9 :

 49 Figure 4.9: Architecture of LUT method to calculate polynomial

Figure 4 . 10 :

 410 Figure 4.10: Implementation of optimal DVR model by LUT

Figure 4 .

 4 Figure 4.11: NMSE according to variation of N Q

Figure 5 . 1 :

 51 Figure 5.1: Block diagram of dual-input PA

Figure 5 . 2 :

 52 Figure 5.2: Architecture of Doherty PA with frequency dependency compensating circuit

Figure 5 . 3 :

 53 Figure 5.3: Photo of dual-input Doherty PA

Figure 5 . 4 :Figure 5 . 5 :

 5455 Figure 5.4: Block diagram of testbench of dual-input Doherty PA

Figure 5 . 8 :

 58 Figure 5.8: Block diagram of digital splitter

Figure 5 .

 5 Figure 5.10: ESC scheme

Figure 5 . 11 :Algorithm 13 :

 51113 Figure 5.11: Block diagram of HHSC

Figure 5 . 12 :

 512 Figure 5.12: Flowchart of the proposed auto-tuning approach

Figure 5 . 14 :

 514 Figure 5.14: Behavior of EVM C and PAPR of v(n) versus µ

Figure 5 . 15 :

 515 Figure 5.15: Evolution of FOMs when sweeping ϕ

Figure 5 . 18 :

 518 Figure 5.18: Evolution of free-parameters over HHSC iterations

Figure 5 . 19 :

 519 Figure 5.19: Dual-input Doherty PA characteristics with 20MHz LTE signal

Figure 5 .

 5 Figure 5.20: AM-AM characteristic using RF input-output signals

Figure 5 .

 5 Figure 5.21 illustrates the HC algorithmŠs evolution in terms of C, Cond, and N , where there are a total of 1659 DVR model structures tested through 9 HC iterations. The blue dots present the neighbors tested by the HC algorithm. The red dots indicate the search path taken by the HC algorithm. The green diamond highlights the best solution.

Figure 5 . 21 :

 521 Figure 5.21: Evolution of the HC algorithm in 3D in terms of N , C and Cond for sizing the DVR model

Figure 5 . 22 :

 522 Figure 5.22: Evolution of NMSE according to DPD iterations

  Figure 5.23 shows the improvement of EVM DD and EVM CDD according to the DPD iterations. EVM C is presented as well.

Figure 5 .

 5 Figure 5.23: EVM of CFR, DPD+DUT, and CFR+DPD+DUT versus DPD iterations

  (a) IQ constellation without DPD -EVM=8% (b) IQ constellation with DPD -EVM=2.97%

Figure 5 .

 5 Figure 5.24: IQ constellation to compute EVM CDD without and with DPD

Figure 5 .

 5 Figure 5.25: ACPR L1 and ACPR U 1 versus DPD iterations

Figure 5 . 26 :

 526 Figure 5.26: Spectra of the output signal of DUT without and with DPD

Figure 5 . 27 :

 527 Figure 5.27: Characteristics of dual-input Doherty PA without and with DPD Since the DVR model is optimally sized to ensure a good trade-off between performance linearization, model complexity, and numerical stability of the identiĄcation process. The numerical properties of the optimal DVR model at the Ąnal DPD iteration are: C = 50 Cond = 10 8 ϵ = 4 where ϵ is the dynamic range of the model coefficients deĄned in 2.23.The impressive numerical properties of the DVR model used as a predistorter in this study can be conĄrmed in Figure5.28, where the real and imaginary parts of the model coefficients are depicted.

Figure 5 . 28 :

 528 Figure 5.28: Real and imaginary parts of the model coefficients Linearity FOMs are greatly improved by DPD, which closely meets the targeted linearity FOMs under the optimal conĄguration Θ opt,HHSC . However, DPD applies a BO to the dual-input Doherty PA, which may deteriorate the efficiency. At the Ąnal DPD iteration, the efficiency FOMs are:    PAE = 30.75% P out = 34.09 dBm

w 1 , 1 2 =

 112 = 0.05 ; w 1,2 = 0.01 ; w 1,3 = 0.01 w coefficients w reĆect each FOMŠs impact in the value of the cost function J, Figure 5.29 presents the weighting contribution of each FOM, of which Figure 5.29a illustrates the contribution with the initial weighting coefficients, and Figure 5.29b illustrates the contribution after updating w.

Figure 5 . 29 :

 529 Figure 5.29: Impact distribution of FOMs in the design of J

Figure 5 . 30 :

 530 Figure 5.30: Optimal pruning of free-parameters based on HC algorithm

Figure 5 . 31 :

 531 Figure 5.31: Selected Free-parameter over HC iterations

Figure 5 . 32 :

 532 Figure 5.32: Evolution of cost function J and Θ ′ over HHSC iterations

Figure 5 .

 5 33 illustrates the Ąnal architecture of the dual-input Doherty PA with CFR and DPD. The Ąnal optimal conĄguration is also highlighted. The AM-AM and AM-PM characteristics of the whole system (CFR+DPD+DUT) are shown in Figure 5.33, along with the spectra of the input-output signals and the IQ constellation of u(n) plotted in blue and y(n) plotted in red.

Figure 5 . 33 :

 533 Figure 5.33: Final optimal conĄguration with DPD

Figure 1 :

 1 Figure 1 : Compromis linéarité-rendement

Figure 2 :

 2 Figure 2 : Principe de la DPD

Tableau 1 :

 1 Comparaison des mod' eles DPD étudiées

  principe de lŠapproche proposée est basé sur la recherche GS. Pour un nombre de segments K donné, le nombre de seuils β K est K -1 : β = β 1 , β 2 , ..., β K-1 . LŠapproche proposée a été validée sur le mod' ele DVR avec T S = [T 1,0 T 1,1 T 2 T 3 T 4 ], M lin = 2, et M = 1. La Ągure 3 présente les avantages de la segmentation optimisée en comparant avec la segmentation uniforme en termes de performance de linéarisation (NMSE et ACPR) pour différents nombres de segments K.

Figure 3 :

 3 Figure 3 : Comparaison de la segmentation uniforme et la segmentation optimisée en fonction des performances de linearisation

Figure 5 : 4 k=1 1 i=0c

 541 Figure 5 : Algorithme HC avec α = 0.05

  LŠefficacité des SysApp proposées est validée en utilisant les mêmes conditions expérimentales que celles rapportées pour le dimensionnement et lŠoptimisation des seuils.La Ągure 6 compare le comportement de chaque SysApp avec lŠalgorithme HC avec une segmentation uniforme, o' u Ored-SysApp est nettement sous-optimal de 10 ' a 30 coefficients, tandis que Senv-SysApp, Sitr-SysApp et Ogen-SysApp sont presque optimaux ' a partir de 15 coefficients. En revanche, lŠapproche directe (D-SysApp) est optimale ' a partir de 10 coefficients. Les résultats de la Ągure 6 sont obtenus avec α = 0.05.

Figure 6 :

 6 Figure 6 : Comparaison du comportement de différentes SysApps

Figure 7 :

 7 Figure 7 : Implementation du modele DVR optimal par LUT

Figure 8 :

 8 Figure 8 : Block diagram of dual-input PA

Figure 9 :

 9 Figure 9 : Bloc diagrame de lŠarchitetcure proposée

Figure 11 :

 11 Figure 11 : Spectre de signal de sortie de DIDPA sans et avec DPD

Figure 12 :

 12 Figure 12 : Constellation des donnees I/Q : (a) Sans DPD; (b): Avec DPD

  2.1 Comparison of DPD models complexity . . . . . . . . . . . . . . . . . . . . 2.2 Comparison of linearization performances . . . . . . . . . . . . . . . . . . . 2.3 Comparison of implementation properties . . . . . . . . . . . . . . . . . . . 2.4 Parameters of GMP and DVR model for each bandwidth at 44 dBm output power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Linearization performance of GMP and DVR-6 with 39 coefficients . . . . 2.6 Numerical properties of GMP and DVR-6 with 39 coefficients . . . . . . . 2.7 GMP structure obtained by hill-climbing algorithm for each output power . 2.8 Comparison for each output power level of GMP and DVR models . . . . .

  Chouaib Kantana, Rui Ma, and Mouhacine Benosmane, ŞDeep Learning-based Online Adaptation of Digital Pre-Distortion and Power AmpliĄer SystemsŤ, submitted for US patent application in December 2020. 3. Chouaib Kantana, Rui Ma, and Mouhacine Benosmane, ŞInteractive Online Adaptation for Digital Pre-Distortion and Power AmpliĄer System Auto-TuningŤ, submitted for US patent application in December 2020. Mouhacine Benosmane, ŞA System Approach for the Linearization and Efficiency Enhancement of Dual-Input Doherty Power AmpliĄerŤ, submitted to IEEE Transactions on Microwave Theory and Techniques. 4. Chouaib Kantana, Olivier Venard and Genevieve Baudoin, ŞA System Approach to Design an Optimal Digital Predistortion for Power AmpliĄer LinearizationŤ, in preparation to submit to IEEE Transactions on Microwave Theory and Techniques. 5. Chouaib Kantana, Olivier Venard and Genevieve Baudoin, ŞOn the Impact of the Nonlinearity Orders on the Segmentation Approach for Digital Predistortion through the Model Sizing AlgorithmŤ, in preparation to submit to IEEE Microwave and Wireless Components Letters. Chouaib Kantana, Rui Ma, and Mouhacine Benosman, ŞA Hybrid Heuristic Search Control Assisted Optimization of Dual-Input Doherty Power AmpliĄerŤ, accepted to EuMW 2021. 2. C. Kantana, O. Venard and G. Baudoin, ŤDecomposed Vector Rotation Model Sizing by Hill-Climbing Heuristic for Digital Predistortion of RF Power AmpliĄers,Ť 2020 IEEE Topical Conference on RF/Microwave Power AmpliĄers for Radio and Wireless Applications (PAWR), San Antonio, TX, USA, 2020, pp. 22-25. 3. C. Kantana, O. Venard and G. Baudoin, ŤLinearization Performance vs Implementation Properties of Digital Predistorter Modelling for Three-Way Doherty Power AmpliĄer,Ť 2019 29th International Conference Radioelektronika (RADIOELEK-TRONIKA), 2019, pp. 1-6. (Best paper award). 4. C. Kantana, O. Venard and G. Baudoin, ŤComparison of GMP and DVR modelsŤ 2018 International Workshop on Integrated Nonlinear Microwave and Millimeterwave Circuits (INMMIC), Brive La Gaillarde, 2018, pp. 1-3.

	ACPR ADC AWG CPWL CCDF CDMA Publications and Patents Adjacent Channel Power Ratio Analog Digital Converter Arbitrary Waveform Generator Canonical Piece Wise Linear Complementary Cumulative Distribution Function Code Division Multiple Access Submitted Patents 1. Chouaib Kantana, Rui Ma, and Mouhacine Benosmane, ŞProposal linearizer-splitter for multiple-input power ampliĄersŤ, submitted for US patent application in April 2021. 2. International Conferences 1.
	CFR	Crest Factor Reduction
	CDF	Cumulative Distribution Function
	DAC	Digital Analog Converter
	DPD	Digital Predistortion
	DUT	Device Under Test
	DVR	Decomposed Vector Rotation
	DDR	Dynamic Deviation Reduction
	ESC	Extremum-Seeking Control
	EVM	Error Vector Magnitude
	FPGA	Field-Programmable Gate Array
	GA	Genetic Algorithms
	GMP	Generalized Memory Polynomial
	GS	Golden Section
	HC	Hill Climbing
	LTE	Long Evolution Term
	LS	Least Square
	MP	Memory Polynomial
	NMSE	Normalized Mean Square Error
	OFDM	Orthogonal Frequency Division Multiplex
	OP	Operating Point
	PA	Power AmpliĄer
	PAE	Power Added Efficiency
	PAPR	Peak-to-Average Power Ratio
	PDF	Probability Density Function
	QAM	Quadrature Amplitude Modulation
	RF	Radio Frequency
	SA	Simulated Annealing
	SNR	Signal to Noise Ratio
	SDR	Software DeĄned Radio
	VSA	Vector Signal Analyzer

Journals 1. Chouaib Kantana, Olivier Venard and Genevieve Baudoin, ŞThresholds Optimization of Decomposed Vector Rotation Model for Digital Predistortion of RF Power AmpliĄerŤ, Radioengineering, 30(1), 250-258. 2. Hadi Muhammad Usman, Chouaib Kantana, Pier A. Traverso, Giovanni Tartarini, Olivier Venard, Genevi' eve Baudoin, and Jean Luc Polleux. ŤAssessment of Digital Predistortion Methods for DFB-SSMF Radio-Over-Fiber Links Linearization.Ť Microwave and Optical Technology Letters 62 no. 2 (2020): 540-546. 3. Chouaib Kantana, Rui Ma, 6. Chouaib Kantana, Rui Ma, Mouhacine Benosman, Olivier Venard and Genevieve Baudoin, ŞOn the Low-Complex Optimal model of Digital Predistortion: A Stochastic ApproachŤ, in preparation to submit to IEEE Microwave and Wireless Components Letters. 5. Kantana C., Venard O., Baudoin G., ŤDigital Predistortion for RF power ampliĄers with high power efficiency and linearityŤ, Workshop International en Technologies de lŠInformation et de la Communication au service dŠune Ville intelligente, 3-4 May 2018, Marrakech, Maroc.

Table 2 .

 2

		1: Comparison of DPD models complexity
	DPD Model	Parameters	Number of coefficients
	MP	K: nonlinearity order	K

Table 2 .2: Comparison of linearization performances DPD Model NMSE (dB) ACPR L (dB) ACPR R (dB) Without DPD -10.29 -23.32 -22.46

 2

	MP	-32.65	-41.23	-38.90
	GMP	-35.39	-43.02	-42.46
	DDR-1	-29.39	-38.37	-36.26
	DDR-2	-31.81	-39.81	-38.32
	MDDRV-1	-29.39	-38.37	-36.26
	MDDRV-2	-31.82	-39.81	-38.33
	DVR-2	-32.84	-41.60	-38.95
	DVR-6	-34.94	-41.88	-41.81

Table 2 .

 2

		3: Comparison of implementation properties
	DPD	Number	Condition Coefficient dynamic
	Model	of coefficients	number	range
	MP	55	1.12e 27	2.18e 11
	GMP	43	5.24e 23	1.75e 10
	DDR-1	50	7.52e 22	4.59e 8
	DDR-2	90	3.16e 24	2.53e 9
	MDDRV-1	70	7.54e 22	4.83e 8
	MDDRV-2	110	3.72e 24	2.53e 9
	DVR-2	55	5.40e 7	1
	DVR-6	71	2.70e 9	11.65

Table 2 .

 2

	4: Parameters of GMP and DVR model for each bandwidth at 44 dBm output
	power
	Bandwidth	DVR-6

Table 2 .6: Numerical properties of GMP and DVR-6 with 39 coefficients

 2

	Bandwidth	log 10 (Condition Number) Dynamic range DVR-6 GMP DVR-6 GMP
	5 MHz	13	17	109	34887
	10 MHz	14	18	327	19700
	20 MHz	13	18	307	16278

Table 2 .

 2

	7: GMP structure obtained by hill-climbing algorithm for each output power
	GMP parameters 34 dBm 38 dBm 42 dBm 44 dBm
	K a	2	1	2	3
	L a	6	6	5	4

Table 2 .

 2 8: Comparison for each output power level of GMP and DVR models

	Output Power	34 dBm GMP DVR-6 GMP DVR-6 GMP DVR-6 GMP DVR-6 38 dBm 42 dBm 44 dBm
	NMSE (dB)
	log 10 (Condition Number) Dynamic range	9 8	11 42	11 75	10 31	13 474	10 8	17 1.18e 5	10 104

-32.58 -32.34 -36.28 -35.45 -36.34 -35.36 -39.25 -37.73 ACPR L (dB) -50.41 -49.93 -50.44 -49.43 -46.77 -45.70 -48.99 -47.46 ACPR R (dB) -49.38 -49.27 -49.85 -49.11 -46.03 -45.52 -48.35 -47.86

Table 3 .

 3 1: Comparison of linearization performances, optimal solutions and complexity

			Uniform segmentation	Brute-force	GA	GS search
	Segmentation	[0.25 0.5 0.75] [0.58 0.81 0.92] [0.59 0.81 0.92] [0.59 0.81 0.92]
	NMSE (dB)	-35.82	-38.69	-38.69	-38.69
	ACPR(dB)	L1 U1	-43.93 -44.91	-48.52 -48.83	-48.51 -48.83	-48.49 -48.82
	NMSE comp.	1	161700	2404	341

Table 3 .

 3

	2: Comparison of linearization performances, optimal solutions, and complexity
	with random selection
			Uniform	50 trials of random
			segmentation	segmentation
				Best case	Worst case
	Segmentation before starting GS	[0.25 0.5 0.75] [0.61 0.82 0.93] [0.64 0.76 0.82]
	NMSE (dB)	-35.82	-38.68	-37.68
			Apply GS search
	Optimal Segmentation	[0.59 0.81 0.92] [0.59 0.81 0.92] [0.59 0.81 0.92]
	NMSE (dB)	-38.69	-38.69	-38.69
	ACPR (dB)	L1 U1	-48.49 -48.82	-48.49 -49.82	-48.49 -49.82
		GS iter.	4	1	2
	Complexity	Interval upd.	171	43	85
		NMSE comp.	341	135(=50+85) 219(=50+169)

Table 3 .

 3 3: Probability distribution Ątting to number of NMSE computations

		Mean Var X♣p(N< X)=0.99 X♣p(N< X)=0.999
	Gamma	Experiment 1: 10 random trials × 100 225 1495 325	364
	Burr	227 3426	451	701
	Inv. Gaussian 225 1502	330	375
	Log-Logistic	224 1801	352	446
	Gamma	Experiment 2: 30 random trials × 100 221 1580 324	365
	Burr	222 2825	427	654
	Inv. Gaussian 221 1527	328	373
	Log-Logistic	216 1506	333	417
	Gamma	Experiment 3: 50 random trials × 100 171 2078 295	348
	Burr	172 2942	362	559
	Inv. Gaussian 171 2121	306	371
	Log-Logistic	170 2421	330	468

Table 3 .

 3 4: Complexity of proposed approach for each K

	Number of segments K 2	3	4	5	6	7	8	9	10
	GS iterations	2	2	4	4	5	5	7	8	9
	Interval update	21 56 171 221 346 407 642 912 1235
	NMSE computations	41 111 341 441 691 813 1283 1823 2469

Table 3 .

 3 5: Complexity of GS search for increasing K from 2 to 10

	Number of segments K 2	3	4	5	6	7	8	9	10
	GS iterations	2	2	2	3	2	1	1	1	1
	Interval update	21 57 85 162 126 72 75 90 88
	NMSE computations	41 113 169 323 256 143 149 179 175

Table 3 .

 3 6: Complexity of GS search for decreasing K from 10 to 2

	Number of segments K	10	9	8	7	6	5	4	3	2
	GS iterations	9	3	1	2	2	3	3	2	2
	Interval update	1235 309 89 151 129 162 126 57 21
	NMSE computations	2469 617 177 301 257 323 251 113 41

Table 3 .

 3

	7: Number of coefficients, NMSE, and cost function of the best DVR model at
	each HC iteration
	HC iteration Initial	1	2	3	4	5	6	7
	C	1	8	19	27	35	39	43	48
	N	-19.

33 -31.54 -36.53 -38.

02 -38.15 -38.20 -38.53 -38.59 f -19.23 -30.74 -34.63 -35.32 -34.65 -34.3 -34.23 -33.79

Table 3 .

 3 8: Parameters of DVR model structure (initial and best) at each HC iteration

	HC iteration	1	2	3	4	5	6	7	-
	Initial solut.	K	2	3	4	4	4	4	5	5

  Neigh. 64 208 217 152 126 116 127 91

	K	3	4	4	4	4	5	5
	Best solut.

Table 3 .

 3 9: Number of coefficients during the HC algorithm

	Total number of DVR models mean standard deviation
	1101	25	13
	3.5

.8.2.2.4 Comparison with Exhaustive Search

Table 3 .

 3 10: Parameters of optimal DVR model structure for d C variation

	d C	3	4	5	6	7	8	9	10	15	20	30	40	50
	K	3	4	4	4	4	4	4	4	4	4	4	4	4
	Structure

  .06 -38.02 -38.02 -37.93 -37.93 -37.04 -38.06 -38.02 -38.02 -38.02 -38.02 -38.02 f -34.50 -35.36 -35.32 -35.32 -35.23 -35.23 -34.55 -35.36 -35.32 -35.32 -35.32 -35.32 -35.32

			6	T 6	T 6	T 6	T 6	T 6	T 6	T 6	T 6	T 6	T 6	T 6
	C	21	27	27	27	27	27	25	27	27	27	27	27	27
	N	-36.60 -38

Table 3 .

 3 11: Number of neighbors versus HC iteration for each d C

	d C	Total number HC Neighbors per each HC iteration of neighbors iter.	Runtime (min)
	3 4 5 6 7 8 9 10 15 20 30 40 50	443 431 427 693 782 798 458 739 913 1009 1101 1101 1101	16 ¶24;35;40;47;19;36;20;47;48;18;18;18;18;18;18;18;1♢ 16 ¶40;67;44;29;32;25;32;18;18;18;18;18;18;18;18;17;1♢ 6 ¶48;81;79;75;44;48;52♢ 16 ¶56;95;88;58;91;44;26;26;26;26;26;26;24;4;28;14;35♢ 14 ¶64;145;143;107;44;49;35;44;44;41;14;14;14;14;10♢ 14 ¶64;145;147;108;49;44;44;43;47;41;14;14;14;14;10♢ 7 ¶64;152;83;65;26;48;19;1♢ 11 ¶64;159;133;105;40;40;40;40;40;40;37;1♢ 7 ¶64;187;183;136;108;102;71;62♢ 7 ¶64;208;193;149;125;111;90;69♢ 7 ¶64;208;217;152;126;116;127;91♢ 7 ¶64;208;217;152;126;116;127;91♢ 7 ¶64;208;217;152;126;116;127;91♢	1.20 1.09 1.01 2.06 2.69 2.41 1.20 2.54 3.24 3.68 4.30 4.35 4.31

Table 3 .

 3 12: Parameters of the optimal DVR model, C, N , and f with the variation α

	α	0.01	0.025	0.05	0.075	0.1	0.15	0.2	0.3	0.4	0.5
	K	7	6	4	4	4	4	3	4	3	2
	Structure

  -38.70 -38.02 -38.02 -38.02 -38.02 -37.01 -36.53 -34.52 -33.04 f -38.38 -37.67 -36.67 -36.01 -35.32 -33.97 -32.81 -30.83 -29.33 -28.04

										6	T 6
		T 6	T 6	T 6	T 6	T 6	T 6	T 6
	C	50	41	27	27	27	27	21	19	13	10
	N	-38.88

Table 3 .

 3 13: Complexity of HC algorithm for each α

	α	Total number HC Neighbors per HC iter. of neighbors iter.	Runtime Total number (min) of coeff.
	0.01	548	15 ¶48;74;84;49;61;58;53; 27;13;17;4;4;8;8;5♢	2.20	11619
	0.025	588	17 ¶48;74;84;49;44;48;56; 35;22;23;13;17;17;17;	2.44	13571
				17;17;6;1♢
	0.05 0.075 0.1 0.15 0.2	399 399 427 427 697	6 6 6 6 19 ¶48;81;79;75;44;67;47; ¶48;74;84;49;44;48;52♢ ¶48;74;84;49;44;48;52♢ ¶48;81;79;75;44;48;52♢ ¶48;81;79;75;44;48;52♢ 21;26;26;26;26;26;26;	1.18 1.20 1.10 1.15 2.35	14589
				24;4;31;5;14;1♢
	0.3	697	19 ¶48;81;79;75;44;67;47; 21;26;26;26;26;26;26;	2.43	14589
				24;4;31;5;14;1♢
	0.4 0.5	303 303	5 5	¶48;81;79;20;35;40♢ ¶48;81;79;20;35;40♢	0.64 0.65

Table 3 .

 3 14: Parameters of DVR model structure (initial and best) and the number of neighbors, at each iteration

		HC Iteration	1 2 3 4 5	6	-
	Initial solut.	K M lin M T S	2 2 2 3 4 0 1 2 2 2 0 1 1 1 1 -T 6 T 3 T 3 T 3 T 3 T 1,0 5 4 2 2 1 1 T 6 T 6 T 6 T 6 T 3 T 6
	Num. of neighbors 49 74 84 49 44 48	52
		K	2 2 3 4 5	4
	Best solut.				HC stop

Table 3 .

 3 [START_REF] Muller | Ofdm with reduced peak-to-average power ratio by optimum combination of partial transmit sequences[END_REF]: Parameters of DVR model with P = 9 structure at each HC iteration

	HC iteration	1	2	3	4	5	6	7	8	9	10	11	12
	Initial solution

Table 3 .

 3 [START_REF] Bauml | Reducing the Peak to Average Power Ratio of Multicarrier Modulation by Selected Mapping[END_REF]: C and N of the best DVR model with P = 9 at each HC iterations

	HC iteration Initial	1	2	3	4	5	6	7	8	9	10	11
	C	1	6	11	15	19	23	27	31	35	39	43	47
	N	-19.33 -31.52 -36.55 -37.71 -38.35 -38.66 -38.75 -38.78 -38.81 -38.85 -38.9 -38.91

Table 3 .

 3 17: Comparison of DVR-4 1 and DVR-6 9 model structures

			DVR-4 1	DVR-6 9
		K	4	2
	Model structure	β M lin M	[0.25 0.5 0.75] 2 1	[0.5] 2 1

Table 3 .

 3 18: Comparison of DVR-4 1 and DVR-6 9 in terms of modeling accuracy, com-

	plexity, and numerical properties
		DVR-4 1 DVR-6 9
	NMSE	-38.02	-38.67
	Number of coeff.	27	23
	Dynamic range	29	1499
	Condition number	10 9	10 13

Table 3 .

 3 19: Comparison of DVR-4 1 and DVR-6 9 in terms of linearization performance

		DVR-4 1 DVR-6 9
	NMSE (dB)	-38.02	-38.67
	ACPR L1 (dB)	-47.12	-47.91
	ACPR L1 (dB)	-47.59	-48.26
	EVM app (%)	0.97	0.95

Table 3 .

 3 20: Comparison of HC complexity for P = 1 and P = 9

	P = 1	P = 9

Table 4 . 1 :

 41 Optimal segmentation saved through Ored-SysApp

	K Optimal segmentation β opt
	2 [0.84]
	3 [0.59 0.87]
	4 [0.65 0.83 0.93]
	5 [0.36 0.66 0.82 0.93]
	6 [0.04 0.24 0.55 0.83 0.95]
	7 [0.11 0.27 0.51 0.65 0.83 0.95]

Table 4 .

 4

				2: Comparison of optimal solutions of system approaches
			HC algorithm	D-SysApp	Senv-SysApp Sitr-SysApp Ored-SysApp	Ogen-SysApp
		K	4	4	4	3	4	5
	Model structure	β M lin M	[0.25 0.5 0.75] [0.59 0.82 0.93] [0.55 0.78 0.92] [0.26 0.88] [0.66 0.83 0.93] [0.11 0.67 0.84 0.94] 2 2 2 2 2 2 1 1 1 1 1 1

Table 4 .

 4

		3: Complexity comparison of system approaches
		D-SysApp Senv-SysApp Sitr-SysApp Ored-SysApp Ogen-SysApp
	Evaluated DVR model	892	399	256	750	506
	HC iterations	21	6	3	17	16
	NMSE computations	51979	910	356	1100	3866
	Runtime (min)	294	2.1	0.6	4.9	8.54
	Model coefficients	74511	5952	2575	19185	28537

Table 5 .

 5

	1: Free-parameters of the proposed architecture
	Bloc	Free-parameter	Symbol Unit
	CFR	Threshold of PAPR reduction	µ	dB
	Digital splitter	Power ratio Phase shift	α ϕ	% Degree
	Transceiver	Attenuation difference	ψ	dB
	DUT	Main bias voltage Peaking bias voltage	V GS,m V GS,p	Volt Volt

Algorithm 14 :

 14 The update of weighting coefficient w Initialize weighting coefficients w = [w 1,1 w 1,2 w 1,3 w 2 w 3 w 4 ] Apply HHSC Get J opt,HHSC Measure EVM C , EVM CD , EVM CDD , ACPR L1 , and ACPR U 1 before DPD Compute ACPR=f (ACPR L1 ,ACPR U 1 ) as in (5.25) EVM HHSC,DD =EVM DD EVM HHSC,CDD =EVM CDD ACPR HHSC =ACPR Apply DPD using ILA Measure EVM C , EVM CD , EVM CDD , ACPR L1 , and ACPR U 1 with DPD Compute ACPR=f (ACPR L1 ,ACPR U 1 ) EVM HHSC+DPD,DD =EVM DD EVM HHSC+DPD,CDD =EVM CDD ACPR HHSC+DPD =ACPR Compute J HHSC+DPD if J HHSC+DPD < J opt,HHSC then

	n 1 =

EVM HHSC,DD -EVM HHSC+DPD,DD

EVMt

n 2 =

EVM HHSC,CDD -EVM HHSC+DPD,CDD EVMt

Table 5 .

 5 SA 0.62 0.14 164 • 1.54 -1.37 -1.95 0.65 Θ opt,HHSC 0.65 0.15 164 • 1.53 -1.39 -1.95 0.66

			3: Optimal free-parameters
		µ	α	ϕ	ψ	V GS,m V GS,p	J
	Θ 0	0	0.5 40 •	0	-1.5	-2.5 0.57
	Θ opt,

Table 5 .

 5 6: FOMs according to the optimal free-parametersEVM C [%] EVM DD [%] EVM CDD [%] ACPR U 1 [dB] ACPR L1 [dB] PAE [%] P out [dBm] J opt,HHSC

	w/o DPD	1.48	7.26	8	-27.44	-27.15	46.58	35.63	0.81
	w DPD	1.48	2.22	2.97	-48.71	-47.86	30.75	34.09	0.78

  EVM C [%] EVM DD [%] EVM CDD [%] ACPR U 1 [dB] ACPR L1 [dB] PAE [%] P out [dBm] J HHSC+DPD

	Optimal
	pruning	1.48	2.22	2.97	-48.71	-47.86	30.75	34.09	0.81
	of HHSC
	Update
	DPD	1.48	2.04	2.46	-49.21	-50.10	36.11	39.11	0.85
	coeff.

  P (f ) est la densité spectrale de la puissance du signal en sortie du PA.• EVM (en anglais : Error Vector Magnitude) pour mesurer lŠécart de position entre celles des symboles reçus et les positions idéales attendues. La mesure EVM est effectuée sur les données I/Q en bande de base, ce qui nécessite une démodulation du signal RF ampliĄé. Il est déĄni par :

	• ACPR (en anglais : Adjacent Channel Power Ratio) pour quantiĄer les remontées
	spectrales dans les bandes fréquentielles adjacentes au canal principal. Il est déĄni
	par le rapport des puissances entre celle du canal principal et celle dans les canaux
	adjacents, sous la forme suivante :
	ACP R (dB) = 10 log 10	2 BWc P (f )df BWg P (f )df + BW d P (f )df
	o' u

  Le projet était porté par Arelis et labellisé par les pôles Aerospace Valley (pôle principal), Images & Réseaux et Elopsys. Les partenaires impliqués dans ce projet étaient : Centrale Supelec Rennes, ESIEE -Paris, Telerad, Thales Communications & Sécurité, Université Bordeaux IMS, Université Nantes IETR, Université Poitiers XLIM. Au cours de cette th' ese, jŠai obtenu lŠopportunité dŠeffectuer un séjour scientiĄque en tant que chercheur visiteur ' a MERL (Mitsubishi Electric Research Laboratories) ' a Cambridge aux États-Unis. Cette mobilité internationale a été encadrée par le Dr Rui MA, chercheur senior au MERL. Le sujet est intitulé : ŤPrédistorsion numérique par lŠapprentissage automatique pour linéariser le PA Doherty ' a double entréeŤ.

  [START_REF] Bauml | Reducing the Peak to Average Power Ratio of Multicarrier Modulation by Selected Mapping[END_REF] dB. Ce PA est excité par un signal LTE de 20 MHz et un PAPR de 8 dB ' a une fréquence centrale de 2.14 GHz, et par une fréquence dŠéchantillonnage de 200 Ms/s.Les param' etres du mod' ele GMP sont déterminés de mani' ere optimale par lŠalgorithme HC[START_REF] Wang | ŤA Novel Algorithm for Determining the Structure of Digital Predistortion Models[END_REF]. Pour les autres mod' eles de lŠapproche globale, nous avons choisi de limiter lŠordre maximum de non-linéarité et la profondeur mémoire au même maximum que les valeurs du mod' ele GMP, soit : K = 11 et M = 4. Concernant les mod' eles de segmentation, nous Ąxons le nombre de segments K ' a 10 et M = 4 pour le mod' ele DVR-2, et K = 3 et M = 4 pour le mod' ele DVR-6.Le tableau 2 présente une comparaison générale des mod' eles du tableau 1 en fonction des performances de linéarisation, obtenues par une architecture postdistorsion.Le tableau 3 présente le nombre de coefficients de chaque mod' ele de DPD utilisés pour les résultats de linéarisation du tableau 2, leur dynamique et le conditionnement de la matrice qui intervient dans le calcul de la procédure dŠidentiĄcation.Selon les tableaux 2 et 3, le mod' ele GMP atteint les meilleures performances de linéarisation ' a 43 coefficients. Le mod' ele DVR-6 avec 3 segments obtient des performances similaires, mais avec presque le double du nombre de coefficients. Cependant, le mod' ele DVR-6 présente des propriétés numériques impressionnantes qui en font un candidat tr' es intéressant pour lŠimplémentation matérielle.AĄn de valider lŠefficacité de cette étude comparative, les mod' eles GMP et DVR-6 seront spéciĄquement comparés dans différents scénarios en utilisant un PA de technologie LDMOS 50 W avec une fréquence centrale de 1.[START_REF] Yu | ŞDigital predistortion using adaptive basis functions[END_REF] GHz et une fréquence dŠéchantillonnage de 200 Ms/s. Les tests sont effectués avec des signaux LTE avec une largeur de bande de 5, 10 et 20 MHz et un PAPR maximum de 12 dB, et ' a 4 niveaux de puissance de sortie : 34, 38, 42 et 44 dBm. Le tableau 4 présente les performances de linéarisation des mod' eles GMP et DVR-6 pour le même nombre de coefficients (39 coefficients).

	Tableau 3 : Comparaison des propriétés numériques
	Modèles		Nombre	Conditionnement	dynamique
	DPD	de coefficients			des coefficients
	MP		55		1.12e 27	2.18e 11
	GMP		43		5.24e 23	1.75e 10
	DDR-1		50		7.52e 22	4.59e 8
	DDR-2		90		3.16e 24	2.53e 9
	MDDRV-1		70		7.54e 22	4.83e 8
	MDDRV-2		110		3.72e 24	2.53e 9
	DVR-2		55		5.40e 7	1
	DVR-6		71		2.70e 9	11.65
	Tableau 2 : Comparaison des performances de linéarisation
	Modèles de DPD NMSE (dB) ACPR L (dB) ACPR R (dB) Tableau 4 : Performances de linéarisation des mod' eles GMP et DVR-6
	Sans DPD MP Largeur de bande	-10.29 -32.65 NMSE (dB) DVR-6 GMP DVR-6 GMP DVR-6 GMP -23.32 -22.46 -41.23 -38.90 ACPR L (dB) ACPR R (dB)
	GMP 5 MHz		-35.39 -39.66 -39.75 -42.43 -42.53 -40.81 -40.84 -43.02 -42.46
	DDR-1 10 MHz	-29.39 -40.31 -40.76 -44.78 -44.48 -43.43 -43.57 -38.37 -36.26
	DDR-2 20 MHz	-31.81 -38.82 -39.41 -49.38 -49.76 -48.80 -49.94 -39.81 -38.32
	MDDRV-1	-29.39	-38.37	-36.26
	MDDRV-2 Le tableau 5 présente les propriétés numériques utilisées par les deux mod' eles pour -31.82 -39.81 -38.33
	DVR-2 obtenir les résultats du tableau 4. Les propriétés numériques sont présentées par la dy--32.84 -41.60 -38.95
	DVR-6 namique des coefficients et le conditionnement de la matrice impliquée dans la procédure -34.94 -41.88 -41.81
	dŠidentiĄcation.
	Tableau 5 : Propriétés numériques des mod' eles GMP et DVR-6
	Bandwidth	log 10 (Conditionnement) Dynamique des coefficients DVR-6 GMP DVR-6 GMP
	5 MHz	13	17	109	34887
	10 MHz	14	18	327	19700
	20 MHz	13	18	307	16278

Tableau 6 :

 6 Comparaison des mod' eles GMP et DVR-6 pour chaque puissance de sortie 5.15e11 7.91e 11 8.63e 10 2.54e13 1.94e 10 5.86e17 8.47e10

	Puissance de sortie	34 dBm GMP DVR-6 GMP DVR-6 GMP DVR-6 GMP DVR-6 38 dBm 42 dBm 44 dBm
	NMSE (dB) ACPR L (dB) ACPR R (dB) Conditionnement 9.38e 9 Dynamique des coeff. -32.58 -32.34 -36.28 -35.45 -36.34 -35.36 -39.25 -37.73 -50.41 -49.93 -50.44 -49.43 -46.77 -45.70 -48.99 -47.46 -49.38 -49.27 -49.85 -49.11 -46.03 -45.52 -48.35 -47.86 8 42 75 31 474 8 1.18e 5 104

  lŠentrée et la sortie, respectivement, M lin est la profondeur de mémoire du terme linéaire, a i est le vecteur des coefficients et S est lŠensemble de termes T t utilisées dans le mod' ele DVR avec S ⊂ T = [T 1,[0,...,P ] , T 2 , T 3 , T 4 , T 5 , T 6 , T 7 ].

Tableau 7 :

 7 Comparaison des performances de linéarisation, des solutions optimales et de la complexité Dans ce scénario, lŠapproche proposée a convergé vers la même segmentation optimale que la force brute et la recherche GA, cŠest-' a-dire β opt = [0.59 0.81 0.92]. Selon le tableau 7, seules 341 évaluations de fonctions objectif sont effectuées pour optimiser les seuils, tandis que GA nécessite 2404 évaluations et la force brute nécessite 161700 évaluations pour trouver les mêmes résultats optimaux. Quant ' a la segmentation uniforme, le NMSE de lŠapproche proposée a été considérablement amélioré de pr' es de 4 dB. LŠACPR est également amélioré de pr' es de 5 dB. Ceci est conĄrmé dans la Ągure 3 pour K = 4.

			Segmentation uniforme	Force brute	GA	Approche proposée
	Segmentation	[0.25 0.5 0.75] [0.58 0.81 0.92] [0.59 0.81 0.92]	[0.59 0.81 0.92]
	NMSE (dB)	-35.82	-38.69	-38.69	-38.69
	ACPR(dB)	L1 U1	-43.93 -44.91	-48.52 -48.83	-48.51 -48.83	-48.49 -48.82
	Nombre de calculs de NMSE	1	161700	2404	341

  NMSE, notée par N , est utilisé pour évaluer les performances de linéarisation. La complexité du mod' ele est représentée par le nombre de coefficients, noté par C. La fonction de coût, notée par f , est déĄnie en combinant N et C en un seul crit' ere par une combinaison additive f = N + α.C o' u α est un param' etre positif appelé le coefficient de pondération. Une nouvelle déĄnition du voisinage est proposée pour le mod' ele DVR. Pour le mod' ele DVR, il y a quatre param' etres ' a dimensionner : le nombre de segments K, la profondeur de mémoire M lin pour le terme linéaire, la profondeur de mémoire M pour lŠensemble T S , et les termes qui seront choisis dans lŠensemble T.Dans un espace discret U , lŠélément x i qui représente une structure de mod' ele DVR est caractérisé par 3 entiers : K i , M lin,i , M i et par T S i qui est une k combinaison de lŠensemble T avec k = ¶1, ..., P +7♢. Comme ces param' etres peuvent être modiĄés indépendamment, ils composent un espace discret ' a 4 dimensions. Le voisin de lŠélément x i est déĄni comme un nİud (

•

  La motivation derri' ere lŠapproche conjointe optimale, indiquée par O-SysApp (en anglais : Optimal System Approach), est dŠeffectuer lŠalgorithme HC pour dimensionner le mod' ele DVR sur un ensemble de voisins avec une segmentation optimale, cŠest-' a-dire que la segmentation optimale des mod' eles évalués par le HC doit être disponible et stockée dans une LUT. La taille de la LUT dépend du nombre de fois que le processus dŠoptimisation de β a été exécuté. Le processus impliqué derri' ere la LUT pour stocker les seuils optimaux est effectué en conjonction avec lŠalgorithme HC, ce qui en fait une approche conjointe. Bien que le cas parfait soit celui o' u la segmentation optimale de toutes les structures est disponible et stockée, cela ne peut pas être fait puisquŠil nécessite une recherche exhaustive, ce qui conduit ' a D-SysApp. O-SysApp est développé ' a travers deux versions :-Une premi' ere version dŠapproche syst' eme appelée Ored-SysApp (en anglais :Optimal approach on Reduced parameters) qui consid' ere que la segmentation optimale nŠest pas sensible ' a la variation des param' etres M et T S . Autrement dit, les seuils β ne sont optimisés que lorsque le nombre de segments K change lors de lŠalgorithme HC. La segmentation optimale correspondant ' a chaque K est enregistrée dans la LUT. La segmentation optimale pour chaque K est stockée dans la LUT. La taille de la LUT dépend du nombre maximum de segments explorés par lŠalgorithme HC. Une segmentation β opt optimale pour un K donné couvre toutes les structures qui ont le même K, quels que soient les autres param' etres (M , T S ).-Une deuxi' eme version appelée Ogen-SysApp (en anglais : Optimal approach on Generalized parameters) dans laquelle les seuils β sont optimisés lorsque le nombre de segments K ou les termes du mod' ele T S changent par lŠalgorithme HC. Ogen-SysApp représente une version améliorée dŠOred-SysApp, o' u lŠalgorithme est similaire, sauf que lorsque les termes du mod' ele changent, le processus dŠoptimisation des seuils est appelé.

	(K)

Tableau 8 :

 8 Comparaison des solutions optimales des SysAPp opt [0.55 0.78 0.92] [0.59 0.82 0.93] [0.55 0.78 0.92] [0.26 0.88] [0.66 0.83 0.93] [0.11 0.67 0.84 0.94]

			Algorithme HC	D-SysApp	Senv-SysApp Sitr-SysApp Ored-SysApp	Ogen-SysApp
	Structure de mod' ele	K β M lin M T S	4 2 1 T 1,0 T 3 T 6	4 2 1 T 2 T 6	4 2 1 T 1,0 T 3 T 6	3 2 1 T 3 T 6	4 2 1 T 1,0 T 1,1 T 4 T 5	5 2 1 T 3 T 6
		C	27	19	27	15	35	23
		N	-38.02	-38.6	-38.85	-38.01	-38.80	-38.93
		f	-36.32	-37.65	-37.5	-37.26	-37.05	-37.78

  2, 3, 4♢ o' u w i est le poids correspondant ' a FOM i (Ągure de mérite), et FOM t,i est la valeur cible (target) de la Ągure de mérite que lŠutilisateur essaie dŠatteindre. Dans cette fonction de coût, les FOMs utilisés sont ¶EVM;ACPR;PAE;P out ♢. La conception de J pour assurer un bon compromis entre linéarité et efficacité est faite pour que le HHSC maximise J ' a 1 selon les contraintes suivantes : Les cibles de FOMs sont déĄnies selon les spéciĄcations de lŠutilisateur. Dans cette étude, Les cibles de FOMs sont : EVM t = 3%, ACPR t = -50 dB, PAE t = 100 % et P

	4 i=1	w i = 1 and max	FOM i FOM t,i	= 1	(5.33)

o' u [w 1 ; w 2 ; w 3 ; w 4 ] sont initialisés par [0.1; 0.1; 0.4; 0.4]. out,t = 40 dBm.

  1. Une étude comparative est faite entre les mod' eles de la DPD, qui sont classés selon deux approches : lŠapproche globale et lŠapproche de segmentation. La comparaison et lŠanalyse a été réalisée selon trois crit' eres : les performances de la linéarisation (NMSE et ACPR), la complexité du mod' ele présentée par les coefficients du mod' ele et les propriétés numériques présentées par la dynamique des coefficients et le conditionnement de la matrice dŠidentiĄcation. 2. Pour les mod' eles basés sur lŠapproche segmentale, notamment le mod' ele DVR, deux études principales ont été développées : lŠoptimisation des seuils et la détermination de la structure optimale. Une nouvelle approche de lŠoptimisation des seuils est proposée en utilisant une approche itérative basée sur la décomposition du probl' eme dŠoptimisation globale en un ensemble de sous-probl' emes uni-modaux nécessitant une minimisation unidirectionnelle, comme la section dorée. Concernant la détermination de la structure optimale, lŠétude de dimensionnement du mod' ele GMP par lŠalgorithme de hill-climbing a été étendu pour le mod' ele DVR avec deux aspects : le crit' ere de recherche et la déĄnition de voisinage. 3. Une approche syst' eme a été proposée pour concevoir un mod' ele DVR optimal qui est basée sur la combinaison de lŠoptimisation des seuils et du dimensionnement de la structure selon trois versions. La premi' ere version consiste ' a combiner le processus dŠoptimisation des seuils et le dimensionnement de la structure dans une recherche exhaustive globale. La seconde version consiste ' a intégrer le processus dŠoptimisation des seuils dans le dimensionnement du mod' ele mais avec une restriction : les seuils ne sont optimisés que si le nombre de segments dŠune structure de mod' ele change. La troisi' eme version est une mise ' a jour de la deuxi' eme version, dans laquelle les termes du mod' ele sont considérés comme un param' etre conduisant ' a optimiser les seuils lorsque le nombre de segments et les termes du mod' ele sont modiĄés.

my gratitude to the following people for their help and continuous support throughout my PhD.

First and foremost, I would like to thank my supervisor Professor HDR Genevi' eve BAUDOIN and my co-supervisor Professor Olivier VENARD, whose guidance and patience guided my work and assisted me in writing this thesis. They ensured my academic development and continuously supported my work and ideas throughout my three years in Université Gustave Eiffel and ESIEE Paris. his unconventional ideas, guidance, and helpful advice.

I would like to sincerely thank my thesis defense committee

DVR model according to a cost function [START_REF] Kantana | ŤDecomposed Vector Rotation Model Sizing by Hill-Climbing Heuristic for Digital Predistortion of RF Power AmpliĄers[END_REF] representing a trade-off between modeling accuracy and model complexity.

DVR Model

As mentioned in Section 2.3.4, the DVR model is based on a segmentation approach to serve as behavioral PA modeling or a predistorter model.

According to (2.8), the nonlinear basis function is constructed by vector decomposition and phase rotation. The absolute operation from vector rotation is used to generate the nonlinearity. This property allows Ątting many shapes of nonlinear behavior. The structure of the model basis function (n) is presented in Figure 3.1. In [START_REF] Zhai | ŤA ModiĄed Canonical Piecewise-Linear Function-Based Behavioral Model for Wideband Power AmpliĄers[END_REF], a modiĄed CPWL based model is proposed for modeling wideband PA. The proposed model has a structure similar to that of the DVR model but without decomposed vector rotation operation introduced in Section 2.3.4, in which the Ąrst-order basis function (n-i) is replaced by a k-order function (n-i) In another research work, the authors of [START_REF] Yang | ŤA decomposed vector rotation-based sideband distortion suppression model for wideband mmwave transmitter leakage compensation[END_REF] proposed a low-complexity sideband distortion suppression model based on the DVR model concept to construct the distortion component and eliminate the unwanted transmitter leakage.

In [START_REF] Lu | ŤA modiĄed decomposed vector rotation behavioral model for concurrent dual-band envelope tracking RF power ampliĄers[END_REF], another modiĄcation has been made to the DVR model by borrowing the GMP models concept into the structure of the DVR model for dual-band envelope tracking PA. The authors of [START_REF] Lu | ŤA modiĄed decomposed vector rotation behavioral model for concurrent dual-band envelope tracking RF power ampliĄers[END_REF] have proposed a behavioral model by combining the GMP model in (2.6) and the DVR model presented in (2.14). The optimal solution shown in Figure 3.35 corresponds to the following structure:

with C = 23 coefficients and N = -38.67 dB. In the following, this optimal structure is presented as the DVR-6 9 .

HC Complexity

In this search to size the DVR model with P = 9, the HC algorithm has 11 HC iterations, and there are a total of 1067 different DVR model structures tested in 3.73 minutes.

Table 3.15 presents the parameters of the best structures at each HC iteration. Similarly to Table 3.8, each column presents the structure of the initial solution, the number of the neighbors tested by the HC algorithm, and the structure of the best solution.

Experimental Results

The effectiveness of the proposed approaches is validated using the same experimental conditions as those reported in Section 3.3.

The HC algorithm part of the system approaches is performed with C max = 50, α = 0.05, and d C = 5 coefficients. The nonlinearity order P is set to P = 1. The different system approaches have the same stop condition as the HC algorithm with uniform segmentation.

Search Behavioral

In this section, we present the convergence behavior of each system approach where the HC algorithm results with uniform segmentation are plotted in the background.

We remind that the optimal structure returned by the HC algorithm with uniform segmentation, with C max = 50, α = 0.05, d C = 5, and P = 1 is:

with C = 27 coefficients and N = -38.02 dB.

D-SysApp

where 27 adders and 33 multipliers are required to achieve this hardware implementation.

Implementing the optimal DVR model by direct method is explicitly illustrated in Figure 4.7, where the number of multipliers and adders used has not been optimized.

LUT Method

LUT method is an efficient solution to implement the gain function of predistorter in FPGA. Figure 4.8 depicts the basic cell of the LUT method, where the magnitude of the signal is used as an index.

The corresponding value of the gain function is precomputed and retrieved in the memory with the address associated with the quantiĄed magnitude of the input sample.

Conventionally, for the polynomial models, the LUT is built only for each polynomial term [86] [58]. This method is more Ćexible since once the DPD coefficients are identiĄed, the contents of LUT are updated without changing their size. However, more memory is needed to implement LUTs, and additional multipliers and adders are needed to combine all of the polynomial terms into the Ąnal result.

On the other hand, all terms with the same memory depth M can be calculated and saved in one LUT, allowing only M + 1 LUTs. This technique considerably reduces the number of multipliers and adders. Figure 4.9 presents an example architecture of this LUT method to compute the polynomial function

be divided into two input signals, which are different in amplitude and phase, using a splitting function, so-called digital splitter, designed in baseband.

Each block has parameters to be set or controlled, which requires a design of a control engine based on an optimization approach that optimally determines these parameters to ensure an operating point of DUT exhibiting a better trade-off between efficiency and linearity. In this architecture, the DPD block will not be controlled by this control engine since the DPD technique requires linear regression techniques such as the LS method to identify the model coefficients.

The baseband notation of the input and output block is deĄned as follows. The input signal is denoted by v(n), which is the CFR input. The input of DPD is denoted by u(n), which is the CFR output. x(n) is the output of DPD, which presents the input of the digital splitter. The splitter outputs are denoted by x m (n) and x p (n), which are DUTŠs inputs. y(n) is the baseband output of DUT after the acquisition process.

Free-Parameter of CFR

In this work, the CFR technique used to reduce the PAPR is based on peak cancellation [108] [START_REF] Lopez | ŤPeak cancellation and digital predistortion of high-order QAM wideband signals for next generation wireless backhaul equipment[END_REF].

The peak cancellation is carried out through two operations: clipping and Ąltering. Its principle is illustrated in Figure 5.7.

Extremum-Seeking Control

Once SA algorithm has reached J opt,SA , the optimization procedure switches to ESC to Ąne-tune Θ opt,SA .

Generally, the adaptive control process is used to regulate linear and nonlinear systems [START_REF] Benosman | Learning-based iterative modular adaptive control for nonlinear systems[END_REF]. However, in some applications, a control process may be used to optimize an objective function of unknown parameters or keep its cost function at its extremum, as is shown in [START_REF] Ma | ŤMachine-Learning Based Digital Doherty Power AmpliĄer[END_REF]. ESC can solve optimization problems and optimize free parameters according to an unknown or uncertain cost function.

ESC is an online model-free optimization method that relies on feedback from output measurements. It is used to determine and track the optimal operating point of a given system when no model information is available.

The idea of ESC dates back to the Ąrst decades of the previous century [120] [121]. Various ESC techniques and applications have been developed and reported in the literature in the mid-twentieth century. The stability of ESC was rigorously conĄrmed in [START_REF] Krstic | ŞPerformance improvement and limitations in extremum seeking control[END_REF] where the authors proved the local stability of a near-optimal solution for a broad set of dynamical systems. The behavior of ESC is practically oriented by the local gradient of the equilibrium system. A feedback loop is then applied to drive the system to an operating point where the gradient is zero.

Joint Optimization of CFR and DUT

In the second block of our approach, we propose to optimize CFR, digital splitter, transceiver, and DUT jointly with HHSC by using the cost function J designed in the previous section. In the following, we use the acronym DUT to refer to the digital splitter, the transceiver, and DUT together. The DPD is not integrated at this level.

The free-parameters to be optimized are deĄned as:

Their searching spaces are summarized in Table 5.2.

The boundaries Θ min and Θ max are deĄned as well. The interval [Θ min Θ max ] presents the searching range of each free parameter.

Some preliminary tests, or information about the system, especially DUT from previous works, are necessary to determine the optimization interval range and avoid some critical searching range, especially for the phase.

HHSC is then ready to optimize the vector Θ starting from a given initial solution Θ 0 = [µ 0 α 0 ϕ 0 ψ 0 V GS,m 0 V GS,p 0 ], which is arbitrarily deĄned.

The optimized conĄguration is Θ opt,HHSC with its corresponding cost function J opt,HHSC .

DPD Linearization

When HHSC optimizes the free-parameters, it will be decided whether to include DPD in the optimization process or not. If not included, the cost function will be re-designed only based on the efficiency requirement as long as the low linearity speciĄcations can be targeted, assuming that a subsequent application of DPD will meet the system requirements. For the wideband signal scenarios, the linearity speciĄcations will be more challenging to meet. Therefore, it is preferred to include DPD in the auto-tuning approach.

In this study, the DVR model in (3.3) has been chosen to serve as a DPD model as it demonstrates its ability to linearize strong nonlinear behavior with memory. The DVR model has been sized to Ąnd its optimal structure using the HC algorithm presented in Chapter 3 but with uniform segmentation.

The ILA is used to identify the DPD coefficients, which are iteratively estimated by Ąnding the LS solution that minimizes the LS criterion, presented in (2.18). The DPD coefficients to be extracted are not taken as a free-parameters for this study. One of the most impressive properties of using DPD adaptively is that DPD models are linear with respect to their coefficients, allowing the use of a linear regression technique such as

The HC algorithm starts from a given initial element Θ 1 (0) ,HC at the Ąrst iteration and continually moves in the direction of the element with the best cost function value among its neighbors.

In the following, we denote Θ q (i) ,HC by Θ q (i) , and J q (i) ,HC by J q (i) .

At the q th iteration, the search procedure starts from Θ q (0) and test its neighbors Θ q (1) , Θ q (2) , . . . , Θ q (M ) , where M is the number of neighbors of Θ q (0) .

In this study, the neighborhood deĄnition is inspired by the proposed one for the DVR model in Chapter 3. The vector Θ consists of 6 free-parameters: [µ α ϕ ψ V GS,m V GS,p ]. As these free-parameters can have their values changed independently, they compose a 6-dimension space. The neighbor of element Θ is deĄned as an 6-tuple ¶µ×(1+δ µ ) ; α×(1+δ α ) ; ϕ×(1+δ ϕ ) ; ψ×(1+δ ψ ) ;

10 with i = 1, .., 6. The main property of this neighborhood deĄnition is to apply the operation of δ Θ(i) to each free parameter Θ(i) individually.

According to this deĄnition, the element Θ q (0) = [µ α ϕ ψ V GS,m V GS,p ] at the q th iteration has 12 neighbors, which are:

The element Θ q (i) with the maximized cost function J q (i) is the solution denoted by Θ q (s) . With the neighborhood deĄnition, the best solution Θ q (s) can be compared with the initial solution Θ q (0) since only one free-parameter is changed. An efficient way to do the comparison is to subtract Θ q (s) from Θ q (0) , which make it easy to locate the position of the nonzero element in the vector V = Θ q (s) -Θ q (s) . The index of the nonzero element, denoted by idx, will then be stored in LUT, and the HC algorithm moves to the next iteration as long as J q (s) is better than J q-1 (s) . Otherwise, the HC algorithm stops.

The size of LUT is 6 × 2, where 6 refers to the number of free parameters in Θ (6 inputs). The cell corresponding to each free parameter, noted by pos, is incremented when the HC algorithm Ąnds idx. Once the HC algorithm is Ąnished, the free parameter to be used in HHSC, noted by Θ ′ , is determined by the maximum incremented variable in the second column of LUT. If many free parameters have the same number of occurrences in LUT, the algorithm will take them as Θ ′ .

The optimal pruning of free parameters in HHSC is described in Algorithm 15.

Algorithm 15: Optimal pruning of free-parameters in HHSC DeĄne cost function J Choose the initial element Θ 0 = Θ opt,HHSC Set q = 1 Θ q (0) ,HC = Θ 0,HC J q (0) = J 0,HC while (1) do DeĄne number of neighbors M of Θ q (0) for i ← M do Determine the neighbor Θ q (i)

Find index idx of nonzero element in V LUT(q) = idx q = q + 1 Θ q (0) = Θ q-1 (s) else end while loop end end pos = Most frequent values in LUT Θ ′ = Θ(max(pos)) Apply HHSC to optimize J(Θ ′ ) using J with updated w

The cost function that corresponds to the optimized free-parameter Θ ′ opt , denoted by J opt,upd , will be compared to J HHSC+DPD .

If J opt,upd is better than J HHSC+DPD , HHSC has improved FOMs compared to those from the previous optimal conĄguration, and a DPD is required to linearize the DUT. Otherwise, the user must manually check whether the linearity-efficiency speciĄcations are met or not.

Experimental Results

General Consideration

The proposed auto-tuning approach for dual-input Doherty PA is tested using a 64-QAM modulated 20 MHz bandwidth LTE signal with a roll-off factor of 0.6 at f c =3 GHz with 7.5 dB of PAPR.

Following the Ćowchart in Figure 5.12, we deĄne the search range of the free parameters in Table 5.1 by setting the upper and lower bounds for each free parameter. The search range is determined empirically according to some preliminary tests:

• Threshold of PAPR reduction µ: The CFR applies a nonlinear process by clipping the input signal v(n) according to a clipping threshold µ, which causes an EVM C degradation. Figure 5.14 shows the behavior of EVM C and PAPR of the output However, this example reveals the effect of tuning α, but it is not an optimization process since the other free parameters were set to an arbitrary conĄguration and were not involved in this optimization process.

Cost function

At this point, it is necessary to determine the initial weighting coefficients w and the target FOMs.

We initialize the weighting coefficients w = [w 1 w 2 w 3 w 4 ] by:

We attribute more weights to the efficiency since DPD will be included in this test to linearize the DUT. This refers to the fact that the linearity requirements are more relaxed, as it is easier to meet with DPD, unlike efficiency.

Regarding target FOMs, they are deĄned as:

Joint Optimization of CFR and DUT

The initial solution for HHSC is deĄned as:

In this step, HHSC is performed according to Algorithm [START_REF] Popovic | Synthesis of power efficient multi-tone signals with Ćat amplitude spectrum[END_REF]. For the SA algorithm, we set T 0 = 1, T f = 0.01, C = 0.96, k bolt = 1, and SA max = 60.

The convergence of J is shown in Figure 5.17.

Figure 5.17: Evolution of cost function J over HHSC iterations

In HHSC, we set empirically 100 iterations for SA to converge, while ESC requires 20 iterations. As SA is deĄned as a stochastic optimization method, we can see from Figure 5.17 that the stochastic behavior has been exhibited in the Ąrst 60 iterations where J evolves randomly.

After 61 iterations, the SA algorithm returns the optimal solution. After 120 iterations in total, HHSC returns the optimal conĄguration Θ opt,HHSC summarized in Table 5