I wish to extend my profound sense of gratitude to my parents

Abstract

In various radar systems, a great deal of interest has been paid to selecting a waveform and designing a whole processing chain from the transmitter to the receiver to obtain the high range resolution profile (HRRP). For the last decades, radar designers have focused their attentions on different waveforms such as the pulse compression waveforms and the stepped frequency (SF) waveform:

On the one hand, three different types of wide-band pulse compression waveforms have been proposed: the linear frequency modulation (LFM) waveform, the phase coded (PC) waveform, and the non-linear frequency modulation (NLFM) waveform. They are very popular but the sampling frequency at the receiver is usually large. This hence requires an expensive analog-to-digital convertor (ADC). In addition, the PC and NLFM waveforms may be preferable in some high range resolution applications since they lead to peak sidelobe ratio (PSLR) and integrated sidelobe ratio (ISLR) better than the ones obtained with the LFM waveform.

On the other hand, when dealing with SF waveforms, a small sampling frequency can be considered, making it possible to use a cheap ADC.

Pulse compression and SF waveforms can be combined to take advantage of both. Although the standard combination of PC or NLFM with SF leads to the exploitation of a cheap ADC, the performance of the PC waveform or NLFM waveform in terms of PSLR and ISLR cannot be attained. As the PSLR and the ISLR have a great influence on the probability of detection and probability of false alarm, our purpose in the PhD dissertation is to present two new processing chains, from the transmitter to the receiver:

1. In the first approach, the spectrum of a wide-band pulse compression pulse is split into a predetermined number of portions. Then, the time-domain transformed versions of these various portions are transmitted. At the receiver, the received echoes can be either processed with a modified FD algorithm or a novel timewaveform reconstruction (TWR) algorithm. A comparative study is carried out between the different processing chains, from the transmitter to the receiver, that can be designed. Our simulations show that the performance in terms of PSLR and ISLR obtained with the TWR algorithm is better than that of the modified FD algorithm for a certain number of portions. This comes at the expense of an additional computational cost. Moreover, whatever the pulse compression used, the approach we present outperforms the standard SF waveforms in terms of PSLR and ISLR.

2. In the second approach, we suggest approximating the wide-band NLFM by a piecewise linear waveform and then using it in a SF framework. Thus, a variable chirp rate SF-LFM waveform is proposed where SF is combined with a train of LFM pulses having different chirp rates with different durations and bandwidths.

The parameters of the proposed waveform are derived from the wide-band NLFM waveform. Then, their selection is done by considering a multi-objective optimization issue taking into account the PSLR, the ISLR and the range resolution.

The latter is addressed by using a genetic algorithm. Depending on the weights used in the multi-objective criterion and the number of LFM pulses that is considered, the performance of the resulting waveforms vary.

An appendix is finally provided in which additional works are presented dealing with model comparison based on Jeffreys divergence.

INTRODUCTION

Radar has been widely used in various civil and military applications that include police traffic radar, weather radar, earth resource monitoring, tracking of aircrafts and many others. Although hardware and software limitations play a role in the design of a radar system, the goal of each radar application has a great influence on the selection of the radar waveform parameters by the designer. For the last decades, a great deal of interest has been paid to obtaining a high range resolution in various radar applications, from synthetic aperture radar (SAR) and ground penetrating radar (GPR) to radar target recognition. The key way to obtain a high range resolution is to select a wide-band waveform. For this purpose, two families can be considered.

On the one hand, waveforms with large instantaneous bandwidths can be used. The most popular one is the pulse compression waveform. It consists of a train of modulated pulses transmitted on the same carrier frequency. These pulses are internally modulated in phase or in frequency. In the literature, three different pulse compression waveforms exist: the linear frequency modulation (LFM) waveform, the phase coded (PC) waveform, and the non-linear frequency modulation (NLFM) waveform. The received echoes of this family of waveforms are processed by using the matched filter (MF) to produce the high range resolution profile (HRRP). The latter is representative of the reflectivity of the target to an HRR radar waveform projected onto the radar lineof-sight. Its mainlobe width and its sidelobe levels are characterized by the correlation function of the waveform. When dealing with LFM waveforms, the sidelobe levels are high. Hence, an amplitude windowing is generally combined with a MF in the frequency domain to reduce them. This comes at the cost of a smaller signal-to-noise ratio (SNR) at the output of the MF and an increase in the width of the mainlobe of the HRRP, and consequently a degradation in the range resolution [START_REF] Richards | Principles of modern radar[END_REF]. For some PC and NLFM waveforms, there is no need to apply the amplitude windowing to the MF output since the sidelobes of the HRRP are sufficiently low.

Based on Nyquist criterion, the high instantaneous bandwidth of this family of waveforms leads to a high sampling rate at the receiver, and hence, an expensive analog-todigital converter (ADC) is required.

On the other hand, a waveform that consists of a train of externally modulated pulses can be considered. This waveform is known as the stepped-frequency (SF) waveform [START_REF] Lord | Aspects of stepped-frequency processing for low-frequency SAR systems[END_REF] [116] [START_REF] Melvin | Principles of modern radar[END_REF]. In this case, a train of pulses that have a small instantaneous bandwidth is transmitted on different equidistant carriers. This makes it possible to exploit a cheaper ADC due to the small sampling rate. At the receiver, high range resolution is obtained by synthesizing the wide bandwidth from the received echoes. The latter can be processed with the MF to produce the HRRP. However, its computational cost is high. In addition, grating lobes may appear in the HRRP [START_REF] Gladkova | Grating lobes suppression in stepped-frequency pulse train[END_REF]. One alternative can be seen as a kind of stretch processing. It includes three different algorithms, namely the IFFT, the time domain (TD), and the frequency domain (FD) algorithms [START_REF] Lord | Aspects of stepped-frequency processing for low-frequency SAR systems[END_REF]. The FD and the IFFT algorithms have computational costs smaller than that of the MF-based approach. However, they have some limitations. The IFFT algorithm produces ghost targets in the HRRP of extended targets [START_REF] Lord | Aspects of stepped-frequency processing for low-frequency SAR systems[END_REF]. The TD algorithm [START_REF] Lord | High resolution SAR processing using stepped frequencies[END_REF] does not produce ghost targets but suffers from the up-sampling requirement of the pulses. Finally, the FD algorithm can cope with the drawbacks of the TD and the IFFT algorithms, but a DFT must be computed on a relatively large number of samples. Nevertheless, the recent advances in designing and fabricating powerful processors can facilitate its implementation.

The pulse compression and SF waveforms can be combined to take advantage of the features of both and produce a waveform for which the sampling rate and the number of pulses within the coherent processing interval are reduced. The standard combination consists in transmitting the train of pulses of a pulse compression waveform on different equidistant carriers. In the literature, SF-LFM waveform has been well studied [START_REF] Zhai | Application of super-SVA to stepped-chirp radar imaging with frequency band gaps between subchirps[END_REF] [120] [START_REF] Chua | Phase coded stepped frequency linear frequency modulated waveform synthesis technique for low altitude ultra wideband synthetic aperture radar[END_REF] [66] [69] [START_REF] Luo | Micro-Doppler effect analysis and feature extraction in ISAR imaging with stepped-frequency chirp signals[END_REF]. However, to the best of our knowledge, few papers deal with SFPC and SF-NLFM waveforms [62] [37].

For the evaluation of radar waveforms, a detection test is usually designed and its performance in terms of probability of false alarm (PFA) and probability of detection (PD) must be evaluated [START_REF] Richards | Principles of modern radar[END_REF]. Various authors have worked on waveform optimization based on several criteria. The reader may refer to [START_REF] De Maio | Code optimization with similarity and accuracy constraints[END_REF] [75] [START_REF] Rouffet | Combining two phase codes to extend the radar unambiguous range and get a trade-off in terms of performance for any clutter[END_REF] for instance. In this thesis, we focus our attention on three of the performance measures that have great influences on the PD and the PFA. They characterize the HRRP and are known as the peak sideloberatio (PSLR), the integrated sidelobe ratio (ISLR), and the range resolution [32] [119].

In various high-resolution radar applications, PC waveform or NLFM waveforms can be used to take advantage of their features in terms of PSLR and ISLR. Although the standard combination of these waveforms with SF waveform reduces significantly the sampling rate at the receiver, their features in terms of PSLR and ISLR cannot be attained. This is due to the fact that the instantaneous frequency of the concatenated pulses of the transmitted waveform does not correspond to a wide-band PC or NLFM pulse.

In the light of the aforementioned problems, the work done in this thesis is decomposed into two parts: The first one aims at overcoming the shortcoming when combining a SF scheme with a PC waveform or NLFM waveform. The second part proposes another approach to overcome the drawback of the standard combination of SF with an NFLM waveform. To address these two issues, this thesis is organized as follows:

In chapter 1, we briefly recall the principles of radar operation, radar modeling at the transmitter and receiver sides, and the principles of threshold detection. Then, highresolution radar waveforms including pulse compression waveforms and SF waveforms are presented. The way to process them at the receiver is also recalled. Finally, performance measures related to the range profile are defined.

In chapter 2, our first contribution is presented: it consists in designing a processing chain from the transmitter to the receiver dealing with a SF scheme combined with one of the pulse compression waveforms. More particularly, the spectrum of a wide-band pulse compression pulse is split into a predetermined number of portions. Then, the time-domain transformed versions of these various portions are transmitted. At the receiver, a modified FD algorithm and a time waveform reconstruction (TWR) algorithm are proposed to process the received signals. After presenting this chain from a theoretical point of view, various simulation results are given. They make it possible to compare the performance of the different processing chains that can be defined.

The third chapter deals with the design of a variable chirp rate SF-LFM waveform.

More particularly, we suggest approximating a wide-band NLFM by a piecewise linear waveform and then using it in a SF framework. Thus, a variable chirp rate SF-LFM waveform is proposed where SF is combined with a train of LFM pulses having different chirp rates with different durations and bandwidths. In this PhD dissertation, the parameters of the proposed waveform are derived from the wide-band tangent-based NLFM waveform. Then, they are adjusted by means of a multi-objective optimization issue. The multi-objective criterion is defined from the PSLR, the ISLR, and the range resolution that could be obtained with respect to the ones that characterize the wideband NLFM waveform. It should be noted that the optimization issue is addressed by using genetic algorithms.

This dissertation ends up by drawing some conclusions concerning the various approaches that have been tackled and highlighting some perspectives on how to develop our work in the future.

It should be noted that the PhD dissertation also includes an appendix dealing with Jeffreys divergence for model comparison. This additional work consists in analyzing the Jeffreys divergence between different types of processes: sum of complex exponentials disturbed by additive white noises, autoregressive moving average processes (ARMA) and long-memory processes such as fractionally integrated (FI) white noises and ARFIMA processes. Although this topic is not directly related to the core of the PhD dissertation, it is representative of the work done before and at the beginning of the PhD.

CHAPTER 1

Radar waveforms and Signal Processing Overview

Introduction

The word "Radar" was originally an acronym that stood for radio detection and ranging. This acronym summarizes two main tasks of a radar system: detecting a target and determining its position. Nowadays, modern radar systems are not confined to the latter task, but also deal with tracking, identifying and classifying targets. Thus, radar systems are used in civilian and military applications that include police traffic radar, weather radar, air traffic control, collision avoidance, two-and three-dimensional mapping, earth resources monitoring, missile guidance, tracking of aircrafts and ballistic missiles and many others. In this chapter, the basics of a monostatic pulsed radar including operating principles, signal modeling, and threshold detection are briefly reviewed. Then, high range resolution waveforms including pulse compression waveforms and stepped-frequency waveforms are presented. Finally, as we will focus our attention on the range profile, three performance measures, namely the peak sidelobe ratio (PSLR), the integrated sidelobe ratio (ISLR), and the range resolution are defined.

Radar concepts

A radar system radiates electromagnetic waves in a region of interest. If there are some objects, these waves are then reflected back toward the system. Finally, the backscattered signals that are received by the radar antenna are processed. The monostatic radar system consists of different subsystems, as shown in Fig. 1.1. Thus, the transmitter is used to generate electrical radar signals. The antenna is the interface between the radio waves propagating through the space and the transmitter/receiver (T/R). The T/R device plays the role of connecting the transmitter and the receiver to the same antenna by providing isolation between them to protect the sensitive receiver components from the high-powered transmit signal. The electromagnetic waves reflected in the direction of the radar are picked up by the antenna and routed into the receiver in the form of an electrical signal. In the receiver, this signal usually passes through different stages [START_REF] Richards | Principles of modern radar[END_REF]: first, it is amplified by using a low-noise amplifier (LNA). Then, it is converted to an intermediate frequency (IF) by using a mixer and a local oscillator. Then, the IF signal is amplified by means of an IF amplifier. Afterward, the IF signal is down-converted to baseband using the detector. Finally, the baseband signal is digitized using an analog-to-digital converter (ADC) where the output of the receiver is applied to the signal processor. The primary function of the signal processor is to detect the presence of a target in spite of disturbances. The latter may include:

• The thermal noise, the electromagnetic waves reflected from the ocean or the ground, known as clutter,

• An intentional jamming in the form of noise or false targets,

• The electromagnetic disturbance created by other human-made sources such as television broadcast station and communication systems.

Generalities about the transmitted and received signals

In general, a pulsed radar waveform is generated in two steps: first, a train of N p baseband rectangular pulses separated by a pulse repetition time T r is produced. Then, these pulses are modulated with the same sinusoidal carrier. Thus, by assuming that the duration of each pulse is denoted by T p , the transmitted waveform whose total bandwidth is equal to B can be described, for the m th pulse (m ∈ 1, N p ) and for the time instant satisfying (m -1)T r < t < (m -1)T r + T p , as follows:

s tx (t) = Np m=1 s tx,m (t) = Np m=1 v bb,m (t)exp(j2πf c t) (1.1)
where s tx,m (t) is the m th transmitted pulse, f c is the carrier frequency and v bb,m (t) is the m th baseband pulse. When unmodulated pulses are considered, v bb,m (t) is shown in Fig. 1.2 and defined by:

v bb,m (t) = A rect t -(m -1)T r -T p /2 T p (1.2)
where A is the amplitude of each pulse and rect(t) stands for a rectangular pulse equal to 1 for -1 2 ≤ t ≤ 1 2 and zero elsewhere. It should be noted that during the transmission time, the pulsed radar system does not receive any echoes. The receiving time starts at the end of each transmitted pulse. Thus, the minimum range through which the target can be detected by the radar is defined by:

R min = cT p 2 (1.3)
The minimum range is an attractive feature for short-range radar applications such as automotive radar and it should be as small as possible.

Fig. 1.2 Radar transmission waveform

Let us now briefly present what happens when the transmitted pulses hit an obstacle located at a distance R from the radiating antenna. Let us assume that the power at the transmitter is P t and the antenna is isotropic. The power density Q at a distance R from the radiating antenna is the transmitted power divided by the surface area of a sphere of radius R:

Q = P t 4πR 2 (1.4)
If a directional antenna is used instead of an isotropic one, the power density at the center of the antenna beam pattern is higher than the one obtained with an isotropic antenna, because the transmitted power is concentrated onto a smaller area on the surface of the sphere. In this case, the power density is given by:

Q = GP t 4πR 2 (1.5)
where G is the gain of the antenna. The transmitted signal that hits the target induces time-varying currents so that the target becomes a source of radio waves that propagate in various directions and more particularly in the direction of the radar. The power reflected back to the radar denoted P ref l is the product of the Q and the radar cross section (RCS) of the target, denoted as σ. The latter is determined by many parameters such as the physical size, the shape, and the materials of the target. Therefore, the reflected power can be expressed as follows:

P ref l = σQ = σGP t 4πR 2 (1.6)
Then, the power density at the radar receiving antenna Q r is given by:

Q r = P ref l 4πR 2 = σGP t (4π) 2 R 4
(1.7)

The power received by the antenna with an effective area A e is expressed as the product of the received power density and the effective area of the antenna:

P r = Q r A e = σGA e P t (4π) 2 R 4 (1.8)
The relation between the transmitted power and the received power in (1.8) is developed in the ideal case where no losses exist. However, in real cases, losses in the propagation path and losses between transmitter/receiver and the antenna exist. If L represents the total system loss, (1.8) becomes:

P r = Q r A e = σGA e P t (4π) 2 R 4 L = K 2 P t (1.9)
where K is the gain of the received signal. Given the above phenomenon and the time it takes for the signal to propagate a distance R and return, the received signal is a delayed version of the transmitted signal usually affected by a disturbance. As mentioned above, the latter usually includes the thermal noise and might include clutter echoes, electromagnetic disturbance from other transmitting sources, and intentional jamming. Therefore, for a stationary point target1 at range R, the received signal can be modeled as follows:

s rx (t) = Np m=1 s rx,m (t) = Np m=1 K m s tx,m (t -t d ) + η(t) = Np m=1 K m v bb,m (t -t d )exp(j2πf c (t -t d )) + η(t) (1.10)
where s rx,m (t) is the m th received pulse, and K m is the gain of the m th received pulse.

In addition, if c = 3 × 10 8 m/s is the speed of light, t d = 2R c is the time delay that corresponds to the range of the target R. Since the target is stationary, t d is the same from the 1 st pulse to the N th p pulse. Finally, η(t) represents the disturbance. Then, the received signal is down-converted to baseband by multiplying it by the reference signal s ref,m (t) = exp(-j2πf c t). This process is called a demodulation process. After demodulation, the received baseband signal becomes:

s rx,bb (t) = s rx (t)s ref,m (t) = Np m=1 s rx,m (t)s ref,m (t) = Np m=1 s rx,bb,m (t) = Np m=1 K m v bb,m (t -t d )exp -j2πf c 2R c + η(t)exp(-j2πf c t) (1.11)
where s rx,bb,m (t) denotes the m th received baseband pulse.

The received baseband signal in (1.11) is then sampled at a frequency that respects the Nyquist criterion, by using an ADC. If B b is the baseband bandwidth and B p = 2B b is the passband bandwidth, the sampling frequency at the receiver can be set at:

F (Rx) s = B p = 2B b (1.12)
Finally, the sampled data is processed.

Remark: When a train of similar pulses are transmitted on the same carrier frequency, the bandwidth satisfies:

B = B p (1.13)
In most radar systems, the traditional approach to process the sampled data at the receiver consists in applying a matched filter (MF). The latter is a linear filter used to maximize the instantaneous signal-to-noise ratio (SNR). Its impulse response is proportional to the time-reversed and complex conjugated copy of the transmitted signal.

In the absence of Doppler effect, the shape of the MF output corresponds to a shifted version of the autocorrelation function of the transmitted signal. For more information about the derivation of the matched filter impulse response, the reader may refer to [START_REF] Richards | Principles of modern radar[END_REF]. At the MF output, a range profile is created. It is a representative of the reflectivity of the target to a radar waveform projected onto the radar line-of-sight. This 1-D signature makes it possible to estimate the target size and the positions of some scattering points of the target structure. The range profile of a single point scatterer is characterized by a mainlobe and sidelobes. The center of the mainlobe corresponds to the position of the point scatterer whereas the sidelobes are undesirable lobes that should be as low as possible.

Normally, in radar processing, a threshold detection test is applied to the output of the MF to declare the presence or absence of targets. In the next section, an overview of threshold detection is presented.

1.4 Overview of threshold detection 1.4.1 Threshold detection concept

In radar systems, the detection is the decision whether a given signal return is the result of an echo from a target or a disturbance. Detection is done automatically in the signal processor where a threshold level is set based on the voltage of the disturbance. The voltage of the returned signal is compared with the threshold. A target is declared present if and only if the returned signal exceeds the threshold. In some cases when no target is present, the return from the disturbance may exceed the threshold. This leads to the creation of a false alarm. In practice, as the voltage of the disturbance varies, a fixed threshold level cannot be considered. Therefore, the threshold should be set adaptively to obtain a constant false alarm rate [START_REF] Richards | Principles of modern radar[END_REF].

Probabilities of false alarm and detection

In the presence of a target, the input signal at the receiver consists of a delayed version of the transmitted signal combined with the disturbance. The latter can be modeled by a randomly varying voltage. Even if the delayed signal is modeled as a constant voltage, the output voltage of the receiver varies randomly due to the inherited thermal noise. Therefore, the process of detecting the presence of a target on the basis of the received signal voltage is a statistical process. The resulting radar detection performance is usually characterized by the probability that a target is detected, called the probability of detection (PD), and the probability that detection will be declared when no target is present, called the probability of false alarm (PFA). When the detection threshold decreases, both the detection probability and the false alarm probability increase. In order to enhance the PD while maintaining a constant PFA, the signal-to-interference ratio (SIR) must be increased. The latter is defined as the ratio between the power of the transmitted signal and the power of the disturbance signal. Radar detection algorithms are usually designed according to the Neyman-Pearson criterion. This rule fixes the PFA that is allowed by the detection processor and then maximizes the PD, for a given SIR. Let us assume that the disturbance signal present at the receiver corresponds only to the thermal noise. The latter is independent and identically distributed (IID), zero-mean Gaussian with variance σ 2 n . Under H 0 , one has [START_REF] Kay | Fundamentals of Statistical Signal Processing[END_REF]:

p s M F (t te )|H 0 = 1 πσ 2 n exp - |s M F (t te )| 2 σ 2 n (1.14)
Under H 1 , and in the presence of a non-fluctuating target, let us assume that the measurement s M F (t te ) is represented by a complex sample Cexp(jθ) added to the noise sample where C is a constant that represents the amplitude of the MF at t te . It depends on K m and the amplitude of η(t). The phase θ of the target echo can be modeled as a random variable distributed over (0,2π) and independent of C. Using the Bayesian approach for random parameters, it can be shown that p s M F (t te )|H 1 is Rician distributed 2 [START_REF] Kay | Fundamentals of Statistical Signal Processing[END_REF]:

p s M F (t te )|H 1 = 1 πσ 2 n exp - 1 σ 2 n (|s M F (t te )| 2 + C 2 ) J 0 2C|s M F (t te )| σ 2 n (1.15)
where J 0 is the Bessel function of the first kind of order zero defined by:

J 0 (x) = +∞ i=0 (-1) i x 2i 2 2i (i!) 2 , x > 0 (1.16) 2 If x = x 2 1 + x 2 2 with x 1 ∼ N (µ 1 , σ 2 ) and x 2 ∼ N (µ 2 , σ 2 )
, the rician PDF of x is:

p(x) =    x σ 2 exp -1 2σ 2 (x 2 + α 2 ) J 0 αx σ 2 x > 0 0 x < 0 where α 2 = µ 2 1 + µ 2 2 and J 0 (x) = +∞ m=0 (-1) m x 2m 2 2m (m!) 2 with x > 0
Given (1.14) and (1.15), the logarithmic-likelihood ratio test can be written as:

ln(L R ) = ln p s M F (t te )|H 1 p s M F (t te )|H 0 = ln J 0 2C|s M F (t te )| σ 2 n - C 2 σ 2 n ≷ H 1 H 0 ln T
(1.17) where T is the detection threshold. Rearranging the terms leads to:

ln J 0 2C|s M F (t te )| σ 2 n ≷ H 1 H 0 C 2 σ 2 n + ln T (1.18)
In (1.18), ln J 0

2C|s M F (tte)| σ 2
n is a monotonic increasing function. Thus, the same detection results can be obtained by comparing its argument

2C|s M F (tte)| σ 2 n with a modified threshold T m = σ 2 n 2C J -1 0 (T e C 2 σ 2
n ). In this case, the detection test is:

|s M F (t te )| ≷ H 1 H 0 T m (1.19) Let z = |s M F (t te )|, then the detection test becomes z ≷ H 1 H 0 T m .
In this case, the distribution of z under the two hypothesis is required. Under H 0 , z is Rayleigh distributed [START_REF] Kay | Fundamentals of Statistical Signal Processing[END_REF]:

p z (z|H 0 ) =    2z σ 2 n exp(-z 2 σ 2 n ) z ≥ 0 0 z < 0
Therefore, the probability of false alarm is given by:

P F A = +∞ Tm p z (z|H 0 )dz = exp(- T 2 m σ 2 n ) (1.20)
The threshold T m can be calculated by inverting (1.20) as follows:

T m = σ n -ln(P F A ) (1.21)
On the other hand, when H 1 is considered, it can be shown that the PDF of z is given by:

p z (z|H 1 ) =    2z σ 2 n exp -z 2 +C 2 σ 2 n J 0 C 2 z σ 2 n z ≥ 0 0 z < 0 (1.22)
Given (1.22), the probability of detection is hence: . The integral in (1.23) becomes:

P D = +∞ Tm p z (z|H 1 )dz = +∞ Tm 2z σ 2 n exp - z 2 + C 2 σ 2 n J 0 2C 2 z σ 2 n dz (1.
P D = +∞ 2T 2 m σ 2 n xexp - x 2 + a 2 2 J 0 (ax)dx = Q M 2AC 2 σ 2 n , 2T 2 m σ 2 n (1.24)
where Q M is the Marcum's function 3 which can be evaluated iteratively since no closed form is known for it. Finally, using (1.21) and knowing that C 2 σ 2 n is the SNR, (1.24) reduces to:

P D = Q M √ 2SN R, -2 ln(P F A) (1.26)
In Fig. 1.3, PD versus PFA for different values of the SNRs is illustrated. For a given PFA, the PD increases as the SNR increases. Thus, in order to increase the PD, it is required to increase the SNR. Besides the SNR, the range resolution and the level of the sidelobes should also be taken into account since they have a great influence on the PD and PFA. These metrics can be examined using the radar ambiguity function. 3 The Marcum's function is defined by:

Q M (a, b) = +∞ b xexp - x 2 + a 2 2 J 0 (ax)dx (1.25)

Radar ambiguity function

The radar ambiguity function is an analytical tool used by the designers to study different radar waveforms. More particularly, it is used to analyze the response of the matched filter in the presence of moving targets. In addition, it is useful for examining range and Doppler resolutions, sidelobe behaviour, and range and Doppler ambiguities for any radar waveform processed with a matched filter. The ambiguity function of a radar waveform is defined as [START_REF] Richards | Principles of modern radar[END_REF]:

AF (τ, f d ) = +∞ -∞ s tx (t)s tx (t -τ )exp(j2πf d t)dt (1.27)
where f d is the Doppler frequency that corresponds to the velocity of the target, and () is the conjugate operator. For f d = 0, the ambiguity function returns the auto-correlation function of the waveform. It should be noted that the shape of the ideal4 range profile of a stationary point scatterer can be deduced from the radar ambiguity function depending on the range and the velocity of the target. As the radar waveform has an influence on the ambiguity function, let us focus on high range resolution radar waveforms in the following section.

High range resolution radar waveforms

Range resolution is a measure of the ability of a radar to discern between two closely spaced objects along the line of sight of the radar [START_REF] Wehner | High resolution radar[END_REF]. As the range resolution is inversely proportional to the bandwidth of the transmitted waveform, the larger the bandwidth the higher the range resolution. High range resolution is required in various applications from ground penetrating radar (GPR), and synthetic aperture radar (SAR) to target recognition. The key way to attain high range resolution is to select a wide bandwidth waveform. In this case, a highresolution range profile (HRRP) is obtained. For this purpose, two families can be considered:

1. On the one hand, a wide bandwidth waveform exploiting one of the pulse compression techniques can be considered. In this case, it possesses a high baseband bandwidth, and hence based on Nyquist criterion, a high sampling rate and consequently an expensive ADC are required. This type of waveforms is usually processed by using the MF.

2. On the other hand, a stepped-frequency (SF) waveform that exhibits small baseband bandwidth can be considered, and hence a simple ADC can be exploited [START_REF] Lord | Aspects of stepped-frequency processing for low-frequency SAR systems[END_REF] [116] [START_REF] Melvin | Principles of modern radar[END_REF]. In addition, at the transmitter, the waveforms small baseband bandwidth lessens the dispersion effects in some systems (e.g., a wideband, phased array radar that does not employ time delay units) [START_REF] Melvin | Principles of modern radar[END_REF]. With SF waveforms, the wide bandwidth is synthesized at the receiver. This waveform can be processed by using various algorithms.

In the following subsection, let us recall the modeling and some properties of the pulse compression waveforms.

Pulse compression waveforms

In various radar applications, it is desirable to have a high range resolution while maintaining adequate average transmitted power. The latter feature contributes well to generating a low probability of intercept (LPI) radar waveform [START_REF] Pace | Detecting and Classifying Low Probability of Intercept Radar[END_REF]. This can be achieved by using a technique known as pulse compression. In this technique, the pulses are internally modulated in frequency or phase and this modulation is called the intra-pulse modulation. In the literature, three main pulse compression waveforms exist, namely, the linear frequency modulation (LFM) waveform, the phase coded (PC) waveform, and the non-linear frequency modulation (NLFM) waveform. When dealing with a stationary target, the MF output has the shape of the autocorrelation function of the pulse compression used. Let us introduce the three pulse compression waveforms in the subsections to come.

Linear frequency modulation waveforms

A linear frequency modulation waveform consists of a sinusoid whose instantaneous frequency changes linearly over time. It can be represented by (1.1) where the m th baseband pulse is given by:

v bb,m (t) = A rect t -(m -1)T r -Tp 2 T p exp jπγ(t -(m -1)T r - T p 2 ) 2 (1.28)
where the chirp rate γ is given by:

γ = B p T p = (1.13) B T p (1.29)
In addition, the instantaneous frequency of each pulse is equal to:

f (t) = γ(t -(m -1)T r - T p 2 ) (1.30)
The instantaneous frequency of an LFM pulse with B p = 100 MHz and T p = 20 µs is presented in Fig. 1.4. In what follows, we will focus our attention on the spectrum and the correlation properties of the m th pulse. In order to find the spectrum of the m th pulse, let us consider the first pulse v bb,1 (t)= rect( t-

Tp 2
Tp )exp jπγ(t -Tp 2 ) 2 . Its Fourier transform can be expressed as follows:

V bb,1 (f ) = Tp 0 exp jπγ(t - T p 2 ) 2 exp(-j2πf t)dt (1.31)
After substituting the variable u = t -Tp 2 in (1.31), one has:

V bb,1 (f ) = exp(-jπf T p ) Tp 2 - Tp 2 exp jπγu 2 exp(-j2πf u)du (1.32) = exp(-jπf T p ) Tp 2 - Tp 2 exp(jπγ(u 2 -2 f γ u + f 2 γ 2 ))exp(-jπγ f 2 γ 2 )du = exp(-jπf T p ) Tp 2 - Tp 2 exp(jπγ(u - f γ ) 2 )exp(-jπ f 2 γ )du
At this stage, let us set x at √ πγ(u -f γ ). (1.31) becomes:

V bb,1 (f ) = 1 √ πγ exp(-jπ f 2 γ )exp(-jπf T p ) √ πγ( Tp 2 -f γ ) √ πγ(- Tp 2 -f γ ) exp(jx 2 )dx (1.33)
Let us introduce the Fresnel integrals as follows: Combining the above equations, (1.33) becomes:

C(u) = u 0 cos(x 2 )dx (1.
V bb,1 (f ) = 1 √ πγ exp(-jπ f 2 γ )exp(-jπf T p ) × C √ πγ( T p 2 + f γ ) (1.37) + jS √ πγ( T p 2 + f γ ) + C √ πγ( T p 2 - f γ ) + jS √ πγ( T p 2 - f γ )
As v bb,m (t) is a shifted version of v bb,1 (t), the Fourier transform of the m th pulse can expressed in terms of V bb,1 (f ). It is given for m ∈ 1, N p by:

V bb,m (f ) = exp -j2πf (m -1)T r V bb,1 (f ) (1.38)
Therefore, the power spectrum of the m th pulse can be obtained by taking the square of the modulus of (1.38). Thus, the N p LFM pulses have the same power spectrum given by:

V (f ) = 1 πγ C √ πγ( T p 2 + f γ ) + C √ πγ( T p 2 - f γ ) 2 (1.39) + S √ πγ( T p 2 + f γ ) + S √ πγ( T p 2 - f γ ) 2
Let us now express the autocorrelation function of v bb,1 (t). It is defined by:

R v (τ ) = +∞ -∞ v bb,1 (t)v bb,1 (t -τ )dt (1.40)
When -T p ≤ τ ≤ 0, after some mathematical developments and simplifications, one has:

R v (τ ) = τ +Tp 0 exp(-jπγτ 2 )exp j2πγτ (t -T p /2) dt = exp(-jπγτ 2 )exp(-jπγτ T p ) τ +Tp 0 exp j2πγτ t dt = (T p + τ )sinc γτ (T p + τ ) (1.41)
where sinc(x) = sin(πx)/πx. When 0 ≤ τ ≤ T p , similar analysis can be carried out where the integration limits are from τ to T p . In this case, one has R v (τ ) = (T p -τ )sinc γτ (T p -τ ) . Therefore, the expression of R v (τ ) for |τ | ≤ T p is given by:

R v (τ ) = (T p -|τ |)sinc γ|τ |(T p -|τ |) (1.42)
Given (1.42), one can deduce that the HRRP of a stationary target obtained with LFM waveform has the shape of a sinc function multiplied by a triangular function. Thus, the sidelobes of the HRRP are relatively high. To reduce them, an amplitude windowing is generally applied in the frequency domain just before producing the HRRP. This comes at the cost of a smaller SNR at the output of the MF and an increase in the width of the mainlobe of the HRRP [START_REF] Richards | Principles of modern radar[END_REF].

In the following subsection, phase coded waveforms are presented.

Phase coded (PC) waveforms

The phase coded (PC) waveform is represented by (1.1) where the m th baseband pulse is given by:

v bb,m (t) = M -1 i=0 A rect t -iT c -mT r -T c /2 T c exp(jφ i ) (1.43)
where M is the length of the phase code, T c = Tp M is the duration of each subpulse, and φ i (i ∈ 0, M -1 ) represents the phase sequence of the code. It should be noted that φ i intentionally varies from one subpulse to another in order to achieve the desired mainlobe and sidelobe response for the autocorrelation function of the phase code. In the literature, phase codes are grouped into two categories:

• Bi-phase coded waveforms which exhibit two possible phase states: 0 and π [START_REF] Lindner | Binary sequences up to length 40 with best possible autocorrelation function[END_REF] [18] [START_REF] Coxson | Efficient exhaustive search for optimal-peak-sidelobe binary codes[END_REF] [53].

• Polyphase coded waveforms which exhibit more than two phase states. This is the case of the polyphase Barker codes [START_REF] Nunn | Polyphase pulse compression codes with optimal peak and integrated sidelobes[END_REF] [31], Frank, P1, P2, P3, P4, [START_REF] Lewis | A new class of polyphase pulse compression codes and techniques[END_REF] [61], and ZadoffChu codes [START_REF] Frank | Phase shift pulse codes with good periodic correlation properties[END_REF].

In this thesis, the polyphase Barker and P4 codes are considered.

1.6.1.2.1 Barker codes These codes are widely used in radar applications. For any length M , the autocorrelation function of the Barker code has equal sidelobes and a well-defined peak at its mainlobe. The peak sidelobe to mainlobe ratio is equal to 1 M . Moreover, the maximum even length is equal to 4 and the maximum odd length is equal to 13 [START_REF] Cook | Radar Signals, An Introduction to Theory and Application[END_REF]. The autocorrelation function 5 of the Barker code with M = 13 is presented in Fig. 1.5. codes, polyphase Barker codes were proposed to obtain peak sidelobe to mainlobe ratio similar to Barker codes with larger M . These codes can be derived by using various search techniques [START_REF] Richards | Principles of modern radar[END_REF]. Thus, the phases of the polyphase Barker codes are either unrestricted or restricted to a P th root of unity such that the allowable phase increment is ∆φ = 2π P and P is an integer greater than 2. In [START_REF] Borwein | Polyphase sequences with low autocorrelation[END_REF], polyphase codes up to length 63 have been identified. Some of these codes used in this thesis are provided in Table . 1.1. In addition, their autocorrelation functions and power spectra are presented in

v(n) of length N is defined for n ∈ 1, 2N -1 by R v (n) = X(n -N ) where X(n) is defined by: X(n) = N -n-1 i=0 v(i + n)v(i) for n ≥ 0 X(-n) for n < 0 (1.44)
φ i = πB T p (iτ chip ) 2 (1.45)
For a rectangular chip, the chip width is the reciprocal of the bandwidth, i.e. τ chip = 1 B . After making the appropriate substitution and subtracting the result from πi, the phase sequence of P4 for i ∈ 0, M -1 reduces to:

φ i = πi 2 M -πi (1.46)
In the next section, let us focus our attention on an alternative type of waveform whose instantaneous frequency varies over time in a non-linear way.

Non linear frequency modulation (NLFM) waveforms

There are some waveforms which are "naturally" NLFM such as the Hybrid NLFM [START_REF] Huang | Matched NLFM pulse compression method with ultra-low sidelobes[END_REF], tangent-based, and the sine-based waveforms [108] [19]. However, others can be designed to meet certain requirements. Thus, the principle of stationary phase can be used to synthesize an NLFM waveform that has a PSD similar to certain well-known windows such as the Taylor and Blackman-Harris windows [START_REF] Melvin | Principles of modern radar[END_REF] [98] [START_REF] Enache | Sidelobe reduction in pulse-compression radar using the stationary phase technique: An extended comparative study[END_REF]. Recently, in [START_REF] Saeedi | Synthetic aperture radar imaging using nonlinear frequency modulation signal[END_REF], a piecewise NLFM (PW-NLFM) waveform was proposed to enhance the imaging quality in SAR applications. In this thesis, the tangent-based NLFM and the PW-NLFM waveforms are considered. Let us give some details about them in the paragraphs below.

v bb,m (t) = A rect t -(m -1)T r -T p /2 T p exp jφ(t -(m -1)T r -T p /2) (1.47)
where the phase φ(t) is given by:

φ(t) = -2π B p T p 4βtanβ ln cos( 2βt T p ) = (1.13) -2π BT p 4βtanβ ln cos( 2βt T p ) (1.48)
It should be noted that:

φ(t) = φ(-t) (1.49)
In (1.48), φ(t) is is related to the instantaneous frequency for -Tp 2 ≤ t ≤ Tp 2 as follows:

f (t) = 1 2π dφ(t) dt = B tan(2βt/T p ) 2tanβ = -f (-t) (1.50)
where β ∈ 0, π/2 is a parameter that can be adjusted by the practitioner. In Fig. 1.10 and 1.11, f (t) and φ(t) are respectively presented for different values of β. Note that the degree of non-linearity increases with β. Moreover, when β tends to 0, this leads to the LFM case. As the instantaneous frequency varies with β, the power spectrum also varies with β. In Fig. 1.12, the power spectrum is provided with B = 100 MHz for β = 0.01, β = 0.95 and β = 1.42. As the instantaneous frequency varies with β, the power spectrum also varies with β. When β tends to 0, the power spectrum tends to be the one of an LFM waveform. The power is more or less "uniformly" distributed in the interval forms have been studied [43] [13]. In this thesis, we focus on the one proposed in [START_REF] Saeedi | Synthetic aperture radar imaging using nonlinear frequency modulation signal[END_REF], the authors called PW-NLFM waveform. It consists of two symmetric sets, where each one is composed of L + 1 concatenated LFM subpulses with different bandwidths and durations. Its instantaneous frequency is shown in Fig. 1.13. In order to derive the analytical expression of the PW-NLFM waveform, let us define the following parameters:

     (τ 1 , ν 1 ) = (0, 0) (τ L+2 , ν L+2 ) = ( Tp 2 , B 2 ) γ (i) = ν i+1 -ν i τ i+1 -τ i with 1 ≤ i ≤ L + 1.
(1.51)

Then, the instantaneous frequency of the PW-NLFM waveform, denoted as f P W (t), is given by:

               f P W (t) = ν i + γ (i) (t -τ i ) for τ i ≤ t ≤ τ i+1 with i = 1, ..., L + 1 f P W (t) = B -f P W (T p -t) = B -ν 2L+3-i + γ (2L+3-i) (t -T p + τ 2L+3-i ) for τ i ≤ t ≤ τ i+1 with i = L + 2, ..., 2L + 2 (1.52) 20
Remark: It should be noted the instantaneous frequency for i = L + 2, ..., 2L + 2 can be written in different ways. In [START_REF] Saeedi | Synthetic aperture radar imaging using nonlinear frequency modulation signal[END_REF], the authors directly use:

f P W (t) = B -ν 2L+4-i + γ (2L+3-i) (t -T p + τ 2L+4-i ) (1.53)
The above expression is valid. This can be shown by subtracting (1.53) to (1.52) and then by using definition of γ (i) . However, in this thesis, (1.52) is used. Then, the instantaneous phase φ P W (t) can be derived by integrating (1.52) and multiplying by 2π. This leads to:

             φ P W (t) = 2π γ (i) t 2 2 + (ν i -γ (i) τ i )t + l i for τ i ≤ t ≤ τ i+1 and i = 1, ..., L + 1 φ P W (t) = 2π γ (2L+3-i) t 2 2 + (γ (2L+3-i) (τ 2L+3-i -T p ) + B -ν 2L+3-i )t + l i for τ i ≤ t ≤ τ i+1 and i = L + 2, ..., 2L + 2
(1.54) where l 1,...,2L+2 are used to preserve the continuity between different stages of the phase function.

Let us now express the set of phases {l i } i=1,...,L+1 . To this end, let us assume that l 1 is set at 0. In this case, using the above expressions, the continuity of the phase at τ 1 leads to:

l 2 = (γ (1) + γ (2) ) τ 2 2 2 -ν 2 τ 2 = γ (1) τ 2 2 2 -ν 2 τ 2 + γ (2) τ 2 2 2 = -ν 2 τ 2 2 + γ (2) τ 2 2 2 (1.55)
More generally one has for i = 1, ..., L:

l i+1 = l i + (ν i -ν i+1 )τ i+1 + (γ (i) + γ (i+1) ) τ 2 i+1 2 -γ (i) τ i τ i+1 (1.56) At this stage, as ν i+1 -ν i = γ (i) (τ i+1 -τ i )
, this leads to:

l i+1 = l i + (γ (i+1) -γ (i) ) τ 2 i+1 2 (1.57)
Then, it can be easily shown that:

l L+2 = l L+1 + T p 2 (2ν L+1 -B) + γ (L+1) ( T p 2 (T p -2τ L+1 )) (1.58)
and for i = L + 2, ..., 2L

l i+1 = l i + (ν 2L+2-i -ν 2L+3-i )τ i+1 + γ (2L+3-i) ( τ 2 i+1 2 + (τ 2L+3-i -T p )τ i+1 ) (1.59) -γ (2L+2-i) ( τ 2 i+1 2 + (τ 2L+2-i -T p )τ i+1 )
By replacing τ i+1 by (T p -τ 2L+3-i ), the above expression can be rewritten as follows:

l i+1 = l i + (ν 2L+2-i -ν 2L+3-i )(T p -τ 2L+3-i ) (1.60) -(γ (2L+3-i) + γ (2L+2-i) ) (T p -τ 2L+3-i ) 2 2 + γ (2L+2-i) (T p -τ 2L+2-i )(T p -τ 2L+3-i ) In addition, by replacing ν 2L+3-i -ν 2L+2-i = γ (2L+2-i) (τ 2L+3-i -τ 2L+2-i )
, the above expression becomes:

l i+1 = l i -(γ (2L+3-i) -γ (2L+2-i) ) (T p -τ 2L+3-i ) 2 2 (1.61)
As all the parameters of the instantaneous phase of the PW-NLFM waveform have been defined, the expression of the transmitted waveform is represented by (1.1) where:

v bb,m (t) = A rect t -(m -1)T r -T p /2 T p /2 exp jφ P W (t -(m -1)T r ) (1.62)
Due to the expression (1.52) of the instantaneous phase, the spectrum of s tx (t) is centered at B 2 . The NLFM waveform employs frequency modulation to shape the spectrum and thus reduces the range sidelobes. These waveforms do not require an amplitude weighting to the MF output which is commonly used with an LFM waveform, and thereby avoid the associated SNR loss.

Comments on the pulse compression waveforms

Even if the LFM remains the most popular waveform probably thanks to its simplicity to be generated and its Doppler tolerance, some phase codes exhibit a Doppler tolerance similar to that of an LFM [START_REF] Richards | Principles of modern radar[END_REF]. In addition, the advent of high-speed and high dynamic range digital-to-analog converter (DAC) and high-speed FPGA has facilitated the generations of sophisticated NLFM waveforms [START_REF] Weinberg | Topics in Radar Signal Processing[END_REF]. Pulse compression waveforms can be used in high range resolution radar application. However, they require a high sampling frequency and consequently an expensive ADC. To overcome the aferomentioned drawback, SF waveforms were proposed. Let us present them in the following section.

Stepped-frequency (SF) waveforms

Stepped-frequency waveform is a waveform that transmits a train of narrow-band pulses with equally-spaced carrier frequencies. At the receiver, a large bandwidth is synthesized by coherently combining the received echoes. This arrangement does not require an expensive and complex ADC since the required sampling frequency at the receiver depends on the baseband bandwidth rather than the total bandwidth covered by the received pulses. In the literature, SF waveforms have been well studied [START_REF] Lord | Aspects of stepped-frequency processing for low-frequency SAR systems[END_REF] [81] [START_REF] Jiang | A new approach for synthesizing the range profile of moving targets via stepped-frequency waveforms[END_REF]. In addition, SF can be combined at the transmitter with one of the pulse compression techniques. The reader may refer to [START_REF] Gladkova | Grating lobes suppression in stepped-frequency pulse train[END_REF] [70] [START_REF] Yuan | Study on I SAR imaging of stepped-frequency chirp signal[END_REF] [69] [START_REF] Li | RFI suppression based on phase-coded steppedfrequency waveform in through-wall radars[END_REF] [17] [START_REF] Liu | Imaging approach for airborne stepped-frequency synthetic aperture radar in the squinted mode[END_REF]. One main advantage of this approach is that it reduces the number of pulses within the coherent processing interval (CPI) 6 . For instance, in radar imaging, the size of the CPI is limited to avoid the degradation caused by the motion through the resolution cells [4]. Moreover, with mechanically scanning search radars, the number of transmitted pulses within the dwell beam is constrained by the speed of rotation of the antenna. A great deal of interest has been paid to SF-LFM [START_REF] Yuan | Study on I SAR imaging of stepped-frequency chirp signal[END_REF] [17] [START_REF] Liu | Imaging approach for airborne stepped-frequency synthetic aperture radar in the squinted mode[END_REF] [69] [START_REF] Luo | Micro-Doppler effect analysis and feature extraction in ISAR imaging with stepped-frequency chirp signals[END_REF]. On the contrary, few researchers have studied the stepped-frequency phase-coded (SFPC) waveform and the stepped frequency non-linear frequency modulation (SF-NLFM) waveform. In [START_REF] Li | RFI suppression based on phase-coded steppedfrequency waveform in through-wall radars[END_REF], the SFPC waveform was used with the through-the-wall radar to enhance the anti-radio frequency disturbance performance. In [START_REF] Gladkova | Grating lobes suppression in stepped-frequency pulse train[END_REF], the author illustrated the advantage of the SF-NLFM waveform over the SF-LFM waveform in terms of reductions of the grating-lobe levels. In the next section, the general waveform modeling of a SF waveform is presented. Then, the modeling of the particular cases SF-LFM and SFPC are presented.

General modeling of SF waveforms

The SF radar transmits a burst of N p > 1 pulses, whose carrier frequency monotonically increases from pulse to pulse by a fixed frequency step size denoted as ∆f . For the m th pulse (m ∈ 1, N p ), the transmitted SF waveform is described as follows:

s tx (t) = Np m=1 s tx,m (t) = Np m=1 v bb,m (t)exp(j2πf (m) c t) (1.63) with f (m) c = f c + ∆f (m) (1.64)
where

∆f (m) = 1 -N p 2 + m ∆f (1.65)
In the above equations, f

(m) c
is the carrier frequency of the m th transmitted pulse, and f c is the central carrier frequency of the complete train of pulses. In addition, usually, ∆f ≤ B p is chosen to avoid creating gaps in the transmitted waveform. When ∆f = B p , the total bandwidth covered by the waveform is B = N p B p . Finally in (1.63), v bb,m (t) corresponds to either modulated pulses or unmodulated pulses. In the sequel and for the sake of simplicity, the equations that are presented only deal with the signal part. The disturbance is omitted. In addition, K m = 1 is considered since the N p pulses are assumed to be subjected to the same channel effect. Thus, the "ideal" received SF signal for a stationary point target at range R can be written as follows:

s rx (t) = Np m=1 s rx,m (t) = Np m=1 v bb,m t -t d exp j2πf (m) c (t -t d ) (1.66)
The reference signal used for the demodulation process is defined by:

s ref,m (t) = exp -j2πf (m) c t (1.67)
Therefore, the received baseband signal is given by:

s rx,bb (t) = Np m=1 s rx,bb,m (t) = Np m=1 v bb,m t -t d exp -j2πf (m) c t d (1.68) 1.6.2.

SF-LFM waveform model

The SF-LFM waveform is derived from the combination of the SF waveform and the LFM waveform. In other words, the carrier frequency of each SF pulse is linearly modulated. In this case, the resulting waveform is represented by (1.63) where v bb,m (t) is defined as in (1.28). Thus, the SF-LFM transmitted waveform whose instantaneous frequency is presented in Fig. 1.14 can be expressed as: Given (1.68), the received baseband SF-LFM signal can be expressed as:

s tx (t) = Np m=1 s tx,m (t) = Np m=1 A rect t -(m -1)T r -T p /2 T p exp jπγ(t -(m -1)T r -T p /2) 2 × exp j2πf (m) c t (1.69)
s rx,bb (t) = Np m=1 rect t -(m -1)T r -T p /2 -t d T p (1.70) × exp jπγ(t -(m -1)T r -T p /2 -t d ) 2 exp -j2πf (m) c t d 1.6.2.

SFPC waveform model

The SFPC waveform is derived from the combination of the SF waveform and PC waveform. In other words, the carrier frequency of each pulse in the PC waveform varies as in the case of the SF waveform. Therefore, the resulting waveform is represented by (1.63) where v bb,m (t) is defined as in (1.43). Thus, the SFPC waveform can be expressed as:

s tx (t) = Np m=1 s tx,m (t) = Np m=1 M -1 i=0 A rect t -iT c -(m -1)T r -T c /2 T c exp(jφ i )exp j2πf (m) c t (1.71)
Given (1.68), the received SFPC baseband signal can be expressed as:

s rx,bb (t) = Np m=1 M -1 i=0 rect t -iT c -(m -1)T r -T c /2 -t d T c × exp(jφ i )exp -j2πf (m) c t d (1.72) 1.6.2.

SF-NLFM waveform

The SF-NLFM waveform is derived from the combination of the SF waveform and the NLFM waveform. Thus, the SF-NLFM waveform is represented by (1.63) where v bb,m (t) is defined by:

v bb,m (t) = A rect t -(m -1)T r -T p /2 T p exp jφ(t -(m -1)T r -T p /2) (1.73)
where φ(t) represents the instantaneous phase of the NLFM used. Thus, the SF-NLFM waveform can be expressed as:

s tx (t) = Np m=1 s tx,m (t) = Np m=1 A rect t -(m -1)T r -T p /2 T p exp jφ(t -(m -1)T r -T p /2) × exp j2πf (m) c t (1.74)
Given (1.68), the received baseband SF-NLFM signal can be expressed as:

s rx,bb (t) = Np m=1 rect t -(m -1)T r -T p /2 -t d T p × exp jφ(t -(m -1)T r -T p /2 -t d ) exp -j2πf (m) c t d (1.75) 1.6.2.

Comments on SF waveforms

Since the instantaneous frequency of an LFM pulse is a linear function of time, the instantaneous frequency of the concatenated pulses of the SF-LFM waveform corresponds to a wide-band LFM pulse. In this case, it is possible to synthesize a wide-band LFM pulse at the receiver and thus the HRRP has the shape of the autocorrelation function of a wide-band LFM pulse. However, with SFPC or SF-NLFM, the instantaneous frequency of the concatenated pulses does not correspond to a wide-band PC or NLFM pulse.

In order to process the SF received baseband signal, different algorithms can be used.

Let us present them in the next section.

Processing SF waveforms at the receiver

The SF waveforms can be processed at the receiver by using the MF. However, it suffers from the upsampling process. In addition, grating lobes may appear in the HRRP [START_REF] Gladkova | Grating lobes suppression in stepped-frequency pulse train[END_REF].

One alternative can be seen as a kind of stretch processing. It includes three different algorithms:

• The frequency domain (FD) algorithm

• The inverse fast Fourier transform (IFFT) algorithm

• The time domain (TD) algorithm

The FD and the IFFT algorithms have computational costs smaller than that of the MFbased approach. However, they have some limitations. The IFFT algorithm produces ghost targets in the HRRP of the extended targets due to the spill-over effect of energy into consecutive coarse range bins [START_REF] Lord | Aspects of stepped-frequency processing for low-frequency SAR systems[END_REF]. As for the TD algorithm [START_REF] Lord | High resolution SAR processing using stepped frequencies[END_REF], it does not produce ghost targets but suffers from the up-sampling requirement of the narrowbandwidth pulses prior to the frequency shift. Finally, the FD algorithm can cope with the drawbacks of the TD and the IFFT algorithms, but a discrete Fourier transform (DFT) must be computed on a relatively large number of samples. Nevertheless, the recent advances in designing and fabricating powerful processors can facilitate its implementations.

In order to process the received echoes of a SF waveform, the baseband signal s rx,bb (t) is sampled at

F (Rx) s
. The number of samples associated with each pulse is equal to

N = F (Rx) s
T p where . is the floor function. Then, the sampled data are processed with one of the existing algorithms. Let us start with the FD algorithm.

1.6.2.6.1 Frequency domain (FD) algorithm It aims at reconstructing a large target reflectivity spectrum by coherently combining the individual spectra of the received SF-LFM pulses in the frequency domain [START_REF] Wilkinson | Stepped-frequency processing by reconstruction of target reflectivity spectrum[END_REF]. The resulting reconstructed spectrum is equivalent to the spectrum of a wide-band LFM pulse and thus the HRRP obtained has approximately the shape of a sinc function. In the following, the steps of the FD algorithm, based on the SF-LFM waveform defined in (1.69), are presented:

1. Sample the received pulses in (1.70) at F (Rx) s = 2B b = B
Np . Thus, each received baseband pulse is defined for n ∈ 0, N -1 by:

s rx,bb,m (n) = exp jπγ(nT (Rx) s -T p /2 -t d ) 2 exp -j2πf (m) c t d (1.76)
where

T (Rx) s = 1 F (Rx) s
is the sampling time at the receiver.

2. A DFT is computed on the received samples associated with each pulse in (1.76) padded with N -1 zeros. Padding by zeros is necessary to make MF in the frequency domain equivalent to a linear convolution in the time domain 7 , which is usually utilized in radars [START_REF] Richards | Principles of modern radar[END_REF]. The result of this transform can be written for the k th frequency bin as:

S rx,bb,m (k) = V pad m (k)exp -j2πf (m) c 2R c exp -j2π kF (Rx) s 2N -1 2R c (1.77)
where

V pad m (k) is the DFT of v bb,m (n) padded with N -1 zeros, and k ∈ 0, 2N -2 . In this case v bb,m (n) is given for n ∈ 0, N -1 by: v bb,m (n) = exp jπγ(nT (T x) s -T p /2) 2
(1.78)

3. A MF is applied to each pulse in the frequency domain. It consists in multiplying (1.77) by V pad m (k). Hence, (1.77) becomes:

Y r,m (k) = V pad m (k) 2 exp -j2πf (m) c 2R c exp -j2π kF (Rx) s 2N -1 2R c (1.79)
It should be noted that taking into account (1.28),

|V pad m (k)| 2 is the same for every m ∈ 1, N p : |V pad m (k)| 2 = |V pad 1 (k)| 2 (1.80) 4.
A compression filter is considered to modify each sub-spectrum [START_REF] Lord | Aspects of stepped-frequency processing for low-frequency SAR systems[END_REF].

It consists in multiplying each sub-spectrum Y r,m (k) by H m (k) = 1 |V pad m (k)| 2 . This leads to: Z r,m (k) = exp -j2π(f (m) c + kF (Rx) s 2N -1 ) 2R c (1.81)
This filter leads to a synthesized magnitude spectrum of rectangular shape, i.e. |Z r,m (k)| = 1.

5.

The sub-spectra in (1.81) are contiguously arranged to synthesize the whole spectrum. Let us denote by Z r,m the vector that contains the 2N -1 samples of the m th subspectra in (1.81):

Z r,m = [Z r,m (1) Z r,m (1) . . . Z r,m (2N -1)] (1.82)
Hence, the total concatenated spectrum is represented as a row vector of size (2N -1)N p as follows:

Z r,total = Z r,1 Z r,2 . . . Z r,Np (1.83)
6. This vector is of interest to deduce the HRRP. Applying an inverse discrete Fourier transform (IDFT) to (1.83) yields:

z r (n) = 1 (2N -1)N p × (2N -1)Np-1 l 1 =0 Z r,total (l 1 + 1)exp j2πl 1 n (2N -1)N p (1.84)
where Z r,total (l) is the l th element of Z r,total . Using (1.64), the above equation can be expressed as follows:

z r (n) = exp -j2π(f c + 1-Np 2 ∆f ) 2R c (2N -1)N p Np m=1 exp -j2π(m -1)(∆f 2R c - n N p ) (1.85) × 2N -2 k=0 exp -j 2πk (2N -1) 2RF (Rx) s c - n N p
Therefore, its modulus represents the HRRP and is equal to:

h F D (n) = |z r (n)| = 1 (2N -1)N p |sin N p π(∆f 2R c -n Np ) | |sin π(∆f 2R c -n Np ) | (1.86) × |sin π 2RF (Rx) s c -n Np | |sin π 2N -1 2RF (Rx) s c -n Np | Providing that F (Rx) s
= ∆f , (1.86) becomes:

h F D (n) = 1 (2N -1)N p sin N p π(F (Rx) s 2R c -n Np ) sin π 2N -1 F (Rx) s 2R c -n Np (1.87)
where n ∈ 0, (2N -1)N p -1 and h F D (n) has its maximum equal to 1 when

n = 2RF (Rx) s Np c = t d F (Rx) s N p . Moreover, if n was not reduced to the interval 0, (2N -1)N p -1 , |z r (n)| would be (2N -1)N p -periodic.

Remark about the compression filter:

If the compression filter 2 was not used, the IDFT of Z r,total would amount to computing the IDFT of the product between the sequence defined from the N p sets {exp -j2π(f This is equivalent to computing the circular convolution between the IDFTs. The first one is given in (1.87) whereas the second can be given by:

H m (k) = 1 |V pad m (k)|
(m) c + kF (Rx) s 2N -1 ) 2R c } k=0,...,
1 (2N -1)N p × Np m=1 2N -2 k=0 V pad m (k) 2 exp j2π((m -1)(2N -1) + k)n (2N -1)N p (1.88)
Given (1.80), this can be rewritten as follows:

1 N p Np m=1 exp j2π(m -1)n N p × 1 (2N -1) 2N -2 k=0 V pad 1 (k) 2 exp j2πkn (2N -1)N p (1.89) In (1.89), 1 Np Np m=1 exp j2π(m-1)n
Np is equal to 0 except when n is a multiple of N p . In addition,

1 (2N -1) 2N -2 k=0 V pad 1 (k) 2 exp( j2πkn (2N -1)Np )
is the autocorrelation function of the LFM waveform interpolated at a rate of N p . Once this sequence is multiplied by the other one which is either equal to 1 or 0, this leads to the autocorrelation function upsampled at a rate of N p . Therefore (1.89) corresponds to a sequence comprising the autocorrelation function of the waveform separated by N p -1 zeros. Given the above mathematical developments, if a compression filter was not used, the IDFT of Z r,total would be the circular convolution between the quantity given in (1.87) and the autocorrelation function upsampled by a factor N p . Therefore, the maximum of |z r (n)| remains reached for n = 2RF (Rx)

s Np c = t d F (Rx) s N p .
As the autocorrelation function of the LFM waveform is far from the Kronecker delta function, the HRRP obtained in (1.87) is modified. In [START_REF] Lord | Aspects of stepped-frequency processing for low-frequency SAR systems[END_REF], the authors did not provide these mathematical details. They motivated the use of the compression filter by the fact that they noticed spectral ripples in the concatenated spectrum, leading to some artifacts in the HRRP in their simulations. Instead of using a compression filter, the authors in [START_REF] Melvin | Principles of modern radar[END_REF] suggest considering overlapped LFM spectra so that the reconstructed spectrum tends to be flat. Remark about a post weighting: Due to (1.87), amplitude weighting can be applied to reduce the range sidelobes. It consists in multiplying (1.83) by another row vector W of the same length containing the spectrum of a shaping window, for instance Hanning.

G r,total = Z r,total . * W (1.90)

where .* stands for a multiplication of vectors element by element. An IDFT is finally applied to the synthesized spectrum in (1.90) to create the HRRP. 1. Sample the received pulses, whose expression is given in (1.68), at

F (Rx) s = 2B b = B
Np . Thus, each received baseband pulse is defined for n ∈ 0, N -1 by:

s rx,bb,m (n) = v bb,m (nT (Rx) s -t d )exp(-j2πf (m) c t d ) (1.91) 2. A MF is applied to the samples given in (1.91). It consists in convolving s rx,bb,m (n) with v bb,m (-n).
3. A sample is collected at each range gate and the process is repeated over the N p pulses.

4. At each range gate, one range profile can be formed by applying an IDFT to the collected samples. Finally, the obtained range profiles can be concatenated to produce the HRRP.

1.6.2.6.3 Time domain (TD) algorithm The TD algorithm was proposed to process a SF-LFM waveform in the time domain [START_REF] Lord | Aspects of stepped-frequency processing for low-frequency SAR systems[END_REF]. It aims at reconstructing a wide-band LFM waveform by combining the narrow-band LFM pulses in the time domain. In the following, the steps of the TD algorithm, based on the waveform defined in (1.69) where the frequency step size is assumed equal to the passband bandwidth of the pulse (i.e., ∆f = B p ), are presented:

1. For each received pulse, sample the baseband pulses in (1.70) by using the sampling frequency

F (Rx) s > 2B b = ∆f = B Np .
2. Interpolate the samples of each pulse so that the new sampling frequency becomes:

F (int) s = 1 T (int) s = B = N p F (Rx) s (1.92)
The interpolation operation can be done in the frequency domain by applying a DFT, zero-padding the result, and then applying an IDFT.

3. Frequency shift each pulse to its proper position. This is done by multiplying each pulse by the proper exponential factor exp(j2π∆f (m) nT

(int) s ) 4.
Add a phase correction term to each pulse to avoid any phase discontinuity in the reconstructed LFM waveform. It is done by multiplying the result of the m (th) pulse in the previous step by exp jπγT 2 p (m + 1-Np 2 ) 2 . This step may be done before the interpolation step, thus requiring fewer multiplications and leading to a faster implementation of the procedure.

Time shift the result obtained by δt

m = m + 1-Np 2 T p .
6. Add the obtained pulses together.

7. Apply a MF in the frequency domain to the reconstructed waveform.

8. Produce the HRRP by applying an IDFT to the output of the MF.

Performance measures

In this section, the definitions of some performance measures that are used in the simulation parts are given. As mentioned in section 1.3, the range profile of a point scatterer is characterized by one mainlobe and several sidelobes. The latter arise from the use of pulse compression techniques and are not desirable. Moreover, as indicated in subsection 1.4.2, the enhancement of the PD while maintaining a constant PFA can be achieved by increasing the SIR. However, when multiple targets are present, besides the SIR, the range resolution and the level of the sidelobes should also be taken into account since they have a great influence on the PD and PFA. As for the level of the sidelobes, they can be quantified using two ratios: the peak sidelobe ratio (PSLR) and the integrated sidelobe ratio (ISLR) [START_REF] Galati | Waveforms design for modern and mimo radar[END_REF] [START_REF] Xu | Range sidelobe suppression technique for coherent ultra wide-band random noise radar imaging[END_REF]. The PSLR is defined as the ratio of the maximum peak magnitude of the sidelobes to the peak magnitude of the mainlobe:

P SLR = 20log h(n s ) h(n p ) (1.93)
where h represents the HRRP, and n s and n p respectively denote the position of the highest sidelobe and the peak value of the mainlobe in h(n). The peak magnitude of the first sidelobe is not necessarily the maximum peak magnitude among the other sidelobes, although it could be the case in some situations. The better performance in terms of PSLR mitigates the masking effect of nearby targets and increase the useful dynamic range. The ISLR is defined as the ratio of the energy of the sidelobes to the energy of the mainlobe:

ISLR = 10log      np-nz n=1 h(n) 2 + np+N +L n=np+nz h(n) 2 n=np+N n=np-nz h(n) 2      (1.94)
where n z denotes the position of the null of the mainlobe in h(n).

The level of significance given for each performance measure, especially the PSLR and the ISLR, depends on the surrounding environment. Whenever the received signal is disturbed by a distributed clutter environment, the ISLR is of high importance and should be maintained as low as possible to enable the detection of weak targets. However, if the received signal is disturbed by a strong discrete clutter, the PSLR is more critical to be kept low. Otherwise, sidelobes may be interpreted falsely as real targets [START_REF] Lellouch | Design of OFDM radar pulses using genetic algorithm based techniques[END_REF].

Finally, let us define the range resolution. In the literature, various criteria are used to define the range resolution. Two of the more common criteria used to define it are [START_REF] Richards | Principles of modern radar[END_REF]:

• The 3-dB width of the mainlobe which defines the range resolution as the separation between the peak point and the point that corresponds to the half of the peak power.

• The Rayleigh criterion which defines the range resolution as the separation between the peak and the null. In this case, the range resolution can be commonly expressed as:

R res = c 2B ef f (1.95)
where B ef f is the effective bandwidth of the waveform.

Waveform B b B B ef f P C M/T p 2M/T p B/2 SF 1/T p (N p -1)∆f +2B b B/2 SF P C M/T p (N p -1)∆f +2B b B/2

Conclusions

In this chapter, generalities about radar have been presented. In addition, we have focused our attention on high range resolution waveforms. After presenting the pulse compression waveforms such as the LFM, the PC, and the NLFM waveforms, SF waveforms have been introduced. The latter can be used to reduce the sampling frequency at the receiver. Three types of algorithms, namely the IFFT algorithm, the TD algorithm and the FD algorithm, which aim at processing the SF waveforms at the receiver have been described. Finally, we have presented some performance measures that have a great influence on the probability of detection and probability of false alarm.

In chapter 2, a processing chain from the transmitter to the receiver is proposed to overcome the drawbacks of the standard combination of PC or NLFM with a SF waveform.

CHAPTER 2

MODIFIED STEPPED-FREQUENCY WAVEFORMS

Introduction

In chapter 1, we have mentioned that pulse compression waveforms can be used in high range resolution radar applications. For some of these applications, the PC and the NLFM waveforms may be more attractive than the LFM waveform since they exhibit smaller PSLR and ISLR. As the sampling rate of such waveforms is high 1 , one of the approaches that can be considered to reduce it is the standard combination of these waveforms with SF. However, when using this combination, the PSLR and the ISLR of the pulse compression used cannot be attained. This is due to the fact that the instantaneous frequency of the concatenated pulses of the resulting waveform does not correspond to a known wide-band PC or NLFM pulse. As a consequence, the obtained HRRP does not have the shape of the autocorrelation function of a known wide-band NLFM or PC pulse. In this chapter, in order to overcome the aforementioned drawback, i.e. in order to reduce the sampling rate while taking the advantage of the features of the pulse compression waveform used in terms of PSLR and ISLR, a new approach is suggested to combine a SF scheme with pulse compression waveforms. More particularly, a general processing chain from the transmitter to the receiver is proposed. It consists in splitting the spectrum of a wide-band modulated pulse into a predetermined number of portions. Then, the corresponding time-domain signals are successively transmitted.

These signals constitute what we call a modified SF waveform in this dissertation. At the receiver, the HRRP can be reconstructed by processing the received echoes either in the frequency domain or in the time domain. Accordingly, a modified version of the FD algorithm and a novel time waveform reconstruction (TWR) algorithm are proposed to reconstruct the HRRP. This chapter is organized as follows: firstly, the processing chain from the transmitter to the receiver is presented. Then, the results of the simulations of the modified SF waveforms in different scenarios are shown.

Our contribution: a processing chain of the modified SF radar waveform combined with a pulse compression technique

In this section, the processing chain from the transmitter to the receiver is presented.

First, the steps that should be done at the transmitter to generate the modified SF waveform are illustrated. Then, the different algorithms that can be considered to process the received echoes at the receiver are presented. Afterwards, the computational costs of the modified FD algorithm and the TWR algorithm are compared. Suggestions to remove some constraints in the modified SF waveform are provided. In addition, some comments on the proposed waveform are presented. Finally, a comparative study is carried out between the different variants that can be considered in the processing chain.

In addition, the modified SF waveforms are compared with the standard SF waveforms in different scenarios.

Generation of the modified SF waveform at the transmitter

The whole processing chain of the modified SF waveform is presented in Fig. 2.1.

Let v • (t) be the continuous-time baseband pulse that exhibits a wide bandwidth, where • stands for NLFM or PC. It is defined for 0 ≤ t ≤ T p by:

v P C (t) = M -1 i=0 A rect t-iTc-Tc/2 Tc exp(jφ i ) v N LF M (t) = A rect t-Tp/2
Tp exp jφ(t -T p /2)

(2.1)
For the generation of the waveform, the discrete version of v • (t) is needed. By introducing the sampling period T (T x) s at the transmitter which satisfies:

1

T (T x) s = F (T x) s = B (2.2)
where

F (T x) s
is the sampling frequency at the transmitter and B represents the bandwidth of the pulse, (2.1) becomes for n ∈ 0, N -

1 with N = Tp T (T x) s : v P C (n) = M -1 i=0 A rect nT (T x) s -iTc-Tc/2 Tc exp(jφ i ) v N LF M (n) = Aexp jφ(nT (T x) s -T p /2) (2.3)
At this level, the following steps are carried out:

1. Apply a DFT to v • (n) to get V • (k),
where k denotes the frequency bin.

2. Split V • (k) into N p non-overlapped equal portions. In the following, N p > 1 and N is necessarily an integer multiple of N p . This provides N p DFT denoted

P • m (k) =    V • (k) (m -1) N Np ≤ k ≤ mN Np -1 0 elsewhere (2.4)
An interpretation of (2.4) could be the following: (2.4) could correspond to portions having equally-spaced carrier frequencies, similarly as in a SF waveform defined in (1.63). Moreover, the frequency band occupied by

P • m (k) is F (T x) s Np .
3. Apply an IDFT to each separated portion P • m (k). This leads to N p sequences,

{p • 1 (n)} n=0,...,N -1 , . . . , {p • Np (n)} n=0,...,N -1 .
It should be noted that these N p sequences and v • (n) satisfy:

v • (n) = Np m=1 p • m (n) (2.5)
Let us now find an expression for p • m (n) in the discrete-time domain. For any k ∈ 0, N -1 , the sequences {P • m (k)} m=1,...Np can be expressed as:

P • m (k) = V • (k)W m (k) m ∈ 1, N p (2.6)
where the window W m (k) is given by:

W m (k) =    1 (m -1) N Np ≤ k ≤ m N Np -1 0 elsewhere (2.7)
Hence, the transmitted pulses can be expressed as the IDFT of

V • (k)W m (k): p • m (n) = v • (n) w m (n) n ∈ 0, N -1 (2.8)
where denotes the circular convolution.

Let us now recall the analytical expression of the sequences {w m (n)} m=1,...Np , i.e. the inverse Fourier transform of W m (k). For one thing, given (2.7), one can deduce that:

w 1 (n) = 1 N N -1 k=0 W 1 (k)exp j2πkn N = 1 N sin( πn Np ) sin( πn N ) exp jπn N -N p N N p (2.9)
Moreover, as W l (k) is a shifted version of W 1 (k), any sequence w m (n) can be expressed in terms of w 1 (n). It is given for m ∈ 1, N p by:

w m (n) = exp j 2πn(m -1) N p w 1 (n) (2.10)
Given (2.10), (2.8) becomes:

p • m (n) = 1 N v • (n) sin( πn Np ) sin( πn N ) exp jπn N (2m -1) -N p N N p (2.11)
4. Scale the amplitudes of the N p sequences {p • m (n)} m=1,...,Np as follows:

p • s,m (n) = 1 α m p • m (n) (2.12)
where

α m = 1 N N -1 n=0 |p • m (n)| 2 . The train of pulses {p • s,m (n)} m=1,.
..,Np represents the modified SF waveform. This step is necessary to guarantee that the powers of the transmitted pulses are equal. In addition, they are equal to 1.

Convert the discrete sequences {p

s tx,m (t) = 1 α m p • m (t)exp j2πf c t (2.13)
In some radar applications, it is of interest to design a processing chain that can prevent the opponent from getting information about the transmitted waveform. Indeed, a radar that is settled on a platform does not necessarily generate different types of waveforms. Thus, identifying the radar waveforms makes it possible to get information about the platform itself. In addition, if the opponent analyzer is able to infer the type of pulse compression used, then some features of the radar itself, such as the range resolution, the PSLR, and the ISLR can be predicted. When the "standard" radar waveforms are used, the time-domain pulses have the same shape, whereas this is not the case with our waveform, as shown in [START_REF] Saleh | A modified stepped frequency phase coding radar waveform designed for the frequency domain algorithm[END_REF]. They hence could be seen as noisy signals. In addition, the signals in (2.13) are usually transmitted in an ordered way, from m = 1 to m = N p . Therefore, a degree of freedom can be added by transmitting the pulses in a pseudo-random way that would be also known at the receiver part. Hence, by constantly varying the shuffling order of the transmitted portions from one scan to another, drawing some conclusions about the capabilities of the radar becomes more challenging for the enemy. This feature is attractive for various applications.

Processing the modified SF waveform at the receiver

After expressing the baseband received signal of the modified SF waveform, we propose two ways to process it in order to deduce the HRRP. The first method is done in the frequency domain whereas the second one operates in the time domain.

Received signal model

As mentioned in the first chapter, the received signal is a delayed version of the transmitted waveform disturbed by the white measurement noise and the clutter. For m ∈ 1, N p , it can be modeled by:

s rx,m (t) = K m s tx,m (t -t d ) + η(t) (2.14)
In the following, the disturbances are not taken into account to focus the reader attention on the signal part. Nevertheless, each step of the algorithm is also applied to the disturbances. Thus, assuming that the environment does not vary much from one transmitted pulse to another, and that the target is stationary, the "ideal" received signal for m ∈ 1, N p is given, using (2.13), by:

s rx,m (t) = K m α l p • m (t -t d )exp j2πf c (t -t d ) (2.15)
Then, the received signal s rx,m (t) is down-converted to the baseband by multiplying it with exp -j2π(f c + (2m-1)B 2Np

)t . The resulting baseband signal is given by:

s bb,m (t) = K m α m p • m (t -t d )exp -j2π( (2m -1)B 2N p )t exp -j2πf c t d (2.16)
In order to process the received waveform, s bb,m (t) is sampled at the sampling frequency F (Rx) s defined by:

F (Rx) s = 2B b = F (T x) s N p = B N p (2.17)
For the sake of simplicity, let us assume that the delay t d is strictly a multiple of the sampling period at the receiver T (Rx) s

. Hence, t d is given by:

t d = d F (Rx) s = dT (Rx) s (2.18)
Given (2.17) and (2.18), the m th baseband received discrete-time signal can be represented for n ∈ 0, N Np -1 by:

s bb,m (n) = K m α m p • m (n -d)N p exp -jπ(2m -1)n exp -j2πf c dT (Rx) s (2.19)
To produce the HRRP, s bb,m (n) can be processed either in the frequency domain by using the modified FD algorithm or in the time domain by using the TWR algorithm.

Let us first give some details about the first approach.

Modified FD algorithm

In chapter 1, we have mentioned that the FD algorithm was proposed to process the received echoes of the SF-LFM waveform. In this algorithm, the compression filter is used to remove the spectral ripples in the concatenated spectrum and thus a magnitude spectrum of rectangular shape is obtained. However, with our proposed waveform, we aim at reconstructing the power spectrum of the pulse compression used. Therefore, the compression filter cannot be used. This leads to proposing a modified version of the FD algorithm to process the proposed waveform. It is presented in the remainder of this section.

In the modified FD algorithm, the MF is applied at the level of each pulse.

p ref,m (n) = IDF T P • m k - (2m -1)B 2N p F (T x) s = p • m (n)exp -j2π (2m -1)B 2N p n F (T x) s = (2.17) p • m (n)exp -jπ 2m -1 N p n (2.20)
Then, the resulting sequence associated with each portion is down-sampled by a factor N p so that it is composed of N Np samples. For n ∈ 0, N Np -1 , it is given by:

p down ref,m (n) = p ref,m (nN p ) = p • m (nN p )exp -jπ(2m -1)n (2.21)
Finally, the output of the down-sampler is padded by N Np -1 zeros to get the sequence p dopad ref,m (n). Then, a DFT is applied to the latter sequence to obtain P dopad ref,m (k) , as shown in Fig. 2.1. The various steps of the modified FD algorithm are illustrated in the following:

1. Apply an inverse scale to the result given in (2. [START_REF] Collins | Active sonar pulse designs[END_REF]) by multiplying each received pulse by αm Km where K m can be estimated by comparing the powers of the transmitted and received signals. Hence, for n ∈ 0, N Np -1 , one has:

p down ref,m (n -d)exp -j2πf c dT (Rx) s exp -jπ(2m -1)d (2.22)
2. Pad each sequence obtained in (2.22) by N Np -1 zeros. This yields a vector of length Q = 2N

Np -1 on which a Q-size DFT is applied. The result is given for k ∈ 0, 2N Np -2 by:

P dopad ref,m (k)exp -j2π k Q d exp -j2πf c dT (Rx) s exp -jπ(2m -1)d (2.23)
3. Apply an MF by multiplying each component of (2.23) by the complex conjugate of P dopad ref,m (k). The output is given by:

Z m (k) = |P dopad ref,m (k)| 2 exp -j2π k Q d exp -j2πf c dT (Rx) s × exp -jπ(2m -1)d (2.24)
The values in (2.24) are then stored in a vector of length Q = 2N Np -1:

Z m = [Z m (0) Z m (1) . . . Z m (Q -1)] (2.25)
4. Construct the whole spectrum by arranging the N p sub-spectra contiguously. The total concatenated spectrum is represented by

H • F D (k) of size N p Q = 2N -N p as follows: H • F D (k) = Z 1 Z 2 . . . Z Np (2.26)
Alternatively, this frequency shift can be achieved by multiplying the time domain version of (2.24) with a linear phase ramp. However, using a cut and paste method in the frequency domain, as it is done here, results in a much more efficient algorithm [START_REF] Lord | Aspects of stepped-frequency processing for low-frequency SAR systems[END_REF].

5. To produce the HRRP, apply an IDFT to H • F D (k). This yields:

h • F D (n) = 1 N p Q Np m=1 Q-1 k=0 H • F D ((m -1)Q + k)exp j 2π((m -1)Q + k)n N p Q (2.27)
A zero padding may be done in the frequency domain to force power of 2-size IFFT, or to interpolate further the HRRP to get a better view. It should be noted that plotting the HRRP by directly using (2.27) constitutes a milestone to validate the results of the simulation section.

As an alternative to the modified FD algorithm, we propose to directly process the received signal in the time domain. This leads to the time waveform reconstruction algorithm presented in the next section.

Time waveform reconstruction (TWR) algorithm

The idea of the TWR algorithm stems from the fact that each received pulse corresponds to a base-band portion of the whole spectrum of v • in the time domain. When these pulses are added together after being shifted to their proper position in the frequency domain, the whole spectrum of v • can be retrieved.

The steps carried out in this algorithm are as follows:

1. Apply an inverse scale to the received baseband signal s bb,m (n) by multiplying each received pulse by αm Km . Thus, for n ∈ 0, N Np -1 , one has: and then interpolating the received signal instead of directly sampling it at the sampling frequency B makes it possible to use a cheap ADC. Taken into account [START_REF] Teh | Piecewise nonlinear frequency modulation waveform for mimo radar[END_REF], the resulting signal denoted as s int r,m (n) can be approximated for n ∈ 0, N -1 as follows:

s r,m (n) = p • m ((n -d)N p )exp -jπ(2m -1)n exp -j2πf c dT (Rx) s ( 2
s int r,m (n) ≈ p • m (n -d)exp -jπ(2m -1) n N p exp -j2πf c dT (Rx) s (2.29)
3. Frequency shift each pulse to its proper position. This is done by multiplying each pulse by the proper exponential factor as follows:

s int,f shif t r,m (n) = s int r,m (n)exp jπ(2m -1) n N p ≈ (2.29) p • m (n -d)exp -j2πf c dT (Rx) s (2.30)
4. Add the N p signals obtained in (2.30). The reconstructed waveform can be written as:

z(n) = Np m=1 s int,f shif t r,m (n) ≈ exp -j2πf c dT (Rx) s Np m=1 p • m (n -d) (2.31)
Apply a MF to the reconstructed waveform. This can be done in two steps: apply a DFT to z(n) after padding it with L zeros to get Z pad (k). Then, by defining

V • pad (k) as the DFT of v • (n) padded with L zeros, multiply the result Z pad (k) by V • pad (k) which represents the complex conjugate of V • pad (k).
It should be noted that L should be at least equal to N -1 to make the MF process in the frequency domain equivalent to a linear convolution in the time domain which is usually utilized in radars [START_REF] Richards | Principles of modern radar[END_REF]. For k ∈ 0, N + L , one has:

H • T W R (k) ≈ Z pad (k)V • pad (k) (2.32) ≈ |V • pad (k)| 2 exp(-j2πf c dT (Rx) s )exp(-j 2πd N + L k) = G |V • pad (k)| 2 N + L exp(-j 2πd N + L k)
where

G = (N + L)exp(-j2πf c dT (Rx) s
).

Remark:It should be noted that

|H • T W R (k)| ≈ |G| |V • pad (k)| 2 N + L = (N + L)S • ref (k) (2.33)
where

S • ref (k) = |V • pad (k)| 2 N +L
is the power spectrum of the time-domain waveform that should be retrieved.

Apply an IDFT to H •

T W R (k) in order to produce the HRRP. For this purpose, let us introduce the correlation function of the sequence v • (n):

R v (n) = v • (n) * v • (-n) (2.34)
where * is the convolution. One has:

h • T W R (n) ≈ IDF T H • T W R (k) (2.35) = GR v (n) δ 2N -1 (n -d)
where δ 2N -1 (n -d) denotes the Kronecker symbol equal to 1 when n = d + r(2N -1) where r ∈ Z. In the practical case, the modulus of h • T W R (n) is considered. The HRRP corresponds to the modulus of the correlation function of the waveform v • , the maximum of which is no longer at 0 but shifted at d. It should be noted that during this step, a standard IDFT is computed. However, a zero padding could be done on H • T W R (k) for the two following reasons: forcing a power of 2-size IFFT or interpolating the HRRP.

Computational cost of the modified FD algorithm vs. that of the TWR algorithm

Table. 2.1 summarizes the computational cost of each step of each algorithm where the parameter L used during the MF step at the receiver is set at its minimum value, i.e. N -1. It also provides the total computational cost of each algorithm. They depend on the number N of samples of the initial waveform v • and the number N p of spectrum portions or equivalently the number of pulses.

Table 2.1 Computational costs of the modified FD algorithm and the TWR algorithm. The symbol / is used when it is not applicable

Steps Modified FD algorithm TWR algorithm 1 N N 2 (2N -Np) 2 Np N p N 2 3 2N -N p N p N 4 / N (N p -1) 5 (2N -N p ) 2 2N (2N -1) 6 / (2N -1) 2 Total F (N, N p ) = N 2 p + N (4N -4N p -1 + 4N Np ) T (N, N p ) = 1 + N (N N p + 8N + 2N p -6)
After computing the difference between the total computational cost of the TWR algorithm denoted as T (N, N p ) and that of the modified FD algorithm denoted as F (N, N p ), one has:

D(N, N p ) = T (N, N p ) -F (N, N p ) = 1 + N p N 2 + 4N 2 + 6N N p -5N -N 2 p - 4N 2 N p (2.36)
The derivate of D(N, N p ) with respect to N is equal to (2N p + 8 -8 Np )N + 6N p -5, which is always positive for any value of N p . Therefore, for a fixed value of the portions, D(N, N p ) is necessarily an increasing function of N . In addition, the derivate of D(N, N p ) with respect to N p is equal to

N 2 + 6N -2N p + 4 N 2 N 2 p . It can be shown that it is positive provided that N p < N 2 +6N

3

. Therefore, in this interval and for a fixed value of N , D(N, N p ) is an increasing function of N p . The evolution of difference D(N, N p ) presented in Fig. 2.2. The computational cost required for the TWR algorithm becomes significantly larger than that of the modified FD algorithm only for large values of N and N p . Although the TWR algorithm requires a computational cost higher than the modified FD algorithm, the practitioner can use the TWR algorithm without paying a high computational tax. This can be done by avoiding selecting the waveform parameters that simultaneously lead to large values of N and N p .

Removing the constraints of the modified SF waveform

Splitting the original spectrum into N p equal portions cannot be done when F (T x) s /N p is not an integer. When the modified FD algorithm is used at the receiver, there are two possibilities:

1. Decomposing the spectrum into N p portions of unequal size: In this case, each portion has a different bandwidth. However, at the receiver, the pulses would

Resorting to an overlapping methodology:

The spectrum is still split into N p equal portions, but the bandwidth of each portion is augmented and overlaps with its neighbor. The percentage of the overlap is the same between every two successive portions. By doing so, N p is maintained within the CPI whereas

F (Rx) s
increases. The larger the overlapping, the higher the F (Rx) s

. Following this methodology, as the second equality in (2.17) no longer holds, some simplifications done for the equations at the receiver part are no longer valid. In this case, (2.22), (2.23), and (2.24) become respectively:

±p down ref,m (n -d)exp -j2πf c dT (Rx) s exp -jπ (2m -1)BdT (Rx) s N p (2.37) ±P dopad ref,m (k)exp -j2π k Q d exp(-j2πf c dT (Rx) s )exp -jπ (2m -1)BdT (Rx) s N p (2.38) ±|P dopad ref,m (k)| 2 exp -j2π k Q d exp(-j2πf c dT (Rx) s ) × exp -jπ (2m -1)BdT (Rx) s N p (2.39)
As a consequence, a trade-off has to be found in the selection of the methodology that would be followed for splitting the original spectrum. Selection relies on the specified application and the limitation one would have in terms of F (Rx) s .

Comments on the modified SF waveform

• About the relation between P dopad ref ,l (k) and V(k) In (2.24), Z m (k) is expressed in terms of P dopad ref,l (k). However, it could be of interest to express Z m (k) in terms of V • (k). This is useful for the simulation of the receiving steps. It also shows the influences of the downsampling and the padding through the psinc function. For this purpose, let us find a relation between P dopad ref,m (k) and V (k). By combining (2.2) and (2.20), P ref,m (k) can be written as follows:

P ref,m (k) = P • m k + 2m -1 2 N N p (2.
P down ref,m (θ) = N Np -1 n=0 p down ref,m (n)exp(-jnθ) (2.41)
and

p down ref,m (n) = 1 N/N p N Np -1 k 1 =0 P down ref,m (k 1 )exp j 2πk 1 n N/N p (2.42)
where θ the normalized angular frequency with respect to the sampling frequency. By substituting (2.42) in (2.41), the latter becomes:

P down ref,m (θ) = N/Np-1 n=0 1 N/N p N/Np-1 k 1 =0 P down ref,m (k 1 )exp j 2πk 1 n N/N p .exp(-jnθ) = N/Np-1 k 1 =0 P down ref,m (k 1 ). 1 N/N p N/Np-1 n=0 exp j( 2πk 1 N/N p -θ)n = N/Np-1 k=0 P down ref,m (k 1 ).exp -j( θ 2 - πk 1 N/N p )(N/N p -1) (2.43) psinc N/Np (θ - 2πk 1 N/N p )
where

psinc N Np (θ) = N p N sin( N Np θ 1 2 ) sin( θ 2 ) (2.44)
The latter is equal to 1 if θ = 0 and to 0 if θ is a multiple of

P dopad ref,m (k) = N Np -1 k 1 =0 P down ref,m (k 1 )exp -jπ( k 2N -N p - k 1 N )(N -N p ) ×psinc N Np 2πN p k 2N -N p - k 1 N (2.45)
The relation between the DTFT of the sequence p ref,m (n) and that of its downsampled version p down ref,m (n) is given by: Np respectively. This can be written for the k th frequency bin as follows:

P down ref,m (f ) = 1 N p Np-1 l=0 P ref,m f -l F (T x) s N p (2.
P down ref,m (f ) =          1 Np P ref,m (f ) if f ∈ [0, F (T x) s 2Np [ 1 Np P ref,m (f + (N p -1) F (T x) s Np ) if f ∈ [ F (T x) s 2Np , F (T x) s Np ] (2.
P down ref,m (k) =          1 Np P ref,m (k) if k ∈ 0, N 2Np -1 1 Np P ref,m (k + (N p -1) N Np ) if k ∈ N 2Np , N Np -1 (
P dopad ref,m (k) = 1 N p N 2Np -1 k 1 =0 V • k 1 + 2m -1 2 N N p exp -jπ( k 2N -N p - k 1 N )(N -N p ) × psinc N Np 2πN p k 2N -N p - k 1 N + 1 N p N Np -1 k 1 = N 2Np V • k 1 + 2N p + 2m -3 2 N N p × exp -jπ( k 2N -N p - k 1 N )(N -N p ) psinc N Np 2πN p k 2N -N p - k 1 N (2.49)
where

k ∈ 0, 2N Np -2 and k 1 ∈ 0, N Np -1 . Finally, to obtain |P dopad ref,m (k)| 2
, one has to take the square modulus of (2.49).

• About the envelope of the modified SF waveform

In the traditional pulsed radar waveforms, the transmitted pulses have a constant envelope even though they may have a phase or frequency modulation in the context of an intra-pulse modulation. However, in our modified SF waveform where the pulses are separated by a certain T r , the resulting samples of each transmitted pulse have not only arbitrary phases but also arbitrary amplitudes, as shown in • About a trade-off between R min and B p in the modified SFPC waveform Radar engineers are always interested in using the minimal possible pulse width in short-range radar applications. In order to obtain high PSLRs with phase coded pulses, it is better to use polyphase codes of large length M . They lead to a tradeoff between R min and B p . On the one hand, exploiting these long codes with a fixed pulse width leads to a dramatic augmentation of B p because the latter is proportional to M . On the other hand, if we augment the pulse width to grasp this long code, R min increases according to (1.3). As a consequence, a trade-off has also to be found between the minimum range and the baseband bandwidth. This is the case of the SFPC waveform. With the modified SFPC waveform and thanks to the methodology followed at the transmitting and receiving sides, the polyphase codes with large length M can be exploited while each pulse of the modified SFPC waveform still has the same width as the original pulse from which it is derived. Hence, for a certain value of B p , the minimum range of the SFPC waveform is smaller than that of the SFPC waveform. It should be noted that this comment does not hold when the modified SF-NLFM waveform is considered. In this case, the minimum range of the SF-NLFM and the modified SF-NLFM is the same. This is due to the fact that B p depends only on the sweep bandwidth rather than the pulse width.

• About the computational cost of modifed SF vs. standard SF For a fair comparison, N p and B p are set at the same values for both modified SFPC and SFPC. As a consequence, each pulse of the SFPC waveforms consists of N samples whereas each pulse of the modified SFPC waveform consists of

/ O(N 2 ) O(N p .N 2 ) O(N ) 2 / / O(N p (2N -1)) O( (2N -Np) 2 Np ) 3 / O((N.N p ) 2 ) O(N p (2N -1)) O(2N -N p ) 4 / O(N N p ) / / 5 / / O(N p (2N -1)) O((2N -N p ) 2 ) 6 / / O(N 2 p (2N -1) 2 ) /
modified SFPC and the SFPC waveforms at the transmitter and receiver sides. At the receiver, the modified FD algorithm is considered for the former waveform.

As for the latter waveform, the FD algorithm is considered.

On the one hand, at the transmitter side, generating the transmitted signal for the SFPC waveform is straightforward and avoids all the procedures done for the modified SFPC waveform. However, all the steps at the transmitter should be executed only once. When the waveform is generated for the first time, the resulting samples are saved in the memory of the radar to be utilized later on for other transmissions. As a consequence, the computational burden at the transmitter is almost negligible compared with that of the receiver. Normally, radar engineers are interested in the latter rather than the former. This is due to the fact that all the steps at the receiver should be repeated every time an echo arrives at its front end.

On the other hand, at the receiver side, given Table . 2.2, the addition of the computational power of the whole steps of each waveform reveals that the modified SFPC waveform requires less computational power than the SFPC waveform. Finally, in our modified SFPC waveform, the generated pulses are multiplied by the same carrier signal which eases the operation of the oscillator whereas in the SFPC waveform the frequency of the carrier signal varies in a fixed step from pulse to pulse.

• About the relation between N p and CPI It is important that all the N p pulses that carry the different portions of the split spectrum are transmitted within the CPI. This highly depends on the velocity of the target. If the target is fixed or moving slowly, the number N p of split portions of the spectrum can be large, and consequently, better performance is achieved. However, if the target is moving fast, then N p should be relatively small in order to guarantee that all the transmitted pulses lie within the CPI. Otherwise, the radar will not be able to receive all the echoes within the CPI. One of the possible solutions for this issue is to resort to compressed sensing techniques that have been proposed to deal with such scenarios where sparsity in the collected data exists [START_REF] He | High-resolution imaging and 3-D reconstruction of precession targets by exploiting sparse apertures[END_REF] [46].

Results and discussions

In this section, the performance of the modified SF-NLFM waveform and the modified SFPC waveforms is investigated. For this purpose, for each waveform, the reconstructed power spectrum and the performance in terms of the PSLR, ISLR, and R res of the waveform processed by the TWR algorithm are compared with those of the waveform processed by the modified FD algorithm. It should be noted that our approach is compared with the standard SF waveform.

Simulation results when dealing with the modified SF-NLFM waveform

In this section, in order to validate the processing chain, three aspects are analyzed. Firstly, the approximation made in (2.29) with the TWR algorithm is studied in order to check if it is relevant for any value of N p . Then, we focus our attention on the reconstructed power spectra obtained with the TWR and the modified FD algorithm. They are compared with the power spectrum S We have launched simulations where β varies between 0 and π/2 with a step equal to 0.03. The results for N p = 4 are presented in Fig. 2.5 and show that the order of magnitude is the same for any value of β. Then, we have focused our attention on β = 1.21.

In Fig. 2.6, the normalized error is presented for N p = 4, 20, and 50 and for m = 1, ..., N p .

For any value of N p , the normalized error is smaller than 10 -13 and has the same order of magnitude. Moreover, for each value of N p , the normalized errors among the N p pulses slightly grow due to the accumulation of errors. Therefore, the approximation done in (2.29) is relevant whatever N p . Based on (2.33), the first one is defined by

|H N LF M T W R (k)| L+N
for the k th frequency bin whereas the second is equal to

|H N LF M F D (k)|
(N/Np+Q) due to (2.26). We choose L = N and Q = N Np so that the number of bins of the different power spectra are the same and correspond to the same frequencies. Let us first have a look at the power spectra obtained by using both algorithms and compare them with the reference power spectrum S

|H N LF M T W R (k)| 2N
and

|H N LF M F D (k)| (2N/Np)
are respectively presented for N p = 4 and 20 in Fig. 2.8 and 2.9. On the one hand, one can notice that when the modified FD algorithm is used and when N p increases, the power spectrum differs more and more from S N LF M ref (k). The quality of reconstruction is no longer good when N p = 20. On the other hand, the power spectra reconstructed with the TWR algorithm are almost the same and very close to

S N LF M ref (k).
Let us now analyze the influence of β on the reconstructed power spectrum by using an objective criterion. Toward this purpose, for β ∈ 0, 1.5 which varies with a step size equal to 0.03, the spectral distances 3 between

{S N LF M ref (k)} k=0,...2N -1
and the reconstructed power spectra

{ |H N LF M T W R (k)| 2N } k=0,...,2N -1 and { |H N LF M F D (k)|
(2N/Np) } k=0,...,2N -1 are computed. Different values of N p have been analyzed. In Fig. 2.10 and 2.11, the spectral distances with the modified FD algorithm and the TWR algorithm are respectively provided for 3 If X and Y are two vectors that respectively store the power spectra X(k) and Y (k) computed at the k th frequency bin with k = 0, ..., 2N -1, the spectral distance between X and Y of length 2N is defined as follows:

Sd = 1 2N ||X -Y || 2

Different comments can be made:

• For each value of N p analyzed, the spectral distances have the same type of be-haviour.

• Whatever N p and the method considered, the order of magnitude of the spectral distance is the same when β is approximately in the interval [0, 1]. This phenomenon can be explained by the fact that when β is in this interval, the instantaneous frequency is close to be a linear function of time and the power spectrum does not vary much as shown in Fig. 1.10.

• When the degree of non-linearity increases, the spectral distance increases.

• Finally, we can clearly observe that whatever the value of β and the number N p , the spectral distance that is computed when the TWR algorithm is used is much smaller than that when the modified FD algorithm is considered. In Fig. 2.12 and 2.13, the HRRPs of a stationary point target located at range R = 240 m are presented. They correspond to the modified SF-NLFM waveform with β = 1.21 combined respectively with the TWR and the modified FD algorithms in a noiseless environment when N p = 4 and N p = 20. The simulations done using MATLAB confirm the theoretical analysis given in the above section. Moreover, the range resolution R res obtained by using both algorithms is equal to 2.6 m which is the same as that of a tangent-based NLFM waveform processed by using the MF. where P m = 1 denotes the power of the m th pulse of the modified SF-NLFM waveform defined in (2.12). In Fig. 2.14 and Fig. 2.15, the evolutions of the PSLR and the ISLR versus the SNR are presented. It reveals that when the TWR algorithm is used, the PSLR converges to the same value at high SNR, for instance at SNR= 30 dB for all N p . For large values of N p (i.e. N p = 20 and N p = 50), the TWR approach outperforms the modified FD algorithm whatever the SNR. For small values of N p (i.e. N p = 4 and N p = 10), the advantage of the TWR algorithm is obvious at low and moderate SNR, whereas it lags behind the modified FD algorithm by a couple of dBs at high SNR. In the same context, the ISLR obtained by using the TWR algorithm is smaller than that obtained with the modified FD algorithm whatever N p . For instance, at SNR=30 dB, The TWR outperforms the modified FD by around 10 dB for N p = 50 and N p = 20. The better results obtained using the TWR algorithm compared to the modified FD algorithm comes at the cost of an additional computational cost. It should be noted that N p = 50 is used to reflect the behavior of our algorithm in a scenario where the proposed waveform can be used in radars designed to detect slowly-moving targets that have small RCS. In this section, the reconstructed power spectra obtained with the TWR and the modified FD algorithms are compared with the power spectrum S P C ref (k). Then, some examples of the modified SFPC waveform processed by both algorithms are provided. Moreover, the performance of the modified SFPC waveform in terms of PSLR , ISLR, and R res is analyzed. Afterwards, the performance of the modified SFPC is compared with that of the PC waveform. Then, its performance is compared with that of the standard SFPC waveform. In the following, the modified SFPC waveform is generated from a polyphase P4 pulse with T p = 1 µs and M = 100. In addition, T r = 40 µs and f c = 3 GHz.

About the reconstructed power spectrum of the modified SFPC waveform

In this section, the reconstructed power spectra of the modified SFPC obtained using the TWR defined by

|H P C T W R (k)| L+N
and the modified FD algorithms defined by

|H P C F D (k)|
(N/Np+Q) for N p = 4 and 20 are respectively presented in Fig. 2.17 and Fig. 2.18. They are compared with the reference power spectrum S P C ref (k) presented in Fig. 2.16 using the spectral distance. Similarly, as done in section 2.3.1.2, L and Q are respectively equal to N and N/N p . For any value of N p , the spectral distances have the same order of magnitude 10 -29 when the TWR algorithm is used. When the modified FD algorithm is used, the spectral distances are much higher than that obtained with the TWR algorithm. They also have the same order of magnitude for any N p . Thus, when N p = 4, Sd = 0.0264 and when N p = 20, Sd = 0.0875. This reflects the fact that the quality of reconstruction of the power spectrum is no longer good when N p is large. In Fig. 2.19 and 2.20, the HRRPs of a stationary target located at range R = 240 m are presented. They respectively correspond to the modified SFPC waveform combined with the TWR and the modified FD algorithm in a noiseless environment. For N p = 20, the range resolution is equal to 1.5 m when both algorithms are used, and is the same as that of a polyphase P4 waveform processed by using the MF. For N p = 4, R res is equal to 1.7 m using the modified FD algorithm which is slightly larger than R res = 1.5 m that obtained using the TWR algorithm. In this section, let us compare the performance of the modified SFPC waveform obtained with the modified FD algorithm with that obtained by using the TWR algorithm.

The SNR is defined as in (2.51) where P m now denotes the power of the m th pulse of the modified SFPC waveform. In Fig. 2.21 and Fig. 2.22, the evolutions of the PSLR and the ISLR verus SNR are presented respectively. Similarly, as it is the case with the modified SF-NLFM waveform, the PSLR obtained with the TWR algorithm is the same at SNR=30 dB for all N p . As the SNR decreases, the PSLR increases. The larger the number of portions, the smaller the PSLR. The same comments can be drawn for the ISLR. When the modified FD algorithm is used, the PSLR and the ISLR depends on N p over the whole SNR range. Contrasting the results of the two algorithms reveals that the PSLR when using the TWR algorithm is smaller than that obtained with the modified FD algorithm for N p = 4. The interpretation of the distinguished result for N p = 4 is provided in [START_REF] Saleh | A modified stepped frequency phase coding radar waveform designed for the frequency domain algorithm[END_REF] partially due to the fact that the cyclic-autocorrelation of the P4 code has zero sidelobes. This feature has been exploited recently to separate overlapping echoes in weather radar [START_REF] Omar | A novel barker code algorithm for resolving range ambiguity in high PRF radars[END_REF]. On the other hand, the ISLR obtained with the TWR algorithm is smaller than that obtained when using the modified FD for all N p . Again, this difference is around 10 dB at SNR=30 dB for N p = 20 and N p = 50.

To summarize the simulation, when the TWR approach is used, the PSLR and ISLR obtained for both waveforms are better than those obtained by using the modified FD algorithm for certain number of portions. This comes at the expense of additional computational cost. In addition, the quality of reconstruction is no longer good for large values of N p . As a consequence, a trade-off exists between the selection of the algorithm, the number of portions and the computational cost. Our purpose is to study the performance of the modified SFPC waveform with stationary targets. An AWGN is considered at the receiver, leading to a specific SNR. Some of the common parameters used in the whole subsection are given in Table A.2, while the others are introduced gradually. In all the simulations, two cases (non-overlap and overlap) are addressed. At the receiver, the PC waveform is processed by MF since it is proportional to the bandwidth of the received signal, the sampling frequency at the receiver for the modified SFPC waveform is N p times smaller than that of the PC. Even if this comparison could be unfair due to this difference, we propose to analyze how much the PSLR and the ISLR vary. In the first illustration, the spectrum is split into non-overlapping portions, whereas in the second one, it is split into overlapping portions. Simulation protocol: The SNR varies from -5 dB to 30 dB with a step equal to 1 dB.

For each SNR, 7000 independent realizations of the noise are generated. The SNR is defined as the ratio between the power of the PC pulse, denoted as P PC , and the variance of the noise, σ 2 n . SN R = 10log 10 (

P P C σ 2 n ) (2.52)
For the non-overlapping case and for a fair comparison, the sum of the powers of the N p pulses of the modified SFPC waveform is equal to P P C . Hence, even when N p changes, the sum of the power of all pulses is kept fixed. As for the phase codes used in this illustration, both P4 code with length M = 100 (B = 200 MHz) and polyphase Barker code with length M = 54 (B = 108 MHz) are considered. It should be noted that in this simulation, large values of N p are used to reflect the behavior of our algorithm in a scenario where the proposed waveform can be used in radars designed to detect targets that have small RCS. In addition, the larger N p is, the larger the time it will take to produce the HRRP is. For the overlapping case, the percentage of the overlap is set at 50%. In this case, the total number of samples of the modified SFPC waveform is larger than that of the PC pulse by a factor of 1.5. Thus, for a fair comparison between both waveforms, the sum of the energy of N p pulses of the modified SFPC waveform is equal to the energy of the 64 PC pulse. Furthermore, only the P4 code with length M = 100 (B = 200 MHz) and T p = 1 µs are considered for all the simulations in this illustration. Hence, both R min and R res are fixed.

Modified SFPC waveform with non-overlapping portions vs. PC waveform

Let us now study the evolutions of the PSLR and the ISLR with respect to the SNR. According to Fig. 2.23, the mean value of the PSLR for both waveforms gets worse as the SNR decreases. At very low SNR, the mean of the PSLR of the modified SFPC waveform is 1.5 dB worse than that of the PC for N p = 2. However, at high SNR the mean of the PSLR of the modified SFPC waveform is highly dependent on N p . This difference appears since an MF is applied separately to each received pulse. Furthermore, the ability of the modified SFPC waveform with N p = 4 to outperform the PC waveform comes at the expense of an increase of the range resolution, as shown in Fig. 2.19b. The good performance is mainly due to the interaction between two phenomena: First, the MF is applied to each pulse separately instead of jointly applying it to all the received echoes. Second, the cyclic-autocorrelation of the P4 code has zero sidelobes. The last feature has been exploited recently to separate overlapping echoes in weather radar [START_REF] Omar | A novel barker code algorithm for resolving range ambiguity in high PRF radars[END_REF]. In Fig. 2.25, the evolution of the PSLR of the modified SFPC waveform using polyphase Barker code is presented. The latter code does not exhibit the zero-sidelobes feature. It reveals that the modified SFPC waveform is no longer capable of outperforming the PC waveform. This result confirms our aforementioned interpretation.

In the same context, Fig. 2.24 provides the mean value of the ISLR at different SNR for both waveforms using a P4 code. At very low SNR, the value of the ISLR of the modified SFPC waveform is 2 dB worse than that of the PC waveform for N p = 2. However, at high SNR, this amount varies with N p .

Modified SFPC waveform with overlapping portions vs. PC waveform

Let us now investigate the performance of our modified SFPC waveform when an overlap between different portions exists. For this purpose, a set of Monte-Carlo simulations is carried out in which a P4 code is used as an intra-pulse modulation. In Fig. 2.26 and Fig. 2.27, the mean values of the PSLR and the ISLR of the modified SFPC waveform are respectively presented as functions of the SNR. The PSLR ranges between -29 dB for N p = 39 and -21 dB for N p = 99. Using the overlapping between the transmitted portions offers the possibility to enhance the performance compared to the non overlapping scenario. For instance, at SNR = 30 dB, when N p = 50 for the non-overlapping scenario and N p = 49 for the overlapping one, the gain in terms of PSLR is around 6.8 dB. Although the number of transmitted pulses is almost the same in both scenarios, B b for each transmitted portion in the non-overlapping scenario is equal to 4 MHz, whereas it is set at 8 MHz in the overlapping one. Therefore, this improvement comes at the expense of increasing the sampling frequency at the receiver. Moreover, as with non-overlapping portions, the PSLR of the modified SFPC waveform is larger than that of the PC waveform due to the same reasons. It is obvious that at low SNR, both the PSLR and ISLR of the modified SFPC waveform for the different number of portions N p converge to the same value.

In order to shed light more on the powerfulness of the overlapping scenario in attaining PSLR and ISLR values better than those obtained with the non-overlapping one, we present a summary of different simulations we have conducted in In this section, the performance of the modified SFPC waveform is compared with that of the SFPC waveform. Both the IFFT and the modified FD algorithms are used, because the first one is usually considered whereas the second has not yet been applied in the literature with the SFPC waveform. Simulation protocol: The SNR in both illustrations is defined by: SN R = 10log10(

P SF P C m σ 2 n ) = 10log10( P m σ 2 n ) (2.

53)

where P SF P C m and P m respectively denote the power of the m th pulse of SFPC and modified SFPC waveforms. For a fair comparison in both illustrations, the following equality holds true:

Np m=1 P SF P C m = Np m=1 P m ∀ N p (2.54)
Furthermore, the power of each transmitted pulse in both waveforms is kept fixed even when N p changes. Concerning the parameters and the phase codes used, there are some variations depending on the algorithm used at the receiver:

(a) The case of the FD algorithm: a Polyphase Barker code with length M = 60 is used instead of the P4 code as an intra-pulse modulation. The reason for this selection is due to the fact that in the FD algorithm, the output of the MF is multiplied by the inverse of the power spectrum (See section 1.6.2.6.1, step 3).

This arrangement cannot work properly with polyphase codes, such as P4, because they exhibit small values in their power spectra, as illustrated previously in Fig. 2.3.1.2. The polyphase Barker code is one of the codes that does not exhibit small values in its power spectrum. The parameters of each waveform are given in Table . 2.5. For a fair comparison, B p for both waveforms should be the same. Moreover, this table shows that B ef f is also the same. This has been achieved by making 50% overlapping between the spectra of the different pulses of the SFPC waveform at the transmitter. At the receiver, the spectrum of each received echo of the SFPC waveform is multiplied by a rectangular window of width B/(2N p ). Hence, at the receiver, only half of the transmitted spectrum of each received echo of the SFPC waveform is exploited. This methodology is used for two reasons:

To guarantee a fair comparison between both waveforms by maintaining the same B ef f , and to reduce the ISLR of the SFPC waveform as much as possible.

(b) The case of the IFFT algorithm: The parameters used in this simulation are given in Table . 2.6. Again, the parameters are selected so that B p is the same for both waveforms. Moreover, the polyphase P4 code with length M = 100 is used.

Modified SFPC vs. SFPC using FD algorithm

First, in Fig. 2.28, the HRRP of the SFPC waveform in the absence of noise when N p = 10 is presented. Then, in Fig. 2.29a, the PSLR versus SNR for both waveforms are depicted for different numbers of transmitted pulses. The PSLR of the SFPC waveform cannot exceed -13.5 dB, in the best case with the absence of noise. This is because its spectrum is transformed into a rectangular form, after applying the compression filter. Therefore, its PSLR is significantly worse than that of the modified SFPC at the whole SNR range, no matter what N p is. In Fig. 2.29b the ISLR of both waveforms is shown. It is obvious that the modified SFPC substantially outperforms the SFPC at low to moderate SNR, whereas the performance is equivalent at high SNR and large N p . Furthermore, Table . 2.5 reveals that the pulse width in modified SFPC does not vary with N p . It exhibits a smaller minimum range compared with the SFPC. This is because the pulse width of the latter waveform grows with N p . The minimum range must be as small as possible for radars that search for targets at a close range. It turns out that the exploitation of the FD algorithm with the SFPC is not favorable due to the non-rectangular shape of any phase coded pulse. As a consequence, the modified SFPC waveform is considered as an alternative to the SFPC waveform when the FD algorithm is used. is much smaller in the modified SFPC waveform, which is a great advantage. Furthermore, the latter approximately achieves the same PSLR, and even better than that of the PC waveform in some situations depending on N p . From the other side, the comparison between the modified SFPC and the SFPC waveforms processed with the FD algorithm, reveals that both waveforms attain the same range resolution and require the same sampling frequency at the receiver for the same number of transmitted pulses. However, the former leads to better PSLR and ISLR. Moreover, R min is much shorter. If the SFPC is processed with the IFFT algorithm instead, R res is enhanced by a factor of two, and also the ISLR becomes much better. Nonetheless, compared with the M.SFPC waveform, it still achieves higher PSLR and R min . In the same context, one may wonder why we do not exploit the amplitude windowing with the SFPC as an alternative approach to the modified SFPC, knowing that both are processed with the FD algorithm. It is true that amplitude windowing reduces the range sidelobes of the HRRP. However, this reduction comes at an increase in the range resolution. To cope with this shortcoming, the modified SFPC waveform is designed to permit reconstruction of a spectrum at the receiver that is close as much as possible to that of the phase code initially used at the transmitter. In this case, there is no need to apply windowing to the reconstructed spectrum.

General comments on the results

In this section, we aim at giving some comments on the processing algorithm and the number of pulses that could be selected by the practitioner. According to the obtained results, three conclusions can be drawn:

1. For a fixed number of pulses, the PSLR and the ISLR depends on the processing algorithm used at the receiver. Moreover, the computational cost of the TWR algorithm becomes significantly larger than that of the modified FD algorithm for a large number of pulses.

2. For any processing method, the larger the number of pulses, the smaller the sampling frequency required at the receiver.

3. With the TWR algorithm, the larger the number of pulses, the smaller the ISLR and the PSLR, and the larger the computational cost.

4. With the FD algorithm, the quality of reconstruction of the power spectrum is no longer good when N p is large.

The processing algorithm and the number of pulses that could be selected by the practitioner highly depend on the application of the radar. On the one hand, in scenarios where the speed of the target is fast, the dwell time 4 is small and only a small number of pulses can be used and thus the selection of the method depends on the required performance measures. On the other hand, in scenarios where the speed of the target is small, a larger number of pulses can be used. In this case, the selection of the method depends on the required performance measures and the computational cost. Therefore, a trade-off exists between the number of pulses, the processing algorithm, the sampling frequency, and the performance measures.

Conclusions

In this chapter, we have presented the processing chain of the modified SF waveform. Different variants can be considered in this chain. On the one hand, at the transmitter, the PC or the NLFM waveform can be used to generate the proposed waveform. On the other hand, at the receiver, the modified FD and the TWR algorithms can be used to reconstruct the HRRP.

The conclusions that can be drawn from the comparative studies that have been presented in this chapter are the following:

1. The proposed processing chain can be used with any type of PC and NLFM waveforms.

2. The proposed scheme offers a degree of flexibility for the practitioner to adapt the waveform to different scenarios. This can be done by varying the number of portions. The larger the number of portions, the better the performance, but the longer the time required for transmitting the pulses.

3. The performance of the proposed waveform processed by using the TWR in terms of PSLR and ISLR is better than that obtained with the modified FD algorithm for a certain number of portions. This comes at the expense of an additional computational cost.

4. The performance of the proposed waveform in terms of PSLR and ISLR is better than that of the traditional SF waveforms processed by either the IFFT, the FD, or the MF. With the modified SFPC waveform, the minimum range is smaller than that of the traditional SFPC waveform.

5. There is considerable room to enhance the performance of the modified FD algorithm when it is applied to the modified SFPC waveform by exploiting the overlapping methodology. This enhancement comes at the expense of a higher computational cost.

6. The transmission of the pulses of the modified SF in a pseudo-random way prevent the opponent signal analyzer from reconstructing the transmitted waveform correctly.

7. In this chapter, we focus our attention on cases where there is no Doppler effect. This serves as a first step in the whole analysis that we aim at conducting.

CHAPTER 3

VARIABLE CHIRP RATE STEPPED-FREQUENCY LFM WAVEFORM

Introduction

In this chapter, a second processing chain is proposed to overcome the drawback of the standard combination of SF with NLFM. More particularly, we suggest approximating the NLFM by a piecewise linear waveform and then using it in a SF framework. Thus, a variable chirp rate SF-LFM waveform is proposed where SF is combined with a train of LFM pulses having different chirp rates, durations, and bandwidths. In this chapter, the latter parameters are derived from a tangent-based NLFM waveform. At the receiver, a generalized version of the TD algorithm is proposed to process the received echoes in order to deduce the HRRP. As our purpose is to obtain an HRRP whose properties such as the peak sidelobe ratio, the integrated sidelobe ratio and the range resolution are of the same magnitude orders as those obtained using a tangent-based NLFM waveform, an optimization issue is addressed by using a genetic algorithm to deduce the parameters of the proposed waveform. This chapter is organized as follows: firstly, the processing chain of the proposed waveform is presented. This includes what is done at the transmitter and at the receiver. Secondly, the optimization issue is addressed. Finally, simulations and results are provided.

Our contribution: a processing chain of the variable chirp rate SF-LFM waveform

In this section, we present two of our contributions: firstly, the proposed waveform is presented. It is a generalization of the standard SF-LFM waveform whose chirp rate varies from pulse to pulse. The latter is called variable chirp rate (VCR) SF-LFM, whose parameters correspond to a set of pulse bandwidths and of pulse durations. We show how to derive them from an NLFM waveform. In this chapter and without loss of generality, we suggest considering the tangent-based NLFM waveform presented in chapter 1 (see section 1.6.1.3.1). Secondly, the whole processing chain from the transmitter to the receiver is detailed. More particularly, we present how the parameters of the waveform are exploited to construct a train of baseband pulses at the transmitter and how the received signals are processed to produce the HRRP.

Generalization of the SF-LFM waveform: the VCR SF-LFM waveform

The proposed waveform is a generalized form of the SF-LFM waveform defined in (1.69) where the pulse duration and the bandwidth changes from one pulse to another, and hence the chirp rate varies from one pulse to another. Its notation becomes

γ (m) = B (m) p T (
   ∆f (1) = -B 2 + B (1) p 2 ∆f (m) = -B 2 + m-1 l=1 B (l) p + B (m) p 2 for m = 2, . . . , N p (3.1)
The resulting waveform will be called "variable chirp-rate SF-LFM waveform" defined by the set of the pulse bandwidths {B

Fig. 3.1 Stepped-frequency linear frequency modulated waveform with variable chirp rate

In the next section, let us see how to generate it from an NLFM waveform.

Generation of the VCR SF-LFM waveform at the transmitter

Let us first present some details of the different steps done at the transmitter.

1. From the tangent-based NLFM waveform to its linear piece-wise approximation, called PW-NLFM: Let us consider a tangent-based NLFM waveform whose instantaneous frequency satisfies:

f tan (t) = f (t -T p /2)) + B/2 (3.2)
where f (t) is defined in (1.50). This instantaneous frequency is illustrated in Fig. 3.2a. Remark: while f satisfies f (t) + f (-t) = 0 since its an odd function, f tan has the following property:

f tan (t) -B/2 = f (t -T p/2) = f odd -f (-t + T p /2) (3.3) = (3.2) -(f tan (-t + T p ) -B/2)
Consequently, one has:

f tan (t) + f tan (T p -t) = B (3.4)
This relation will be useful in the following to express the instantaneous frequency of this PW waveform.

In order to derive the parameters of our waveform, we first suggest approximating f tan (t) by a linear piece-wise function. We could select the time instants between 0 and T p , but as the tangent-based NLFM waveform is odd with respect to T p /2, we suggest selecting the time instants between 0 and T p /2. Then, the time instants between T p /2 and T p as well as the frequencies can be deduced so that the anti-symmetry is maintained. Thus, at different time instants denoted as {0 ≤ τ m ≤ T p /2} m=1,...,L+2 , the instantaneous frequency takes the values {0 ≤ ν m ≤ B/2} m=1,...,L+2 and coincides with f tan (t) as follows:

           (τ 1 , ν 1 ) = (0, 0) (τ L+2 , ν L+2 ) = (T p /2, B/2) ν m = f tan (τ m ) = f (τ m -T p /2) + B/2
for m = 1, ..., L + 2

(3.5)
For T p /2 ≤ t ≤ T p , the time instants and the corresponding frequencies can be deduced by taking into account the symmetry of the instantaneous frequency of the tangent-based NLFM waveform. Thus, one has:

     (τ 2L+3 , ν 2L+3 ) = (T p , B) τ m = T p -τ 2L+3-m for m = L + 3, ..., 2L + 3 ν m = f tan (τ m ) = B -ν 2L+3-m (3.6)
Therefore, given (3.6), the number of couples of parameters required to define a PW-NLFM waveform can be reduced to L+2, namely {(τ m , ν m )} m=1,...,L+2 . It is worthy to note that only L couples are free to be selected by the practitioner since (τ 1 , ν 1 ) and (τ L+2 , ν L+2 ) are fixed as shown in (3. As illustrated in Fig. 3.2b, the two pieces around T p /2 have the same duration and bandwidth. Two possibilities can hence be considered to define the pulses of the proposed waveform from the SC-PW-NLFM waveform: On the hand, the two pieces around T p /2 can be used to generate two pulses of the proposed waveform. In this case, N p = 2L + 2.

On the other hand, the two pieces are combined to represent a single pulse with a duration and a bandwidth equal to twice that of one piece. In this case, N p = 2L + 1.

In the following, we will focus our attention on an approximation where the first possibility is considered. There is no particular piece in this case and there are the same number of pieces characterizing the intervals [0, T p /2] and [T p /2, T p ]. This is also in accordance with the definition of a train of piecewise linear function given in [START_REF] Saeedi | Synthetic aperture radar imaging using nonlinear frequency modulation signal[END_REF]. Thus, the N p LFM baseband pulses shown in Fig. 3.3a can be modeled for m = 1, ..., 2L + 2 as:

s bb,m (t) =rect 1 τ m+1 -τ m (t -(m -1)T r - τ m+1 -τ m 2 ) (3.7) × exp jπγ (m) (t -(m -1)T r - τ m+1 -τ m 2 ) 2
where the term -τ m+1 -τm

T (m) p = τ m+1 -τ m for m = 1, ..., L + 1 T (m) p = T (2L-m+3) p for m = L + 2, ..., 2L + 2 (3.8) the bandwidth B (m) p
of the m th pulse is defined as follows:

B (m) p = ν m+1 -ν m for m = 1, ..., L + 1 B (m) p = B (2L-m+3) for m = L + 2, ..., 2L + 2 (3.9)
and the chirp rate of the m th pulse is given by:

γ (m) = B (m) p τ m+1 -τm for m = 1, ..., L + 1 γ (m) = γ (2L-m+3) for m = L + 2, ..., 2L + 2 (3.10)
Afterwards, the whole train is frequency translated to the carrier frequency f c as shown in Fig. 3.3b. This is done by multiplying s bb,m (t) with the proper exponential term. For the m th pulse (m ∈ 1, 2L + 2 ), the proposed transmitted waveform can be expressed as follows:

s tx,m (t) = s bb,m (t)exp(j2πf (m) c t) (3.11)
where f

(m) c
is defined in (1.64).

By combining (3.7), (3.8) and (3.11), the transmitted waveform can be expressed in a detailed manner as follows:

s tx,m (t) = rect 1 T (m) p (t -(m -1)T r - T (m) p 2 ) (3.12) exp j2π γ (m) 2 (t -(m -1)T r - T (m) p 2 ) 2 + (f c + ∆f (m) )t
In the next section, let us analyze how the receiver part is organized.

Processing the VCR SF-LFM waveform at the receiver

In order to process the back-scattered received echoes in the time domain, there exists a well-defined algorithm in the literature called the TD algorithm. However, the latter deals with a train of pulses having a constant chirp rate, with constant pulse-width and bandwidth, which is not the case in this chapter. Thus, We suggest generalizing the existing TD algorithm to fit with the variable chirp rate scenario. It aims at reconstructing a wide-band NLFM waveform by coherently combining an ensemble of variable narrow-bandwidth chirp waveforms in the time domain. The resulting time-frequency relationship of the reconstructed waveform is equivalent to a baseband version of the wide-band PW-NLFM waveform similar to the one shown in Fig. 3.2, but centered at zero Hz. At the receiver, the signal can be formulated as follows:

s rx,m (t) = α m s tx,m (t -t d ) + n(t) (3.13)
where α m and n(t) respectively denote the m th channel attenuation coefficient associated with the m th pulse, and the additive measurement white Gaussian noise. In the sequel, α m = 1 is considered because the burst that consists of 2L + 2 pulses is assumed to be transmitted during the CPI, hence it is subject to the same channel effect. Furthermore, in order to ease the grasp of the steps of our algorithm, we will focus on the signal part only. This amounts to addressing the ideal case. Subsequently, the "ideal" received m th pulse is down converted to baseband as shown in Fig. 3.4a. This is done by multiplying the received m th pulse with the appropriate sinusoidal signal as follows:

s rx,bb,m (t) = (3.13) s tx,m (t -t d )exp(-j2πf (m) c t) = (3.11) s bb,m (t -t d )exp(-j2πf (m) c t d ) (3.14)
This amounts to saying that:

s rx,bb,m (t) = rect 1 T (m) p (t -(m -1)T r - T (m) p 2 -t d ) (3.15)
× exp j2π γ (m) 2 (t -(m -1)T r - T (m) p 2 -t d ) 2 -f (m) c t d )
Then, we suggest frequency shifting each pulse to its proper position as shown in Fig. 3.4b. This is done by multiplying each pulse by the proper exponential factor as follows:

s f shif t rx,bb,m (t) = s rx,bb,m (t)exp j2π∆f (m) t (3.16)
s f shif t rx,bb,m (t) = rect 1 T (m) p (t -(m -1)T r - T (m) p 2 -t d ) (3.17)
× exp j2π γ (m) 2 (t -(m -1)T r - T (m) p 2 -t d ) 2 + ∆f (m) (t -(m -1)T r - T (m) p 2 -t d ) × exp(-j2πf c t d )exp j2π∆f (m) ((m -1)T r + T (m) p 2 )
By introducing φ shif t rx,bb,m (t) = 2π γ (m) 2 t 2 + ∆f (m) t , the above signal can be rewritten as follows:

s f shif t rx,bb,m (t) = rect 1 T (m) p (t -(m -1)T r - T (m) p 2 -t d ) (3.18) × exp jφ f shif t rx,bb,m (t -(m -1)T r - T (m) p 2 -t d ) × exp(-j2πf c t d )exp j2π∆f (m) ((m -1)T r + T (m) p 2 )
Given the set of received pulses that have been frequency-shifted, the purpose is now to reconstruct the whole waveform. The reconstructed waveform can start at a time T start . Moreover, the time shifts cannot depend on t d for two reasons: t d is unknown and the delay lines that will be used to time shift the signals are pre-defined and cannot be adjusted to t d . Thus, we could consider that the reconstruction of the waveform starts T w seconds after receiving the last pulse. This means that:

T start = (2L + 1)T r + T p -τ 2L+2 + t d + T w (3.19)
In this case, as the first pulse has to start at time T start , it has to be time-shifted by:

δt 1 = T start -t d (3.20) = (3.19)
(2L + 1)T r + T p -τ 2L+2 + T w
The second pulse has to start at time T start + τ 1 . Therefore, it has to be time-shifted by:

δt 2 = T start + τ 2 -(T r + t d ) (3.21)
= (3.19) 2LT r + T p + τ 2 -τ 2L+2 + T w
More generally, the m th pulse has to start at time τ m and hence must be time-shifted by:

δt m = T start + τ m -((m -1)T r + t d ) (3.22) = (3.19)
(2L + 3 -m)T r + T p + τ m -τ 2L+2 + T w
Given (3.22), the time shifted pulses can be written as:

s tshif t rx,bb,m (t) = rect 1 T (m) p (t -(m -1)T r -δt m - T (m) p 2 -t d ) (3.23) × exp jφ f shif t rx,bb,m (t -(m -1)T r -δt m - T (m) p 2 -t d ) × exp(-j2πf c t d )exp j2π∆f (m) ((m -1)T r + T (m) p 2 )
or, equivalently:

s tshif t rx,bb,m (t) = rect 1 T (m) p t -(T start + τ m + T (m) p 2 ) (3.24) × exp jφ f shif t rx,bb,m (t -(T start + τ m + T (m) p 2 )) × exp(-j2πf c t d )exp j2π∆f (m) ((m -1)T r + T (m) p 2 )
In order to reconstruct the PW-NLFM waveform, a continuity in the phases of the pulses should be guaranteed before adding them. For this purpose, one just has to express s tshif t rx,bb,m (T start + τ m+1 ) and s tshif t rx,bb,m+1 (T start + τ m+1 ). At these time instants, the rect function is always equal to 1. Moreover, exp(-j2πf c t d ) does not play a role for the continuity as it appears for each time interval. Therefore, the continuity must be guaranteed between:

exp jφ f shif t rx,bb,m (T start + τ m+1 -(T start + τ m + T (m) p 2 )) (3.25)
× exp j2π∆f (m) ((m -1)T r + T (m) p 2 )
and

exp jφ f shif t rx,bb,m+1 (T start + τ m+1 -(T start + τ m+1 + T (m+1) p 2 )) (3.26)
× exp j2π∆f (m+1) (mT r + T (m+1) p

)

Let us express the phases of both terms:

φ f shif t rx,bb,m ( T (m) p 2 ) + 2π∆f (m) ((m -1)T r + T (m) p 2 ) = 2π γ (m) 2 ( T (m) p 2 ) 2 + ∆f (m) ((m -1)T r + T (m) p ) (3.27)
and φ f shif t rx,bb,m+1 (-

T (m+1) p 2 ) + 2π∆f (m+1) (mT r + T (m+1) p 2 ) (3.28) = 2π γ (m+1) 2 ( T (m+1) p 2 ) 2 + mT r ∆f (m+1)
Let us start first the first case, i.e. m = 1, this leads to:

φ f shif t rx,bb,1 ( T (1) p 2 ) + 2π∆f (1) T (1) p 2 = 2π γ (1) 2 ( T (2) p 2 ) 2 + ∆f (1) T (1) p (3.29)
φ f shif t rx,bb,2 (-

T (2) p 2 ) + 2π∆f (2) (T r + T (2) p 2 ) = 2π γ (2) ( T (2) p 2 ) 2 + ∆f (2) T r (3.30)
Let us introduce a phase compensation u 2 so that:

φ f shif t rx,bb ,1 ( T (1) p 2
) + 2π∆f (1) T

(1) p 2 = φ f shif t rx,bb,2 (- T (2) p 2 ) + 2π∆f (2) (T r + T (2) p 2 ) + 2πu 2 (3.31)
This means that s tshif t rx,bb,2 (t) is multiplied by exp(j2πu 2 ). By combining the three equations written above, one can deduce u 2 .

u 2 = - γ (2) 2 ( T (2) p 2 ) 2 -∆f (2) T r + γ (1) 2 ( T (1) p 2 ) 2 + ∆f (1) T (1) p (3.32)
More generally, one has:

           u 1 = 0 u m = u m-1 -γ (m) 2 ( T (m) p 2 ) 2 -(m -1)∆f (m) T r + γ (m-1) 2 ( T (m-1) p 2 ) 2 + ∆f (m-1) (m -2)T r + T (m-1) p for 2 ≤ m ≤ 2L + 2 (3.33)
At this stage, the reconstructed waveform should be processed by a MF in order to produce the HRRP. So far, the signal processing steps have been illustrated in the continuous-time domain. In the sequel, the way to process the signal in the discrete-time domain is presented. Instead of directly sampling the received signal at a sampling frequency equal to B, we propose to operate in two steps: sampling at a lower rate and then interpolating. The advantage of the proposed processing chain over the one used for the SC-PW-NLFM waveform is to avoid a high sampling frequency thanks to the SF methodology used. More particularly, the following steps are carried out:

1. For each received pulse, sample the baseband signals in (3.14) by using the sampling frequency that satisfies:

F (Rx) s = 1
T (Rx) s = B η (3.34)
where η is defined by:

η =     B 2 max 1≤m≤L+2 (B (m) p )     (3.35)
with . the floor integer part. This means for instance that:

η = 2 if B 6 < max 1≤m≤L+2 B (m) p ≤ B 4 (3.36)
and

η = 3 if B 8 < max 1≤m≤L+2 B (m) p ≤ B 6 (3.37)
In this case, the number of samples that represent each pulse is equal to

F (Rx) s T (m) p . If F (Rx) s T (m) p
is not an integer, the duration of the pulse cannot be retrieved exactly and hence must be approximated. Thus, the truncated duration and bandwidth of the (m + 1) th sampled received pulse must be considered and are respectively denoted as:

T (m) p,trn = F (Rx) s T (m) p F (Rx) s (3.38) B (m) p,trn = ν m+1,trn -ν m for m = 1, ..., L + 1 B (m) p,trn = B (2L-m+1) p,trn for m = L + 2, ..., 2L + 2 (3.39)
where ν m+1,trn = f tan (τ m+1,trn ) and τ m+1,trn is defined as:

τ 1,trn = 0 τ m+1,trn = τ m + T (m)
p,trn for m = 1, ..., L + 1

(3.40)
It should be noted that (3.38) and (3.40) will be used in the following steps when

F (Rx) s T (m) p
is not integer to avoid any problems in the reconstruction of the HRRP. Then, for the sake of simplicity, let us address the example where the delay is strictly a multiple of the sampling period, at the receiver, i.e.:

t d = dT (Rx) s (3.41) Hence, using (3.11), (3.14) becomes for n ∈ 0, N m -1 with N m = T (m) p F (Rx) s : s rx,bb,m (n) = s bb,m (n -d)T (Rx) s exp -j2πf (m) c dT (Rx) s (3.42)
2. Interpolate s rx,bb,m (n) so that the new sampling frequency becomes:

F (int) s = 1 T (int) s = B = ηF (Rx) s (3.43)
This can be done either in the time domain or in the frequency domain 1 . The resulting signals is denoted s int rx,bb,m (n) for n ∈ 0, ηN m -1 . Sampling at F (Rx) s and then interpolating the received signal instead of directly sampling it at the sampling frequency B makes it possible to use a cheap ADC.

Remark: For a proper interpolation of s rx,bb,n (n), N m should be greater than 2.

Thus, the minimum duration between any two consecutive time instants should be greater than a threshold . For any m ∈ 1, L + 1 , this leads to:

T (m) p,trn ≥ T (m) p ≥ = 2
F (Rx) s (3.44)
3. Frequency shift each pulse to its proper position so that it occupies a frequency band shifted by -B/2 Hz compared with its corresponding one in the PW-NLFM waveform shown in Fig. 3.2. This is done by multiplying each pulse by the proper exponential factor as follows:

s int,f shif t rx,bb,m (n) = s int rx,bb,m (n)exp j2π∆f (m) nT int s (3.45)
where ∆f (m) is defined in (

is not integer, ∆f (m) satisfies: is not integer, δt m now satisfies:

         ∆f (1) = -B 2 + B (1) p,trn 2 ∆f
(m) = -B 2 + m-1 l=1 B (l
δt m = (2L + 3 -m)T r + T p + τ m,trn -τ 2L+2 + T w (3.47)
5. Add a phase correction term to each pulse to avoid any phase discontinuity in the reconstructed PW-NLFM waveform. It is done by multiplying (3.45) with an appropriate exponential term, as follows:

s int,f shif t,pc rx,bb,,m (n) = s int,f shif t rx,bb,m (n)exp(j2πu m ) (3.48)
where u m is defined in (3.33). When F

(Rx) s T (m) p
is not integer, u m satisfies:

                 u 1 = 0 u m = u m-1 -γ (m) 2 T (m-1) p,trn - T (m) p 2 2 -(m -1)∆f (m) T r + γ (m-1) 2 T (m-1) p 2 2 +∆f (m-1) (m -2)T r + T (m-1) p,trn for 2 ≤ m ≤ 2L + 2 (3.49)
6. Add the obtained pulses together. Therefore, the reconstructed waveform denoted as s P W -N LF M rx (nT

(int) s -dT (int) s
), is a delayed discrete baseband version of a PW-NLFM waveform whose instantaneous frequency is similar to the one given in Fig. 3.2 but its spectral components are centred around the zero-component instead of B/2.

7. Apply a MF in the frequency domain to the reconstructed waveform. It consists in applying an DFT to s P W -N LF M rx (nT

(int) s -dT (int) s
) after padding it with ηN m -1 zeros. Then, multiply the result by S P W -N LF M tx,bb (k) , where . denotes the conjugate,

S P W -N LF M tx,bb (k) is the DFT of s P W -N LF M tx,bb,pad (n). The sequence s P W -N LF M tx,bb,pad (n)is a padded version of s P W -N LF M tx,bb (n)
with ηN m -1 zeros. Thus, the obtained result can be written for k ∈ 0, 2ηN m -2 as:

Z(k) = |S P W -N LF M tx,bb (k)| 2 exp -j2π k 2ηN m -1 2R c (3.50)
8. Produce the HRRP by applying an IDFT to (3.50).

In this section, the processing chain from the transmitter to the receiver has been presented. Nevertheless, the performance are not optimized in terms of PSLR, ISLR, and range resolution. The latter are mainly related to the selection of the parameters of the waveform, namely the time instants {τ m } m=2,...,L+1 . In the following, we present a way to choose them by using a genetic algorithm.

Optimizing the parameters of the VCR SF-LFM waveform

The performance in terms of PSLR, ISLR, and range resolution of the proposed waveform depend on the set of parameters {(τ m , ν m )} m=2,...,L+1 . Our purpose is to select the parameters that lead to performance the closest to that of the tangent-based NLFM waveform. As the set of frequencies {ν m } m=2,...,L+2 can be deduced from the set of time instants {τ m } m=2,...,L+2 by using (3.5), only {τ m } m=2,...,L+1 have to be optimized. For this purpose a multi-objective optimization problem can be considered. It can take into account the performance of the processing chain in terms of PSLR, ISLR, and range resolution. This can be done by minimizing a single fitness function that consists of a weighted sum of the normalized errors on the PSLR, the ISLR, and the range resolution. It is defined as follows:

F (x) = (1 -λ 1 -λ 2 ) P SLR(x) -P SLR ref P SLR ref (3.51) + λ 1 ISLR(x) -ISLR ref ISLR ref + λ 2 R res (x) -R res,ref R res,ref
where 0 ≤ λ 1 ≤ 1 and 0 ≤ λ 2 ≤ 1 -λ 1 are the weights. x denotes the vector storing the time instants {τ m } m=2,...,L+1 . P SLR(x), ISLR(x), and R res (x) are respectively the PSLR, the ISLR, and the range resolution obtained when using the waveform parameters x. In addition, P SLR ref , ISLR ref , and R res,ref are respectively the ISLR, the PSLR, and the range resolution of the tangent-based NLFM waveform. Various optimization algorithms have been proposed to solve optimization problems when locating the global optimum is a difficult task. A grid search approach can be used to explore the solution space in order to find the best candidate. Nevertheless, the computational cost is high. To alleviate this problem, alternative solutions exist such as evolutionary algorithms (EAs). The principle is still the same: given a random population of candidates selected in the space of solution, new populations are created in order to explore the space of solutions and evolve towards the candidate that minimizes a beforehand defined fitness function. Several families of approaches have been proposed in the literature and used in a wide range of applications: simulated annealing, antcolony algorithms, particle swarm optimization, differential evolution (DE) and genetic algorithm (GA) [79] [110]. In this chapter, we focus our attention on this last algorithm.

Generalities about genetic algorithm

GA is a stochastic global search-based optimization technique based on the principle of genetics and natural selection [START_REF] Goldberg | Genetic Algorithms in Search, Optimization and Machine Learning[END_REF]. It has been extensively used as search and optimization tool in many fields and problem domains including sciences, manufacturing and engineering [START_REF] Chin | Robust genetic algorithm and fuzzy inference mechanism embedded in a sliding-mode controller for an uncertain underwater robot[END_REF] [1]. In Fig. 3.6, the different steps of the GA are presented in a flowchart. GA is an iterative process. It starts with an initial population consisting of a certain number of candidates. The relevance of each candidate is evaluated according to a beforehand defined fitness function. Then, new solutions are generated using selection, crossover, and mutation operators.

1. Selection operator: a selection criterion is applied to select a certain number of candidates, each correspond to a vector of parameters, from the population according to their fitness values. Thus, candidates with the best fitness values have more chance to be selected for crossover and mutation.

2. Crossover operator: it consists in generating new candidates from two selected candidates called parents. In the literature, different types of crossover exist such as single-point, double-point, multi-point, matrix, uniform, etc [START_REF] Sivanandam | Introduction to genetic algorithms[END_REF]. In this chapter, we focus on the single-point crossover. It consists in randomly choosing a locus and exchanging the parameters between the parents to create two candidates.

3. Mutation operator: the mutation operator is applied individually to each candidate. It alters each parameter randomly with a probability p M . In this chapter, we use a specific mutation.

After mutation, a new population that consists of the best candidates is thus created. This process is iterated until convergence. In this section, we focus our attention on the way the steps of the GA are implemented with the proposed waveform when taking into account the fact that there is a constraint on the minimum duration between any two consecutive time instants. See (3.44) in section 3.2.3. In this case, we operate as follows:

• Generation of the initial population: As τ 1 = 0 and τ L+2 = T p /2, the initial population is generated so that for m ∈ 1, L , the time instants satisfy:

τ m + < τ m+1 < T p /2 -(L -m)
where was introduced in (3.44). As a consequence, the initial population corresponds to a set of Q p vectors of size L.

• Selection: It consists in randomly selecting two vectors, namely

C f i = [τ f 2 , τ f 3 , ..., τ f k-1 , τ f k , ..., τ f L+1 ] and C m i = [τ m 2 , τ m 3 , ..., τ m k-1 , τ m k , ..., τ m L+1
] from the population at the i th generation.

• Crossover: After using the single-point crossover, the resulting candidates, called children, are:

C 1 = [τ f 2 , τ f 3 , ..., τ f k-1 , τ m k , ..., τ m L+1 ] C 2 = [τ m 2 , τ m 3 , ..., τ m k-1 , τ f k , ..., τ f L+1 ]
The time instants of one of the obtained candidates necessarily respect the constraint. Thus, if τ m k -τ f k-1 > , C 1 is chosen. Otherwise, C 2 is kept. Before belonging to the population of candidate vectors at the (i + 1) th iteration, a mutation will be done on the choosen child vector. Remark: When L is equal to 1, the above crossover cannot be applied. Therefore, we suggest using a linear crossover. In this case, the child is defined as follows:

C 1 = 1 2 C f i + 1 2 C m i (3.52)
• Mutation: Let us assume that C 1 was chosen by the crossover operator. We suggest mutating each element, or equivalently each time instant, of C 1 separately in a way that guarantees that the resulting candidate respect the constraint (3.44). It is done as follows:

The mutations on the time instants are done in a successive manner. Let us start with the first time instant C 1 (1). A random number r uniformly distributed between 0 and 1 is drawn. If r is smaller than the probability of mutation p M , the mutation leads to the time instant equal to C 1 (1) + x 1 , where x 1 is a random number that belongs to the interval ] -C 1 (1) + , C 1 (2) -C 1 (1) -[. Thus, after mutating the first time instant,

C 1 becomes [τ f 2 + x 1 , τ f 2 , ..., τ f k-1 , τ m k , ..., τ m L+1 ]. More generally, for m ∈ 1, L , the mutation leads to the m th time instant equal to C 1 (m) + x m where x m ∈]C 1 (m -1) -C 1 (m) + , C 1 (m + 1) -C 1 (m) -[ with C 1 (0) = 0. If r > p M ,
the value of the time instant is not changed.

The selection, crossover, and mutation are repeated until L new candidates are created. Then, the best L candidates among the new candidates and those of the i th population are selected to create the population at the (i + 1) th generation.

Results and discussions

In this section, the results obtained using GA are presented. This section is organized as follows: firstly, the simulation protocol is given. Then, three illustrations are considered. The first one deal with L = 1, the second with L = 2, and the last one with L = 10.

Simulation protocol

In the simulations, we focus on the tangent-based NLFM that has the following parameters: T p = 20 µs, β = 1.22 and B = 100 MHz. In addition, the reference measures that are considered in the optimization issue are given in Table 3.1. Concerning GA, the size of the population Q p is equal to 200 for L = 1, 2 and 1000 for L = 10. The probability of mutation p M is set at 0.1 and = 0.12 µs. Without taking into account optimization criteria based on the properties of the HRRP, various a priori approximations of the non-linear function describing the time evolution of the instantaneous frequency could be considered. Among them, the first method would amount to searching for the time instant τ 1 that reduces the following criterion:

D 1 (τ 2 ) = τ 2 0 f tan (t) - f tan (τ 2 ) τ 2 t dt (3.53) + Tp 2 τ 2 f tan (t) - f tan ( Tp 2 ) -f tan (τ 2 ) Tp 2 -τ 2 (t -τ 2 ) -f tan (τ 2 ) dt
As the function f tan (t) is concave in the interval [0, Tp 2 ] (its second derivate is negative on this interval), the differences in the integrals are positive. By denoting F tan (t) the primitive of f tan (t), this leads to:

D 1 (τ 2 ) = F tan ( T p 2 ) -F tan (0) - τ 2 2 f tan (τ 2 ) (3.54) -( T p 2 -τ 1 )f tan (τ 2 ) - 1 2 f tan ( T p 2 ) -f tan (τ 2 ) ( T p 2 + τ 2 ) + τ 2 f tan ( T p 2 ) -f tan (τ 2 )
By reorganizing the terms, one has:

D 1 (τ 2 ) = F tan ( T p 2 ) -F tan (0) - T p 4 f tan (τ 2 ) (3.55)
+ 1 2 (τ 2 - T p 2 )f tan ( T p 2 )
Then, by deriving with respect to τ 2 , one must find τ 2 so that:

- T p 4 df tan (τ 2 ) dτ 2 + 1 2 f tan ( T p 2 ) = 0 (3.56)
or equivalently:

β tanβ 1 cos 2 2β(τ 2 -T p /2)/T p -2 = 0 (3.57)
Based on the above reasoning, this means that τ 1 would satisfy:

0 ≤ τ 2 = - T p 2β cos -1 ( β 2tanβ ) + T p 2 ≤ T p 2 (3.58)
The second approach would be following. Let us denote by D 2 (t) the difference between f tan (t) and its linear trend which corresponds to the linear time evolution of the instantaneous frequency defined by B Tp t. The determination of the time instant between 94 0 and Tp 2 would amount to find when the derivative of D 2 (t) is equal to 0, i.e.:

d dt D 2 (t) = B T p ( β tanβ 1 cos 2 2β(t -T p /2)/T p -1) (3.59)
0ne can deduce that D 2 (t) has its maxima at:

0 ≤ τ 2 = - T p 2β cos -1 ( β tanβ ) + T p 2 ≤ T p 2 (3.60)
The performance measures obtained with both methods are provided in Table . 3.2. In addition, the sampling frequency at the receiver in both cases is equal to 100 MHz. Let us first address the case when the weights are selected as follows: λ 1 = 0.4 and λ 2 = 0.2. This means that we consider that the PSLR and ISLR has the same level of importance and greater than that of R res .

The time instants as well as the PSLR, the ISLR, and the range resolution that have been obtained using GA are given in Table .3.3. In addition, the sampling frequency at the receiver is equal to 100 MHz. Moreover, the time evolution of instantaneous frequency of the tangent-based waveform as well as the piecewise-linear approximation optimized by taking into account the criterion (3.51) are provided in Fig. 3.7. Finally, the corresponding HRRP of a stationary target located at R = 8000 m is given in Fig. 3.8. Let us compare the performance measures presented in Table . 3.3 with those given in Table . 3.2. We can notice that the PSLR, the ISLR, and the range resolution are much smaller when considering the optimization issue.

Waveform parameters based on the multi-objective criterion deduced by GA and corresponding performance measures: any set of weights

Let us now address the case when the weights vary. Thus, λ 1 varies between 0.1 and 1, and λ 2 varies between 0 and 1 -λ 1 , both with a step equal to 0.1. In Fig. 3.9-3.12, the time instant, the PSLR, the ISLR, and the range resolution are respectively presented as functions of λ 1 and λ 2 . by the normalized difference on the PSLR. When the sum of λ 1 and λ 2 becomes closer to 1, F (x) is mainly defined by the normalized differences on the ISLR and the range resolution and one can notice that the PSLR increases. For instance, for λ 1 = 0.1. and λ 2 < 0.6, the PSLR remains unchanged. When λ 1 increases, the PSLR increases. The performance measures presented in Fig. 3.9-3.12 make it possible to define the Pareto front for the multi-objective optimization issue. Moreover, based on the time instants presented in Fig. 3.9, the sampling frequency at the receiver is equal to 100 MHz for any set of weights.

When comparing the values of the PSLR, the ISLR and the range resolution we obtain with the performance measures of reference, it is true that there is a loss. The latter is probably due to the strong approximation that is made when only one time instant is used. Thus, increasing L should reduce it. For this reason, in the next section, L = 2 is considered.

Simulation results and comments, L = 2

The same methodology as above is considered. In this simulation, λ 1 still varies between 0.1 and 1, and λ 2 still varies between 0 and 1 -λ 1 , both with a step equal to 0.1. In Fig. 3.14-3.17, the PSLR, ISLR, the range resolution, and the sampling frequency at the receiver are respectively presented as functions of λ 1 and λ 2 when L = 2. For a fixed value of λ 1 , when λ 1 + λ 2 = 1, the PSLR takes the maximum value. This is coherent with the fact that the criterion to be optimized no longer depends on the PSLR. When λ 1 = 0.1 and λ 2 = 0, the PSLR is the closet one to P SLR ref .

In this case, a great importance is given to the PSLR. Let us now look at λ 1 = 0.3. For values of λ 2 smaller than 0.7, the PSLR is almost the same. Given Fig. 3.15, the ISLR takes the maximum value when λ 1 = 0.1 and λ 2 = 0.9 and the minimum value when λ 1 = 1 and λ 2 = 0. According to Fig. 3.17, the sampling frequency at the receiver depends on λ 1 and λ 2 . It takes two distinct values: 50 MHz and 100 MHz. As a conclusion, when L increases to 2, the performance measures become more closer to the reference ones. Moreover, the sampling frequency at the receiver could be reduced.

We may a priori imagine that increasing the number of instants more and more should lead to better performance measures and a smaller sampling frequency at the receiver.

• For a fixed number of time instants, the sampling frequency, the time instants, and the performance measures vary with the weights used in the multi-objective criterion

• Increasing the number of time instants leads to performances closer to the reference ones and reduces the sampling frequency at the receiver.

In practice, the CPI is limited by several factors. They depend on the application of the radar. For instance, in radar target classification, the CPI is limited by the speed of the target. In scenarios where the speed of target is fast, the dwell time2 is small and thus only a small number of time instants can be used. On the contrary, in scenarios where the speed of target is small or moderate, a larger number of time instants can be used. Therefore, a trade-off exists between the number of time instants on the one hand, and the performance measures and the sampling frequency on the other hand.

Conclusions

In this chapter, a processing chain, from the transmitter to the receiver, of the VCR SF-LFM waveform is proposed. It is based on a piecewise linear approximation of the NLFM waveform and a generalized version of the TD algorithm. Our goal is to produce an HRRP whose features are as close as possible as the ones of the HRRP of a wideband NLFM waveform. To this end, the parameters defining the piecewise linear approximation of the NLFM waveform are selected by considering a multi-objective optimization problem taking into account the performance of the waveform in terms of PSLR, ISLR and range resolution. More particularly, the criterion to be minimized is a linear combination of three normalized differences. For a given set of weights, the set of parameters is obtained by using a genetic algorithm. Although a tangent-based NLFM is used, the proposed methodology can be applied to any NLFM waveform. The simulation results reveal that the performance of the resulting waveforms varies with the weights used in the multi-objective criterion and the number of LFM pulses that is considered.

CHAPTER 4

Conclusions and perspectives

In this thesis, our purpose was to design different processing chains dedicated to high range resolution radar waveforms:

1. A processing chain from the transmitter to the receiver has been proposed to overcome the drawbacks of the standard combination of SF waveform with either a PC or an NLFM waveform. It consists in splitting the spectrum of a PC or NLFM pulse into a predetermined number of portions, then transmitting the corresponding time-domain signals. At the receiver, the modified FD and the TWR algorithms are proposed to process the received echoes. In this processing chain, we have also added one degree of freedom related to the pseudo-random transmission way to prevent the opponent signal analyzer from reconstructing the transmitted waveform correctly.

A comparative study was done between the different variants that can be considered. For the proposed SFPC and SF-NLFM waveforms, the TWR algorithm outperforms the modified FD algorithm in some cases using a certain number of portions. This comes at the expense of an additional computational cost. In addition, the quality of reconstruction of the power spectrum is no longer good when N p is large. Moreover, a comparative study has been done between the proposed and the traditional SFPC and SF-NLFM waveforms. On the one hand, we have shown that the performance of the proposed SFPC in terms of PSLR and ISLR obtained with either the modified FD or the TWR algorithm is better than that obtained with the traditional SFPC waveform. The latter is processed by the state of the art algorithms namely, IFFT and FD algorithms.

On the other hand, we noticed that the proposed SF-NLFM waveform avoids producing grating lobes that usually appears when the traditional SF-NLFM is processed with the MF. As a consequence, they have the potential to detect small targets in the presence of large targets. This has been illustrated by conducting some simulations using scenarios representative of real cases. This distinguishable feature is attractive from a practical point of view.

In the future, we aim at investigating it from different viewpoints:

• The first will deal with its effect on the robustness of the transmitted waveform if heavily distorted portions are received. In other words, we plan to evaluate the relevance of our approach when the received echoes are disturbed by an additive white noise and a clutter.

• The second will aim at investigating the Doppler tolerance of the proposed waveform. More particularly, we will analyze the influence of the velocity of the target on the HRRP.

• The third will be related to the challenges that the opponent may face to analyze the waveform and to infer the capabilities of its radar, in addition to the prediction of its platform.

• The last will deal with the performance of these approaches in terms of probability of detection and probability of false alarms.

2. Another processing chain has been proposed to overcome the drawbacks of the standard combination of SF with NLFM waveform. In this thesis, we have focused our attention on one type of the NLFM waveforms namely the tangentbased NLFM waveform. The latter is approximated by a piecewise linear waveform and then used in a SF framework. This leads to proposing a variable chirp rate SF-LFM waveform where SF is combined with a train of LFM pulses having different chirp rates, durations, and bandwidths. The parameters of the proposed waveform are deduced using a genetic algorithm and a multiobjective criterion. Among the perspectives that could be considered, we suggest:

• launching additional simulations in order to evaluate the best compromise between the number of the LFM pulses to be considered and the performances in terms of PSLR, ISLR, and range resolution. This will lead to computationally intensive simulations when the number of time instants increases as the size of the population used in the genetic algorithm will need to be large.

• studying alternative NLFM waveforms such as the sine-based NLFM waveform.

• Optimizing the waveform parameters without any a priori constraints of the waveform.

• addressing the optimization issue by using alternative evolutionary algorithms such as non-dominated genetic algorithm (NSGA-II) [START_REF] Deb | A fast and elitist multiobjective genetic algorithm: Nsga-ii[END_REF], the differential evolution algorithm [START_REF] Storn | Differential evolution -a simple and efficient adaptive scheme for global optimization over continuous spaces[END_REF], etc.

A.1 Introduction

In the field of signal and image processing or even in the field of control, models or processes are often compared. This is, for instance, the case when dealing with identification issues where the estimated model parameters are compared with the true ones in order to analyze the estimation accuracy [27] [9]. Model comparison also occurs when designing a Bayesian estimation approach based on Kalman filtering, H ∞ filtering or particle filtering [START_REF] Najim | Modeling, estimation and optimal filtering in signal processing[END_REF]. In this case, a priori modeling the system under study is necessary and leads to the state space representation (SSR) of the system. However, several problems may arise.

On the one hand, the performance of the estimation algorithm depends on how good the SSR fits the system. As it is not necessarily easy to set it properly, estimation approaches combining different models, or equivalently different SSRs, can be considered. This leads to multiple-model approaches such as the interactive multiple models [76] [63]. In this case, selecting dissimilar models is suggested by Bar-Shalom in [5]. Therefore, a way to a priori compare models has to be designed. On the other hand, the practitioners may prefer to use models whose parameter estimation may be "easier" and which lead to SSRs that can be written in a "simple" way. For instance, when dealing with short-memory processes, they can propose to use an autoregressive (AR) model, a moving average (MA) model or more generally an ARMA model. Therefore, comparing ARMA models can be useful. When dealing with long-memory processes 1 , autoregressive fractionally integrated with moving average (ARFIMA) models can be compared. Process comparison can also be of interest, especially in the field of image processing when textures are compared [START_REF] Bombrun | Multivariate texture retrieval using the SIRV representation and the geodesic distance[END_REF]. In biomedical applications or flood forecast, change detection can be useful. In this latter case, the problem is to detect whether the statistical properties of a process change over time.

In the above situations, one can see that statistical properties have to be analyzed.

To address this issue, one could consider the spectral distance measures which include the log-spectral distance (LSD), the Itakura-Saito divergence (ISD), the Itakura divergence (ID), the model distance measure proposed by Itakura and their symmetric versions as well as parametric spectral distances such as the cepstral distance [6]. In [START_REF] Magnant | Classifying autoregressive models using dissimilarity measures: A comparative study[END_REF], a comparative study was recently done between them. The l-norm (with l = 1, 2, ∞) between the true model parameter vector and the estimated one could be also used, as well as the COSH distance which is the symmetric version of the Itakura-Saito divergence. [START_REF] Gladston | Improving speech intelligibility in Cochlear implants using Vocoder-centric acoustic models[END_REF].

As an alternative, general distance measures [6] can be considered. After the pioneering works of Pearson [START_REF] Pearson | On the criterion that a given system of deviations from the probable in the case of a correlated system of variables is such that it can reasonably be supposed to have arisen from random sampling[END_REF] in 1900, Hellinger [START_REF] Hellinger | Neue begrndung der theorie quadratischer formen von unendlichvielen vernderlichen (in german)[END_REF] in 1909, Bhattacharyya in 1943 [START_REF] Bhattacharyya | On a measure of divergence between two statistical populations defined by their probability distributions[END_REF] and Shannon [START_REF] Shannon | A mathematical theory of communication[END_REF] in 1948 where the measure of entropy and the mutual information were introduced, several researchers focused their attentions on quantifying how close two distributions are from one another: Kullback-Leibler divergence (KL) divergence [START_REF] Kullback | On Information and Sufficiency[END_REF], also known as the relative entropy, generalized the notion of mutual information. In 1952, Chernoff [START_REF] Chernoff | Measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations[END_REF] introduced another measure of divergence called Chernoff distance of order λ. In 1961, Rényi [START_REF] Renyi | On measures of entropy and information[END_REF] suggested an extension of the entropy of order α for discrete probabilities. Then, the Rényi divergence of order α, also called α-divergence was introduced. In the 60ies, another degree of generalization was proposed through the so-called f -divergences where the probability density function (pdf) ratio is weighted by a function f . They are also known as Csiszar f -divergences, Csiszar-Morimoto divergences or Ali-Silvey distances. Depending on the choice of the function f , one can retrieve specific cases. Finally, f -dissimilarities have been introduced when more than two pdfs are considered. The reader may refer to [7] [28] for more details and for information about recent works.

Among the above measures, the KL divergence remains one of the most popular. Several authors analyzed it in various fields of applications, for classification, identification or change detection [START_REF] Bombrun | Multivariate texture retrieval using the SIRV representation and the geodesic distance[END_REF], [START_REF] Murthy | Touchless monitoring of breathing function[END_REF], [START_REF] Magnant | Classifying autoregressive models using dissimilarity measures: A comparative study[END_REF], [START_REF] Georgiou | A convex optimization approach to arma modeling[END_REF] and [START_REF] Schutz | Centroid-based texture classification using the generalized gamma distribution[END_REF]. Meanwhile, the estimations of the KL between two pdfs that are not necessarily Gaussian, by using sets of data, were studied in [21] [22]. When dealing with Gaussian processes, the expression of the KL depends on the logarithm of the ratio between the covariance matrix determinants. Secondly, the KL divergence is not a distance: it is not symmetric and does not satisfy the triangular inequality.

For the above reasons, a great deal of interest has been paid to the symmetric KL divergence, known as Jeffreys Divergence (JD) [START_REF] Jeffreys | Theory of probability[END_REF]. When dealing with the JD, the symmetry conditions are satisfied. As the logarithms compensate each other, they no longer appear in the expression of the JD for the Gaussian case. Given all these considerations, we focus our attention on the JD in the following. When dealing with k-dimensional Gaussian random vectors of size k, the JD amounts to computing the sum of two traces of matrices, that can be expressed as the k × k covariance matrix of the first process pre-multiplied by the inverse of the covariance matrix of the second process. When k increases, the resulting computational cost of the JD increases because the standard computational burden of a generic k × k matrix inversion is usually O(k 3 ) [START_REF] Golub | Matrix Computations[END_REF]. To address this problem with processes that are ergodic wide sense stationary (w.s.s.) AR and/or MA, eigenvalue decomposition could be considered. Analytical expressions of the eigenvalues and the eigenvectors exist for 1 st -order MA processes [START_REF] Noschese | Tridiagonal Toeplitz matrices: properties and novel applications[END_REF]. Concerning 1 st -order AR processes, estimates of eigenvalues have been proposed for a large correlation matrix [START_REF] Stroeker | Approximations of the eigenvalues of the covariance matrix of a first-order autoregressive process[END_REF]. However, to the best of our knowledge, these estimates do not exist for higher-order AR processes. For this rea-son, for a p th -order AR process, a LDL factorization could be rather used and requires the parameters of the AR process where the order varies between 1 and p. Its computational cost is of O( 2k 3 3 ) [START_REF] Golub | Matrix Computations[END_REF]. Alternative approaches have been proposed. Thus, taking advantage of the Markovian properties of the AR process, the JD between the pdfs of the k successive samples of two p th -order time-varying AR (TVAR) processes or AR processes can be recursively computed [START_REF] Magnant | On computing Jeffrey's divergence between time-varying autoregressive models[END_REF]. In this case, the expression of the JD for k-dimensional vectors only depends on matrices of size p, which significantly reduces the computational cost. Then, this method has been used to classify more than two AR processes in different subsets [START_REF] Magnant | Jeffreys divergence between state models: Application to target tracking using multiple models[END_REF]. The analytical expression of the JD between ergodic w.s.s. 1 st -order MA processes, that can be real or complex, noise-free or disturbed by additive white Gaussian noises, has also been studied in [START_REF] Legrand | Jeffrey's divergence between moving-average models that are real or complex, noise-free or disturbed by additive white noises[END_REF]. For this purpose, the authors use the analytical expression of each element of the tridiagonal-correlation-matrix inverse [START_REF] Yueh | Explicit inverses of several tridiagonal matrices[END_REF]. Unlike p th -order AR processes, no recursive expression of the JD can be obtained for 1 st -order MA processes. Finally, comparing ergodic w.s.s. 1 st -order AR and ergodic w.s.s. 1 st -order MA processes by using the JD has been proposed in [START_REF] Legrand | Jeffrey's divergence between moving-average and autoregressive models[END_REF]. It is based on the expression of the inverses of the AR correlation matrices [START_REF] Cernuschi-Frias | A derivation of the gohberg-semencul relation [signal analysis[END_REF].

Concerning the above cases, we can summarize the results obtained as follows:

1. Links with Rao distance [START_REF] Rao | Information and the accuracy attainable in the estimation of statistical parameters[END_REF] have been proposed when it was possible. It was confirmed that the square of the Rao distance was approximately twice the value of the JD, except when a 1 st -order MA process is considered whose zero is close to the unit-circle in the z-plane.

2. The JD tends to have a stationary regime. The difference between two JDs computed for k and (k -1)-dimensional random vectors tends to a constant when k increases. When comparing two 1 st -order MA processes, there is one exception if one of the MA processes has its zero on the unit-circle in the z-plane, different from the zero of the second MA process. This difference is called asymptotic JD increment when k becomes high. In previous papers [START_REF] Magnant | On computing Jeffrey's divergence between time-varying autoregressive models[END_REF] [58] [START_REF] Legrand | Jeffrey's divergence between moving-average and autoregressive models[END_REF], analytical expressions of the asymptotic JD increment are provided for AR and/or 1 st -order MA processes. They depend on the parameters of the processes.

3. The asymptotic JD increment can be used to compare the random processes instead of the JD between k successive samples of the processes.

4. As the asymptotic JD increment does not depend on k, the selection of the number of variates k is no longer a problem for the practitioner.

In this appendix, we present complementary results. Thus, the JD betweeen two noisy sum of complex exponentials (NSCE) is studied. More particularly, the influence of the process parameters on the JD is analyzed. Then, a NSCE process is compared with an AR process. Moreover, an interpretation of the asymptotic JD increment for ergodic w.s.s. ARMA processes is proposed. Finally, the asymptotic JD increment between ARFIMA processes based on the proposed interpretation is analyzed.

In the following, the upperscripts T , * and H denote the transpose, the conjugate and the hermitian.

x k 1 :k 2 = (x k 1 , ..., x k 2 ) is the collection of samples from time k 1 to k 2 .
A.2 Definition of the Jeffreys divergence (JD)

The Kullback-Leibler (KL) divergence between the joint distributions of k successive values of two random processes, denoted as p 1 (x 1:k ) and p 2 (x 1:k ), can be evaluated to study the dissimilarities between the processes [START_REF] Kullback | On Information and Sufficiency[END_REF].

KL (1,2) k = x 1:k p 1 (x 1:k )ln p 1 (x 1:k ) p 2 (x 1:k ) dx 1:k (A.1)
When the processes are both Gaussian and real with means µ 1,k and µ 2,k and covariance matrices Q 1,k and Q 2,k , it can be easily shown, by substituting p 1 (x 1:k ) and p 2 (x 1:k ) with the expressions of real Gaussian multivariate distributions, that the KL satisfies2 [START_REF] Rasmussen | Gaussian processes for machine learning[END_REF]:

KL (1,2) k = 1 2 Tr(Q 2,k -1 Q 1,k ) -k -ln detQ 1,k detQ 2,k (A.2) + (µ 2,k -µ 1,k ) T Q -1 2,k (µ 2,k -µ 1,k ) .
where Tr denotes the trace of a matrix and det is the determinant of a matrix. When dealing with zero-mean processes, (A.2) reduces to:

KL (1,2) k = 1 2 Tr(Q -1 2,k Q 1,k ) -k -ln detQ 1,k detQ 2,k . (A.3)
However, the KL is not symmetric. To address this issue, different approaches can be considered. The first one would consist in selecting the minimum value between KL . In some papers, the definition is slightly different and corresponds to the sum of KL . In the following, for the sake of simplicity, the symmetric version of the KL is defined as :

JD (1,2) k = KL (1,2) k + KL (2,1) k (A.4) A.3.1.

Correlation properties of the NSCE processes

When storing k successive samples of a NSCE process in a vector, the corresponding covariance matrix can be expressed as follows, with l = 1, 2:

Q N SCE,l,k = S l,k P l S H l,k + σ 2 l I k . (A.7)
where I k is the identity matrix of size k, P l is a diagonal matrix of size M l × M l whose main diagonal is defined as:

P l =       γ l,1 0 . . . 0 0 γ l,2 . . . 0 . . . . . . . . . . . . 0 0 . . . γ l,M l       (A.8)
In addition, S l,k is a matrix of size k × M l defined as follows:

S l,k = S 1 l,k . . . S M l l,k (A.9) =       1 . . . 1 e jθ l,1
. . . e jθ l,M l . . . . . .

         1 k S m l,k H S m l,k = 1 1 k S m l,k H S n l,k = 1 k k-1 t=0 e j(θ l,m -θ l,n )t = sin( k(θ l,m -θ l,n ) 2 ) ksin( (θ l,m -θ l,n ) 2 ) e j (k-1)(θ l,m -θ l,n ) 2 (A.10) As lim k→+∞ | sin( k(θ l,m -θ l,n ) 2 ksin( (θ l,m -θ l,n ) 2
)| = 0 for any set of normalized angular frequencies satisfying

θ l,n = θ l,m in the interval [-π, π[, (A.10) becomes:    1 k S m l,k H S m l,k = 1 lim k→+∞ 1 k S m l,k H S n l,k = 0 (A.11)
The same properties hold when we are dealing with one normalized angular frequency of the first process and another one with the second. The convergence speed to tend to this limit when k increases depends on the difference between the normalized angular frequencies. The smaller the difference, the slower the convergence. Given (A.7) and using the matrix inversion lemma3 , one has for l = 1, 2:

Q -1 N SCE,l,k = 1 σ 2 l I k -S l,k σ 2 l P l -1 + S H l,k S l,k -1 S H l,k (A.12)
In (A.12), let us now apply again the inversion matrix lemma on the matrix S H l,k S l,k + σ 2 l P -1 l -1 . This leads to:

(S H l,k S l,k + σ 2 l P -1 l ) -1 = (S H l,k S l,k ) -1 -(S H l,k S l,k ) -1 (σ -2 l P l + (S H l,k S l,k ) -1 ) -1 (S H l,k S l,k ) -1 (A.13)
Given (A.13), the matrix S l,k σ 2 l P -1 l + S H l,k S l,k -1 S H l,k appearing in (A.12) can be rewritten as the sum of two matrices, namely V k and U k :

T k = V k + U k (A.14)
with:

V k = S l,k S H l,k S l,k -1 S H l,k U k = -S l,k (S H l,k S l,k ) -1 (σ -2 l P l + (S H l,k S l,k ) -1 ) -1 (S H l,k S l,k ) -1 S H l,k (A.15)
When k increases, due to (A.10), (A.15) becomes:

   lim k→+∞ V k = 1 k S l,k S H l,k lim k→+∞ U k = -σ 2 k 2 S l,k P -1 l S H l,k (A.16)
Therefore, given (A.16), the limit of Q -1 N SCE,l,k when k increases and tends to infinity is given by:

lim k→+∞ Q -1 N SCE,l,k = 1 σ 2 l (I k - 1 k S l,k S H l,k + σ 2 l k 2 S l,k P -1 l S H l,k ) (A.17)
This result will be useful when the behavior of the JD will be studied. In the next subsection, let us give some information about ARMA processes.

These processes are therefore known to be short-memory processes. The Toeplitz covariance matrices of the vector storing k consecutive samples x n:n+k-1 are denoted as Q ARM A,l,k for l = 1, 2. They are defined by the covariance-function coefficients r ARM A,l,τ with τ = 1 -k, ..., k -1. More particularly, the k elements of its main diagonal correspond to the autocorrelation function of the ARMA process for a lag equal to 0, i.e. r ARM A,l,0 . The τ th subdiagonal is defined by the autocorrelation function with lag τ , i.e. r ARM A,l,τ . According to [START_REF] Gray | Toeplitz and circulant matrices: a review[END_REF], when the normalized covariance matrices belong to the Wiener class Toeplitz matrices, or equivalently have their elements which are absolutely summable, i.e. τ r ARM A,l,τ r ARM A,l,0 is finite, the covariance matrices are nonsingular even if the PSDs of the processes are equal to zero at some frequencies. They are hence invertible. Nevertheless, it should be noted that when studying an infinitesize Toeplitz covariance matrix, the covariance matrix is no longer invertible when the corresponding transfer function of the ARMA process has unit roots. Let us now give some additional information about the covariance matrix Q AR,l,k of an AR process. There are various ways to compute the inverse of the covariance matrix of the AR process. An eigenvalue decomposition could be considered but there is no explicit form for the eigenvalues of an AR covariance matrix. It is true that there are some approaches that provide approximations of the eigenvalues when the size of the AR covariance matrix is large and when the order of the AR process is equal to 1 [START_REF] Stroeker | Approximations of the eigenvalues of the covariance matrix of a first-order autoregressive process[END_REF]. As an alternative, an analytical expression based on the AR parameters exists [START_REF] Cernuschi-Frias | A derivation of the gohberg-semencul relation [signal analysis[END_REF]: [START_REF] Daniels | The approximated distribution of serial correlation coefficients[END_REF] with:

Q -1 AR,l,k = 1 σ 2 u,l (F F H -GG H ) (A.
F = I k + p i=1 a i,l J k i and G = p i=1 a i,l J k k-i (A.26)
where J k is the "shift" matrix which has ones on the first sub-diagonal and zeros elsewhere.

The LDL factorization of Q AR,l,k can be also used. It involves the product between a lower unit triangular matrix L and a diagonal matrix D defined from the AR parameters and the variance of the driving process of AR processes whose order varies from 0 to k -1. They are respectively denoted as {a ν i,l } i=0,...,ν and {σ 2 u,l,ν } i=0,...,ν with ν the order. In the latter case, the inverse of Q AR,l,k can be deduced and is given by:

Q -1 AR,l,k = L H -1 D -1 L -1 (A.27)
where

D -1 = diag( 1 σ 2 u,l,k-1 . . . 1 σ 2 u,l,0 ) (A.28)
where diag(x) is the diagonal matrix whose main diagonal is defined by x.

Finally, one has:

L H -1 =          1 0 . . . . . . 0 a k-1 1,l 1 0 . . . 0 a k-1 2,l a k-2 1,l 1 . . . . . . . . . . . . 0 a k-1 k-1,l a k-2 k-2,l . . . . . . 1          (A.29)
However, as the AR process is assumed to be a strict p th -order AR process, the matrices D -1 and L H -1 become:

D -1 = diag( 1 σ 2 u,l,p . . . 1 σ 2 u,l,p k-p 1 σ 2 u,l,p-1 . . . 1 σ 2 u,l,0 ) (A.30)
and

L H -1 =                1
               (A .31)
A.3.2.3 Minimum-phase filter and inverse filter associated to the ARMA processes In this appendix, our purpose is to compare processes by using their pdfs. However, 2 q ARMA processes of order (p, q) have the same pdf. Indeed, the transfer function is always defined by the same denominator, because the poles have to be inside the unit disc in the z-plane in order to guarantee the asymptotic stability. As for the numerators of the transfer functions, 2 q numerators can be defined depending on whether the zeros or their inverses are chosen. Once the numerator and the denominator are defined, the variance of the driving process can be deduced. Therefore, comparing two ARMA processes by using their pdfs amounts to comparing the ARMA processes whose transfer functions have their zeros inside the unit-disc in the z-plane, i.e. whose transfer functions are minimum phase. As we will see in the rest of the analysis, this equivalency will have the advantage of using stable inverse filters associated to the ARMA processes. Let us define this last notion in the rest of this subsection. The inverse filters are defined by the transfer functions H -1 l (z). Thus, when dealing with a real 1 st -order MA process defined by the MA parameter b 1,1 , the inverse filter is stable if |b 1,1 | < 1:

H -1 1 (z) = 1 σ u,1 (1 + b 1,1 z -1 ) if |b 1,1 | < 1 (A.32) If |b 1,1 | > 1,
the following relationship must be taken into account:

b 1,1 1 + 1 b 1,1 z -1 = b 1,1 + z -1 1 + b 1,1 z -1 1 + b 1,1 z -1 (A.33)
where b 1,1 +z -1 1+b 1,1 z -1 can related to the Blaschke product 4 [START_REF] Colwell | Blaschke products: bounded analytical functions[END_REF], corresponding to the transfer function of an all-pass filter. The inverse filter of the "minimum-phase" MA process is defined in this case by:

H -1 1 (z) = 1 σ u,1 b 1,1 (1 + 1 b 1,1 z -1 ) if |b 1,1 | > 1 (A.34)
It should be noted that when |b 1,1 | = 1, the zero of the MA process, equal to -b 1,1 , has also its modulus equal to 1. Therefore, the inverse filter is unstable. This means that the filter output is not bounded, except when the filter input is an ARMA process having a zero equal to -b 1,1 . Indeed, the simplest illustration is when the filter input is a 1 st -order MA process with MA parameter b 1,1 = b 1,2 and driving-process variance equal to σ 2 u,2 . The equivalent transfer function of the whole processing chain of a zero-mean unit-variance white noise is given by:

σ u,2 H 2 (z)H -1 1 (z) = σ u,2 σ u,1 (A.35)
The influence of the inverse-filter pole is compensated by the MA-process zero. The output of the inverse filter H -1 1 (z) is hence a white noise with variance

σ 2 u,2 σ 2 u,1
. After discussing the stability of the inverse filter associated to the MA process, the inverse filter associated to the AR process is necessarily stable since it corresponds to a finite-impulse response (FIR) filter. More generally, when dealing with a minimum-phase ARMA(p,q) process, the inversefilter transfer function is defined as follows: its zeros correspond to the poles of the ARMA transfer function. Concerning its poles, they follow the same rules as the ones presented for MA processes. This means that unit-zeros can lead to unstable inverse filtering. 4 A Blaschke product is defined as:

B(a, z) = |a| a a-z 1-a * z = 1 |a| 1-az -1 1-1 a * z -1
, where a * denotes the conjugate of a. By taking a = a * = -1 b1,1 , one retrieves the expression b1,1+z -1 1+b1,1z -1 .

The above relation will be useful in the simulation part to reduce the computational cost and to avoid numerical issues. Taking into account the results introduced at the end of section A.3.3, it should be noted that x n,l is mean square summable and stationary when d < 1 2 . Using (A.37) and the standard binomial expansion (A.38) of (1 -z -1 ) g with g = -d, x n,l can be expressed as an infinite-order finite-power causal MA process, whose i th MA parameter is defined by:

b i,l = Γ (i + d) Γ (i + 1)Γ (d) (A.45)
Using D'Alembert criterion to analyze the convergence of

i |b i,l |, |b i+1,l | |b i,l | = i+d i+1 < 1 as long as d < 1.
The corresponding infinite impulse response (IIR) filter whose transfer function is H F I,l (z) = i b i,l z -i is hence stable in the bounded-input bounded-output sense. Moreover, thanks to (A.43) and (A.45), it can be shown that the process power is equal to:

P x,l = (A.45) σ 2 u,l +∞ i=0 b 2 i,l = σ 2 u,l +∞ i=0 Γ 2 (i + d) Γ 2 (i + 1)Γ 2 (d) (A.46) = r F I,l,0 = = (A.43) Γ (1 -2d) Γ 2 (1 -d) σ 2 u,l When d > -1 2 ,
x n,l is invertible and can be expressed as an infinite-order causal AR process, the i th AR parameter of which is defined by:

a i,l = Γ (i -d) Γ (i + 1)Γ (-d) (A.47) If -1 2 < d < 0, x n,l is a short-memory process. If 0 < d < 1 2 , r F I,l,τ
r F I,0,τ is not absolutely summable. In this case, x n,l is a longmemory process. Taking into account (A.44), the normalized autocorrelation function, i.e.

r F I,l,τ r F I,l,0 , is proportional to Γ (1-d)
Γ (d) τ 2d-1 when τ becomes high. It decays at an hyperbolic rate. Let us now look at the properties of the Toeplitz correlation matrix, denoted as

Q F I,l,k . The determinant | Q F I,k
r F I,l,0 | of the Toeplitz normalized correlation matrix can be expressed in terms of the partial correlation coefficients6 [START_REF] Daniels | The approximated distribution of serial correlation coefficients[END_REF] as follows:

Q F I,l,k r F I,l,0 = k-1 τ =1 (1 -ρ 2 τ,l ) k-τ (A.48)
where the partial correlation coefficients are here equal to ρ τ,l = d τ -d according to [START_REF] Palma | Long-Memory Time Series, Theory and Methods[END_REF].

ond term of C (2,1) in (A.70) tends to zero. Therefore, C (2,1) tends to the following expression:

lim k→+∞ C (2,1) = - σ 2 1 σ 2 2 Tr (I M 2 ) = - σ 2 1 σ 2 2 M 2 (A.71)
Depending on the normalized angular frequencies of the second process, the convergence speed is more or less fast when k increases. The closer the normalized angular frequencies are, the slower the convergence speed is.

Finally, let us study the fourth term. It is defined as follows:

D (2,1) = - 1 σ 2 2 Tr S H 1,k S 2,k σ 2 2 P 2 -1 + S H 2,k S 2,k -1 S H 2,k S 1,k P 1 (A.72)
In the above equation (A.72), let us denote:

T 1/2,k = S 2,k σ 2 2 P 2 -1 + S H 2,k S 2,k -1 S H 2,k S 1,k (A.73)
As we already did for C (2,1) , (A.73) can be rewritten by using (A.69) as follows:

T 1/2,k = S 2,k (S H 2,k S 2,k ) -1 S H 2,k S 1,k (A.74) -S 2,k (S H 2,k S 2,k ) -1 (σ -2 2 P 2 + (S H 2,k S 2,k ) -1 ) -1 (S H 2,k S 2,k ) -1 S H 2,k S 1,k = S 1/2,k + U 1/2,k
where:

S 1/2,k = S 2,k S H 2,k S 2,k -1 S H 2,k S 1,k U 1/2,k = -S 2,k (S H 2,k S 2,k ) -1 (σ -2 2 P 2 + (S H 2,k S 2,k ) -1 ) -1 (S H 2,k S 2,k ) -1 S H 2,k S 1,k (
A.75) When substituting the expression (A.74) of T 1/2,k into (A.72), D (2,1) can be expressed as the sum of two traces. In the following, we propose to evaluate both:

1. The first one is induced by S 1/2,k : The matrix S 1/2,k stores the orthogonal projections of the columns of S 1,k onto the space spanned by the columns of S 2,k . Given the "asymptotic"' properties (A.11) of orthogonality when k becomes large, the orthogonal projection of the m th column S m 1,k of S 1,k onto S 2,k is a null column vector except when the two processes have common normalized angular frequencies. Indeed, if there exists n ∈ [[1, M 2 ]] such as θ 1,m = θ 2,n , this leads to:

lim k→+∞ S 2,k S H 2,k S 2,k -1 S H 2,k S m 1,k = S m 1,k (A.76)
Therefore, lim k→+∞ S 1/2,k can be approximated by:

S 1/2 = lim k→+∞ S 1/2,k ≈ S 1 1,k δ 1 1,2 . . . S M 1 1,k δ M 1 1,2 (
lim k→+∞ - 1 σ 2 2 Tr(S H 1,k S 2,k (S H 2,k S 2,k ) -1 S H 2,k S 1,k P 1 ) ≈ - 1 σ 2 2 Tr S H 1,k S 1/2 P 1 (A.78) = - k σ 2 2 M 1 m=1 M 2 n=1 γ 1,m δ θ 1,m ,θ 2,n
This term is null if the processes do not have at least one common normalized angular frequency.

2. The second one is induced by U 1/2,k : When k increases, due to the "asymptotic orthogonality properties" (A.11), some simplifications can be done. Indeed, one has:

lim k→+∞ U 1/2,k = -lim k→+∞ 1 k 2 S 2,k (σ -2 2 P 2 + 1 k I M 2 ) -1 S H 2,k S 1,k (A.79) = -lim k→+∞ σ 2 2 k 2 S 2,k P -1 2 S H 2,k S 1,k
As we aim at calculating the trace of -1 σ 2 2 Tr S H 1,k U 1/2,k P 1 , one obtains:

lim k→+∞ 1 k 2 Tr S H 1,k S 2,k P 2 -1 S H 2,k S 1,k P 1 = M 1 m=1 M 2 n=1 γ 1,m γ 2,n δ θ 1,m ,θ 2,n (A.
(Q -1 N SCE,2,k Q N SCE,1,k
) can be expressed as follows when k becomes high:

Tr(Q -1 N SCE,2,k Q N SCE,1,k ) ≈ k σ 2 2 M 1 m=1 M 2 n=1 γ 1,m (1 -δ θ 1,m ,θ 2,n ) + (k -M 2 ) σ 2 1 σ 2 2 (A.81) + M 1 m=1 M 2 n=1 γ 1,m γ 2,n δ θ 1,m ,θ 2,n
Remark: When the processes have the same covariance matrix, one has:

M 1 m=1 M 2 n=1 γ 1,m γ 2,n δ θ 1,m ,θ 2,n = M 1 = M 2 (A.82)
equal to 0. For this purpose, instead of looking only at the JD, we propose to study the asymptotic JD increment. In the following, let us show how the JD increment evolves when k increases for a specific case, and let us analyze the influence of each parameter. We propose to start the analysis by the additive-noise variances. is the power of the AR process filtered by an all-pass filter whose transfer function is 1 σ l , 2. σ l 2 σ 2 u,l ,p a l p 2 is the power of the additive white noise with variance σ 2 l that has been filtered by the finite-impulse response (FIR) "AR inverse filter" whose transfer function is A p (z) σ u,l ,p , 3. the third term depends on the sum of the ratios between the NSCE process power and the AR-process PSD at the normalized angular frequencies {θ l,m } m=1,...,M . Nevertheless, by using (A.20), each term

γ l,m
S AR,l ,p (θ l,m ) can be also seen as the power of the complex exponential at θ l,m which has been filtered by the AR inverse filter. Therefore, this third term corresponds to the power of the deterministic part of the NSCE process filtered by the AR inverse filter.

The above results lead to some comments:

1. when M = p = 0, (A.105) reduces to: and σ l 2 .

∆JD (AR,N SCE) ≈ -1 + 1 2 ( σ 2
2. The smaller the three terms in (A.105) are, the smaller the asymptotic JD increment is. However, they do not necessarily have the same behavior with respect to the process parameters. Thus, if p = M and if the arguments of the AR poles correspond to the normalized angular frequencies {θ l,m } m=1,...,M , resonances may appear in the transfer function H l (z). The closer to the unit circle in the z-plane the poles {p i,l } i=1,...,p are, the sharper the resonances of H l (z) are. In this case, the 1 st and the 2 nd terms in (A.105) can grow 9 . Meanwhile, the 3 rd term in (A.105) tend to be smaller and smaller.

In the next section, we suggest illustrating the way the JD evolves when comparing two processes. We propose to look at the influence of the process parameters on the JD. This will help us define some clues to give an interpretation of the JD.

A.5.5 Illustrations and comments

In this section, for the sake of simplicity, we suggest comparing a 1 st -order AR process with a NSCE process. The influences of the different parameters can be also illustrated. Note that r AR,1,0 = , which shows the asymptotic JD increment as a function of the argument of a 1,1 , the asymptotic JD increment reaches its minimum when the argument of a 1,1 is equal to -π/2. In this case, the resonance in the AR process power spectrum appears at the same angular frequency as the one of the NSCE process. Following our interpretation, we could have obtained the same results by avoiding several mathematical developments presented in [START_REF] Magnant | On computing Jeffrey's divergence between time-varying autoregressive models[END_REF]. Indeed, the term ∆T (AR 1 ,AR 2 ) which is equal to R u 1-2a 1,1 a 1 1,2 +a 2 1,1

1-a 2 1,2 corresponds to the power of the 2 nd AR process filtered by the inverse filter that corresponds to the 1 st AR process. This amounts to computing the autocorrelation function of this filtered AR (fAR) process, for the lag equal to 0, denoted as r f AR,2,0 . This latter can be deduced from the definition of the autocorrelation function and the set of equations that define the second AR process and the filtering step. More particularly, one has: .

r f AR,2,0 = 1 + a 2 1,1 σ 2 u,
Concerning the term ∆T (AR 2 ,AR 1 ) which is equal to , it is equal to the power of the 1 st AR process filtered by the inverse filter that corresponds to the 2 nd AR process. The reasoning is similar. Since illustrations of the JD between AR processes have already been proposed, interested readers can refer to [START_REF] Magnant | On computing Jeffrey's divergence between time-varying autoregressive models[END_REF] for further details. A.6.2.4 JD between two real 1 st -order MA processes In [START_REF] Legrand | Jeffrey's divergence between moving-average models that are real or complex, noise-free or disturbed by additive white noises[END_REF], by deducing the expression of the inverse of the correlation matrix of the real 1 st -order MA process 10 , it was shown after long mathematical developments that the asymptotic JD increment was constant when |b ].

= -1 + 1 2 ∆T (M A 2 ,M A 1 ) + ∆T (M A 1 ,M A 2 )

However if |b 1,l | = 1, the increment was not finite, except when the MA processes share the same zeros.

We are now able to explain the reason why this phenomenon appears. By developing and reorganizing the terms, we can show that the term ∆T (M A 2 ,M A 1 ) which is equal to 10 See [START_REF] Sutradhar | The inversion of correlation matrix for MA(1) process[END_REF] for more details.

Using the binomial expansion (A.38), the transfer function characterizes an IIR filter whose impulse response is h i,l,inv . Given the D'Alembert criterion, one has:

|h i+1,l,inv | |h i,l,inv | = i -d l l + 1 < 1 (A.134)
This means that the i th inverse filter is also BIBO stable.

Let the first process be filtered by the inverse filter of the second, which is defined by

1 σ u,2 (1-z -1 ) d 2 .
This amounts to filtering a white noise with a variance Given the analysis of the JD we presented in previous papers for ARMA processes, the difference between two consecutive JDs should converge to a constant called the asymptotic JD increment. Based on table I, let us draw three comments :

1. If (d 1 , d 2 ) ∈ (-0.5, 0), the processes to be compared are short memory. The outputs of the inverse filters are stationary F I white noises: the one with differencing order equal to -|d 1 -d 2 | is a short-memory process, whereas the other with differencing order equal to |d 1 -d 2 | is a long-memory process. Given (A.43), the powers of the inverse-filter outputs for (l, l ) = (1, 2) or (l, l ) = (2, 1) are given by:

P (l,l ) = Γ (1 -2(d l -d l )) Γ 2 (1 -(d l -d l )) σ 2 u,l σ 2 u,l (A.135)
2. When (d 1 , d 2 ) ∈ (0, 0.5), the processes to be compared are long memory. The inverse-filter outputs are stationary F I white noises and the same conclusions as in the previous case can be drawn.

3. When d 1 and d 2 have opposite signs, |d 1 -d 2 | can be smaller or larger than 1 2 . If the difference is smaller than 1 2 , the same comments as in the two previous cases can be drawn. If it is larger, one of the filter outputs should correspond to a F I white noise with an differencing order larger than 1 2 . It is a non-stationary process with an infinite variance [START_REF] Granger | An introduction to long-memory time series models and fractional differencing[END_REF].

In what follows, we propose to give some illustrations on the JD between F I white noises, by looking at the evolution of the JD with respect to the number of variates k. Then, we analyze the influences of the process parameters on the asymptotic JD increment ∆JD (F I 1 ,F I 2 ) . We also check if the above theoretical analysis matches the simulation results.

A.7.1.2 Illustrations and comments

JD between two FI white noise processes

Let us study the evolutions of the JD and the JD increment with respect to the variate number, through four different situations depicted in Table A . Let us now study their influences on the asymptotic JD increment. A.7.2 JD between two ARFIMA processes A.7.2.1 Theoretical analysis of the JD between ARF IM A(p, d, q) processes based on inverse filtering interpretation

As it was done with FI white noises in the previous section, let us now investigate the validation of the inverse-filtering interpretation. For this purpose, let us consider two ARFIMA processes defined by their orders (p l , d l , q l ) l=1,2 with {d l } l=1,2 ∈ (-1 2 , 1 2 ), their driving-process variances {σ 2 u,l } l=1,2 , their poles {p j,l } j=1,...,p l , and their zeros {z j,l } j=1,...,q l with l = 1, 2.

Filtering the first ARFIMA process by the inverse filter of the second amounts to filtering a white noise with a variance

σ 2 u,1 σ 2 u,2
by the following transfer function:

H equ (z) = (1 -z -1 ) d 2 -d 1 × (A.139) q 1
l=1 (1 -z l,1 z -1 )

p 1 l=1 (1 -p l,1 z -1 ) p 2 l=1
(1 -p l,2 z -1 )

q 2 l=1 H -1 l,2 (z)
where {H -1 l,2 (z)} l=1,...,q 2 are defined by following the rules we presented in (A. when the input are the ARFIMA processes to be compared. They are also ARFIMA processes: one is an ARF IM A(p 1 + q 2 , d 1 -d 2 , q 1 + p 2 ), whereas the other is ARF IM A(p 2 + q 1 , d 2 -d 1 , q 2 + p 1 ). In addition, they are stationary since |d 1 -d 2 | < 0.5. More particularly, the one with a differencing order equal to -|d 1 -d 2 | is a short-memory process whereas the other with a differencing order equal to |d 1 -d 2 | is a long-memory process. In this case, given (A.139), the inverse-filter output power P (ARF IM A 1 ,ARF IM A 2 ) , with (l, l ) = (1, 2) or (l, l ) = (2, 1), can be seen as the power of an ARF IM A(p l +q l , d l -d l , q l +p l )

process where the variance of the driving process is A.7.4 Some comments for the practitioner to apply this theory in practical cases Interested users would like to make use of this theory in practical cases. In this subsection, we propose some clues. In the field of change detection, to compare two segments, assumed to be of size N , by using the type of analysis we presented above, the covariance matrices for both segments have first to be estimated for two consecutive sizes k -1 and k by using a robust estimator. The JD are then computed for both sizes and their difference is computed. Once smoothed over time, the difference between both JD makes it possible to have an idea of the statistical change in the signal under study. Remark 1: the selection of the size k corresponds to a compromise. On the one hand, k must be sufficiently large so that the difference between the estimated JD can be considered as an approximation of the asymptotic JD increment. On the other hand, it must be much smaller than the number of samples N since the estimates of the covariance matrices must be as accurate as possible. Remark 2: the estimations of the differencing orders {d l } l=1,2 can also be of interest and can be useful to help the practitioner to give an interpretation of the results. They are related to the Hurst exponents, denoted as {H l } l=1,2 = {d l } l=1,2 + 1 2 . Different families of methods have been proposed in the literature to estimate them:

1. The time-domain estimators gather the so-called rescaled range analysis, the aggregated variance method, the absolute-value method and the variance of residuals method [START_REF] Taqqu | On estimating the intensity of long range dependence in finite and infinite variance time series. A practical guide to heavy tails: statistical techniques and applications[END_REF].

2. The frequency-domain estimators consist in analyzing the power spectral density of the time series [START_REF] Sun | Fractional order signal processing: techniques and applications[END_REF]. This is for instance the case of the local Whittle method, the periodogram method and the wavelet-based method.

3. The detrended fluctuations analysis (DFA) [START_REF] Peng | Mosaic organization of DNA nucleotides[END_REF] and some variants such as the adaptive fractal analysis (AFA) [START_REF] Riley | A tutorial introduction to adaptive fractal analysis[END_REF], the detrended moving average (DMA) method either centered or backward, the fluctuation analysis (FA) [START_REF] Shao | Comparing the performance of FA, DFA and DMA using different synthetic long-range correlated time series[END_REF] have also shown good performance, especially to estimate the Hurst exponent of a pure mono-fractal time series. It has been used in various applications to analyze electroencephalograms (EEG) [START_REF] Sanyal | Detrended fluctuation and power spectral analysis of alpha and delta EEG brain rhythms to study music elicited emotion[END_REF] and electrocardiograms (ECG) [90] [93].

In other applications where the purpose is classification, the difference between the estimated differencing orders and the difference between two consecutive JDs that have been estimated as mentioned above can be considered as features for the classification. K-nearest neighbors (KNN) or support vector machine support vector machine (SVM) can then be used.

A.8 Conclusions and perspectives

The following conclusions and perspectives can be drawn:

1. We compared noisy sums of complex exponentials (NSCE) by means of the JD. The difference between two consecutive JD was analyzed. In this case, the asymptotic JD increment and the convergence speed towards the asymptotic JD increment must be taken into account by the practitioner to draw conclusions about the similarity between the processes under study. Besides, we could a priori guess that the JD between an NSCE process, defined by the normalized angular frequencies of its complex exponentials, and an AR process which has sharper and sharper resonances in its spectrum at the same normalized angular frequencies becomes smaller and smaller. Nevertheless, in practice, the results that can be obtained do not necessarily confirm this statement as the JD also takes into account the powers of the processes to be compared.

2. Without developing tedious calculations, our interpretation of the asymptotic JD increment based on inverse filtering makes it possible to explain the atypical cases that were identified in previous works and for which the asymptotic JD increment was not finite. The results we present in this appendix allow us to generalize our analysis to the JD between AR processes, ARMA, MA, etc. Now, we are able to prove that all ARMA processes of any order and whose PSD is null for one or more frequencies can give rise to non-finite asymptotic JD increments. Moreover, there are now several ways to compute the JD: considering the definition or taking advantage of the interpretation we propose in this paper, i.e. using inverse filtering.

3. We studied the JD between ergodic w.s.s. ARFIMA processes. We have shown that the derivative of JD with respect to the number of variates k tends to a constant when the difference between the differencing order is absolutely smaller than 0.5. Moreover, we have analyzed the influence of the process parameters on the value of the asymptotic JD increment. Finally, it was shown that the interpretation of the asymptotic JD increment based on inverse filtering is also valid for these processes.

4. It could be of interest to use this type of approach in various applications, from biomedical applications to radar applications when analyzing the statistical properties of the clutter. In the latter case, it could be of interest to see whether it is homogeneous or not. This can be done by looking at secondary data. This type of analysis could be a preamble to estimate the correlation matrix of the clutter that would be used for space-time adaptive processing (STAP).

1 . 1 2

 11 Keywords: radar waveform, stepped frequency, HRRP, phase coded waveform, NLFM waveform, waveform optimization. RESUME Dans divers systèmes radar, un grand intêrét a été porté à la sélection d'une forme d'onde et à la conception d'une chaîne de traitement complète, de l'émetteur au récepteur, afin d'obtenir un profil distance haute résolution (HRRP, acronyme de High Range Resolution Profile en anglais). Au cours des dernières décennies, les concepteurs d'algorithmes de traitement du signal radar ont concentré leur attention sur différentes formes d'onde telles que les techniques de compression d'impulsion et les systèmes à bande synthétique (SF acronyme de stepped frequency, en anglais). D'une part, trois types de formes d'onde de compression d'impulsions large bande ont été proposés dans la littérature : la forme d'onde modulée linéairement en fréquence (Linear Frequency Modulation), celle à codes de phase (Phase Coded) et la forme d'onde modulée non linéairement en fréquence (Non Linear Frequency Modulation). Ces approches sont très populaires, mais elles requièrent une fréquence d'échantillonnage généralement élevée au niveau du récepteur, et par voie de conséquence un convertisseur analogique-numérique coûteux. De plus, les formes d'onde PC et NLFM peuvent être préférables dans certaines applications à haute résolution, car elles conduisent à de meilleures performances en termes de PSLR et ISLR que celles obtenues avec la forme d'onde LFM. D'autre part, lorsqu'il s'agit de schémas SF, une fréquence d'échantillonnage moins élevée peut être envisagée, ce qui permet d'utiliser un convertisseur analogique numérique (CAN) meilleur marché. Ces deux approches peuvent être combinées pour tirer avantage des deux familles. Bien que la combinaison standard mène à l'exploitation d'un CAN bon marché, les performances en termes de PSLR et ISLR ne sont pas nécessairement adaptées.Comme le PSLR et l'ISLR ont une grande influence sur la probabilité de détection et la probabilité de fausse alarme, notre objectif est de trouver des solutions alternatives. Ainsi, notre contribution dans ce mémoire de thèse consiste à proposer deux nouvelles chaînes de traitement, de l'émetteur au récepteur : Dans la première approche, le spectre de la forme d'onde à large bande est décomposé en un nombre prédéterminé de portions. Puis, les versions temporelles de ces dernières sont successivement transmises. Le signal rec ¸u est alors traité soit en utilisant un algorithme FD (pour frequency domain en anglais) mod-ifié, soit un algorithme de reconstruction de forme d'onde réalisé directement dans le domaine temporel (TWR pour time wave reconstruction). Dans cette thèse, les formes d'ondes PC et NLFM ont été sélectionées. Une étude comparative est alors menée entre les différentes chaînes de traitement, de l'émetteur au récepteur, que l'on peut constituer. Nos simulations montrent que les performances obtenues à partir de l'algorithme TWR sont le plus souvent meilleures que celles de l'agorithme FD modifié. La contre-partie est une augmentation du coût calculatoire. De plus, que ce soit avec une forme d'onde PC ou NLFM, l'approche présentée fournit de meilleurs résultats en termes de PSLR et ISLR que les formes d'onde SF classiques. 2. La seconde démarche proposée consiste à approximer une forme d'onde NLFM à large bande par une forme d'onde LFM par morceaux, puis de la combiner avec une approche de type SF. Cela donne lieu à une forme d'onde combinant SF et un train d'impulsions LFM ayant différentes durées et largeurs de bande. La sélection des paramètres de cette forme d'onde est faite en minimisant un critère multi-objectif, tenant compte du PSLR, de l'ISLR et de la résolution distance. Cette estimation est opérée par algorithmes génétiques. Selon les poids utilisés dans le critère multi-objectif et le nombre d'impulsions LFM pris en compte, les performances les formes d'onde résultantes varient. Une annexe est en outre fournie qui présente des travaux complémentaires sur la comparaison de modèles à partir de la divergence de Jeffreys. Mots-cls: forme d'onde radar, système à bande synthétique, HRRP, forme d'onde PC, forme d'onde NLFM, optimisation de formes d'onde. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Our contribution: a processing chain of the variable chirp rate SF-LFM waveform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Generalization of the SF-LFM waveform: the VCR SF-LFM waveform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Generation of the VCR SF-LFM waveform at the transmitter . . . 3.2.3 Processing the VCR SF-LFM waveform at the receiver . . . . . . 3.3 Optimizing the parameters of the VCR SF-LFM waveform . . . . . . . . 3.3.1 Generalities about genetic algorithm . . . . . . . . . . . . . . . . 3.3.2 Selection of the VCR SF-LFM parameters using GA . . . . . . . 3.4 Results and discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Simulation protocol . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Simulation results and comments, L = 1 . . . . . . . . . . . . . . 3.4.2.1 Waveform parameters based on a priori selection and corresponding performance measures . . . . . . . . . .3.4.2.2 Waveform parameters based on the multi-objective criterion deduced by GA and corresponding performance measures: predefined values of the weights . . . . . . . 3.4.2.3 Waveform parameters based on the multi-objective criterion deduced by GA and corresponding performance measures: any set of weights . . . . . . . . . . . . . . . 3.4.3 Simulation results and comments, L = 2 . . . . . . . . . . . . . . 3.4.3.1 Waveform parameters based on the multi-objective criterion deduced by GA and corresponding performance measures: predefined weights . . . . . . . . . . . . . . 3.4.3.2 Waveform parameters based on the multi-objective criterion deduced by GA and corresponding performance measures: any set of weights . . . . . . . . . . . . . . . 3.4.4 Simulation results and comments, L = 10 . . . . . . . . . . . . . 3.4.5 General comments on the results . . . . . . . . . . . . . . . . . . 3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Conclusions and perspectives LIST OF PUBLICATIONS . . . . . . . . . . . . . . . . . . . . . . . . Appendices A JEFFREYS DIVERGENCE FOR PROCESS COMPARISON: Properties, asymptotic analysis, physical interpretation and way to use it in practical cases A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2 Definition of the Jeffreys divergence (JD) . . . . . . . . . . . . . . . . . A.3 Presentation of the processes under study . . . . . . . . . . . . . . . . . . A.3.1 About the sum of complex exponentials (SCE) disturbed by an additive noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3.1.1 Definition and spectral properties of the NSCE processes A.3.1.2 Correlation properties of the NSCE processes . . . . . . A.3.2 About ARMA processes . . . . . . . . . . . . . . . . . . . . . . LIST OF FIGURES 1.1 Block diagram of a monostatic radar . . . . . . . . . . . . . . . . . . . 1.2 Radar transmission waveform . . . . . . . . . . . . . . . . . . . . . . . 1.3 Probability of detection vs. probability of false alarm, for different SNRs 1.4 Instantaneous frequency of the LFM waveform with B p = 100 MHz and T p = 20 µs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Autocorrelation function of the Barker code of length 13 . . . . . . . . 1.6 Autocorrelation function of the polyphase Barker code of length 54 . . . 1.7 Power spectrum of the polyphase Barker code of length 54 where B = 54 MHz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8 Autocorrelation function of the polyphase Barker code of length 60 . . . 1.9 Power spectrum of the polyphase Barker code of length 60 where B = 60 MHz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.10 Instantaneous frequency of the tangent-based NLFM waveform for different values of β with B = 100 MHz and T p = 20 µs. . . . . . . . . . 1.11 Phase of the tangent-based NLFM waveform for different values of β . 1.12 Power spectrum of the tangent-based NLFM pulse, based on the DFT with B = 200 MHz, when (a) β = 0.45 (b) β = 0.95 (c) β = 1.42. . . . 1.13 Instantaneous frequency of the PW NLFM waveform . . . . . . . . . . 1.14 Stepped-frequency linear frequency modulated waveform . . . . . . . . 1.15 Block diagram of the IFFT algorithm . . . . . . . . . . . . . . . . . . . 2.1 Block diagram of the whole processing chain of the modified SF waveform at both transmitter and receiver sides . . . . . . . . . . . . . . . . 2.2 Evolution of the difference of the computational costs D(N, N p ) with N and N p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Real and imaginary parts of the modified SFPC pulses with B = 200 MHz, T p = 1 µs, and N p = 4 . . . . . . . . . . . . . . . . . . . . . . . 2.4 Real and imaginary parts of the modified SF-NLFM pulses with B = 200 MHz, T p = 1 µs, and N p = 4 . . . . . . . . . . . . . . . . . . . . 2.5 Normalized error vs. β when N p = 4 . . . . . . . . . . . . . . . . . . . 2.6 Normalized error vs. the m th pulse with β = 1.21, when (a) N p = 4 (b) N p = 20 (c) N p = 50 . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Power spectrum of v N LF M (n) padded with N zeros when β = 1.21 . . 2.8 Reconstructed power spectrum of the modified SF-NLFM waveform when β = 1.21 and N p = 4 using (a) the modified FD algorithm (b) the TWR algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9 Reconstructed power spectrum of the modified SF-NLFM waveform when β = 1.21 and N p = 20 using (a) the modified FD algorithm (b) the TWR algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.10 Spectral distance vs. β when the TWR algorithm is used for (a) N p = 4 (b) N p = 50. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.11 Spectral distance vs. β when the modified FD algorithm is used for (a) N p = 4 (b) N p = 50 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.12 HRRP of the modified SF-NLFM waveform when N p = 4 using (a) the TWR algorithm, (b) the modified FD algorithm . . . . . . . . . . . . . 2.13 HRRP of the modified SF-NLFM waveform when N p = 20 using (a) the TWR algorithm, (b) the modified FD algorithm . . . . . . . . . . . 2.14 Mean value of the PSLR vs. SNR using N p portions, when the modified FD and the TWR algorithms are used. SF-NLFM case . . . . . . . . . . 2.15 Mean value of the ISLR vs. SNR using N p portions, when the modified FD and the TWR algorithms are used. SF-NLFM case . . . . . . . . . . 2.16 Power spectrum of v P C (n) padded with N zeros when M = 100 . . . . 2.17 Reconstructed power spectrum of the modified SFPC waveform when M = 100 and N p = 4 using (a) the modified FD algorithm (b) the TWR algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.18 Reconstructed power spectrum of the modified SFPC waveform when M = 100 and N p = 20 using (a) the modified FD algorithm (b) the TWR algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.19 HRRP of the modified SFPC waveform when N p = 4 using (a) the TWR algorithm, (b) the modified FD algorithm . . . . . . . . . . . . . 2.20 HRRP of the modified SFPC waveform when N p = 20 using (a) the TWR algorithm, (b) the modified FD algorithm . . . . . . . . . . . . . 2.21 Mean value of the PSLR vs. SNR using N p portions, when the FD and the TWR algorithms are used. SFPC case . . . . . . . . . . . . . . . . 2.22 Mean value of the ISLR vs. SNR using N p portions, when the FD and the TWR algorithms are used. SFPC case . . . . . . . . . . . . . . . . 2.23 Mean value of PSLR vs. SNR using N p portions of the polyphase P4 code (M = 100) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.24 Mean value of ISLR vs. SNR, when using N p portions of the polyphase P4 code (M = 100) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.25 Mean value of the PSLR vs. SNR using N p portions of the polyphase Barker code (M = 54) . . . . . . . . . . . . . . . . . . . . . . . . . . 2.26 Mean value of the PSLR vs. SNR using N p overlapping portions of the polyphase P4 code (M = 100) . . . . . . . . . . . . . . . . . . . . . . 2.27 Mean value of the ISLR vs. SNR using N p overlapping portions of the polyphase P4 code (M = 100) . . . . . . . . . . . . . . . . . . . . . . 2.28 HRRP of the SFPC waveform when using the FD algorithm. The phase code used is the polyphase Barker code with M = 60 . . . . . . . . . . 2.29 Mean values of the PSLR and the ISLR vs. SNR for the SFPC, treated with the FD algorithm, and for the modified SFPC waveforms using a polyphase Barker code (M = 60) . . . . . . . . . . . . . . . . . . . . . 2.30 Mean values of the PSLR and the ISLR vs. SNR for the SFPC, treated with IFFT, and for the modified SFPC, treated with the FD algorithm, when using a polyphase P4 code (M = 100) . . . . . . . . . . . . . . . 3.1 Stepped-frequency linear frequency modulated waveform with variable chirp rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Instantaneous frequency of (a) the tangent-based NLFM waveform with β = 1.21 (b) the PW-NLFM waveform . . . . . . . . . . . . . . . . . . 3.3 Instantaneous frequency of (a) the train of baseband chirp pulses (b) the transmitted variable chirp rate SF-LFM waveform with center frequency f c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Instantaneous frequency of (a) the train of received baseband chirp pulses (b) the train of received chirp pulses shifted in frequency . . . . . . . . 3.5 Instantaneous frequency of the reconstructed PW-NLFM waveform . . . 3.6 Evolutionary process of GA . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Instantaneous frequency of the tangent-based NLFM waveform and the approximated PW-NLFM waveform when L = 1, λ 1 = 0.4 and λ 2 = 0.2. τ 2 = 1.6099µs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 HRRP of a stationary target located at R = 8000 when L = 1, λ 1 = 0.4 and λ 2 = 0.2. τ 2 = 1.6099µs . . . . . . . . . . . . . . . . . . . . . . . 3.9 Value of the time instant versus λ 1 and λ 2 . . . . . . . . . . . . . . . . 3.10 Value of the PSLR versus λ 1 and λ 2 . . . . . . . . . . . . . . . . . . . 3.11 Value of the ISLR versus λ 1 and λ 2 . . . . . . . . . . . . . . . . . . . . 3.12 Value of the range resolution versus λ 1 and λ 2 . . . . . . . . . . . . . . 3.13 Instantaneous frequency of the tangent-based NLFM waveform and the approximated PW-NLFM waveform when L = 2. . . . . . . . . . . . . 3.14 Value of the PSLR versus λ 1 and λ 2 when L = 2. . . . . . . . . . . . . 3.15 Value of the ISLR versus λ 1 and λ 2 when L = 2 . . . . . . . . . . . . . 3.16 Value of the range resolution versus λ 1 and λ 2 when L = 2 . . . . . . . 3.17 Value of the sampling frequency versus λ 1 and λ 2 when L = 2 . . . . . 3.18 Instantaneous frequency of the tangent-based NLFM waveform and the approximated PW-NLFM waveform when L = 10, λ 1 = 0.4 and λ 2 = 0.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1 JD (N SCE 1 ,N SCE 2 ) k defined in (A.5) for random vectors of dimension (a) k = 5, (b) k = 15, (c) k = 50 . . . . . . . . . . . . . . . . . . . . . . . A.2 Asymptotic JD increment and JD derivative between NSCE processes, whose parameters are given in table A.1 . . . . . . . . . . . . . . . . . A.3 Asymptotic increment vs increment with three simulations where σ 2 2 is modified . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.4 JD and its approximation with three simulations where θ 2,1 becomes closer and closer to θ 1,1 = -π/5 . . . . . . . . . . . . . . . . . . . . . A.5 Asymptotic increment vs increment with three simulations where θ 2,1 becomes closer and closer to θ 1,1 = -π/5 . . . . . . . . . . . . . . . . A.6 JD and its approximation with three simulations where the processes share 1, 2 and then 3 normalized angular frequencies . . . . . . . . . . A.7 Asymptotic increment vs increment with three simulations where the processes share 1, then 2 and then 3 normalized angular frequencies . . A.8 Evolution of the asymptotic JD increment as a function of the argument of a 1,1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.9 JD derivative vs number of variates, second simulation with six cases where the modulus of a 1 varies . . . . . . . . . . . . . . . . . . . . . . A.10 Evolution of the asymptotic JD increment as a function of a 1,1 where the dotted points corresponds to the cases addressed in Fig. A.9. . . . . A.11 Evolution of the asymptotic JD increment as a function of σ 2 2 . . . . . . A.12 JD between white noises vs relative difference between the white-noise variances ∆σ 2 u . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.13 Asymptotic JD increment between an AR process and a white noise as a function of the AR parameter and the relative difference between the noise-variances ∆σ 2 u . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.14 Frequency representation of the two 5 th -order MA processes . . . . . . A.15 JD derivative vs number of variates. The parameters of the two 5 th -order MA processes are given in table A.8 . . . . . . . . . . . . . . . . . . . A.16 Evolutions of JDs and JD derivatives between two short-memory F I white noises vs number of variates k . . . . . . . . . . . . . . . . . . . A.17 Evolutions of JDs and JD derivatives between two long-memory F I white noises vs number of variates k . . . . . . . . . . . . . . . . . . . A.18 Evolutions of JDs and JD derivatives between long-memory F I and short-memory F I white noises vs number of variates k . . . . . . . . . A.19 Asymptotic increment of the JD between F I white noises as a function of ∆d = d 1 -d 2 and ρ = σ1 2 σ 2 2 . . . . . . . . . . . . . . . . . . . . . . . A.20 Evolutions of JD and JD derivatives between long-memory F I and short-memory F I white noises vs number of variates k, |∆d| > 0.5 . . . A.21 Steps to follow to compare two unit-power F I white noises . . . . . . . A.22 Asymptotic JD increment as a function of d 1 and d 2 . . . . . . . . . . . A.23 Evolutions of JDs and JD derivatives between two short-memory ARF IM A processes vs number of variates k . . . . . . . . . . . . . . . . . . . . A.24 Evolutions of JDs and JD derivatives between two long-memory ARF IM A processes vs number of variates k . . . . . . . . . . . . . . . . . . . . A.25 Evolutions of JDs and JD derivatives between long-memory ARF IM A and short-memory ARF IM A processes vs number of variates k, |∆d| < . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.26 Evolutions of JD and JD derivatives between long-memory ARF IM A and short-memory ARF IM A white noises vs number of variates k, ∆d > 1 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.27 Evolutions of JDs and JD derivatives between short-memory ARF IM A process and ARMA process vs number of variates k . . . . . . . . . . A.28 Evolutions of JDs and JD derivatives between long-memory ARF IM A process and ARMA process vs number of variates k . . . . . . . . . .LIST OF TABLES1.1 Polyphase Barker codes . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Relation between the bandwidths of different waveforms . . . . . . . . 2.1 Computational costs of the modified FD algorithm and the TWR algorithm. The symbol / is used when it is not applicable . . . . . . . . . . 2.2 Computational costs of the modifed SF and the standard SF waveforms at the transmitter and receiver sides. The symbol / is used when it is not applicable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 General parameters used in the simulation section . . . . . . . . . . . . 2.4 PSLR and ISLR of the modified SFPC waveform for overlapping vs. corresponding non-overlapping cases, using the polyphase P4 code (M = 100) in a noiseless scenario . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Parameters for contrasting the modified SFPC and SFPC waveforms when using the FD algorithm. The polyphase Barker code (M = 60) is used as an intra-pulse modulation . . . . . . . . . . . . . . . . . . . . . 2.6 Parameters for contrasting the modified SFPC and SFPC waveforms when using the IFFT algorithm. The polyphase P4 code is used as an intra-pulse modulation . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Summary of the performance of various waveforms . . . . . . . . . . . 3.1 Reference measures taken into account for the optimization issue based on GA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Performance measures of the approximated PW-NLFM using one time instant based on a priori selection. . . . . . . . . . . . . . . . . . . . . 3.3 Performance measures and value of the time instant of the approximated PW-NLFM when L = 1 and GA are used with λ 1 = 0.4 and λ 2 = 0.2. . 3.4 Performance measures and the values of the time instants of the approximated PW-NLFM when L = 2. . . . . . . . . . . . . . . . . . . . . .

v 2 F 2 l

 22 bb,m (t) m th baseband pulse c Speed of light equal to 3 × 10 8 m/s Q Power density at a distance R from the radiating antenna Chapter (n) m th down-sampled discrete reference signal R v (n) Autocorrelation function of the sequence v(n) T (T x) s Sampling frequency at the transmitter v • (t) Continuous-time baseband pulse where • stands for NLFM or PC Chapter 3 δt m Duration used to time shift the m th pulse to its proper position ν m Instantaneous frequency at the m th time instant of the piece-wise function τ m m th time instant of the piece-wise function x Vector storing the time instants B (m) Passband bandwidth of the m th pulse in Hz B (m) p,trn Bandwidth of the m th truncated pulse in Hz F (x) Fitness function evaluated for the vector x F (int) s Sampling frequency after interpolation f tna (t) Instantaneous frequency of a specific tangent-based NLFM waveform p M Probability of mutation s tshif t rx,bb,m (t) m th time-shifted pulse s f shif t rx,bb,m (t) m th frequency-shifted received pulse T (int) s Sampling time after interpolation T (m) p,trn Duration of the m th truncated pulse T (m) p Duration of the m th pulse T start start time of the reconstructed waveform u m m th phase compensation term Appendix γ l,m Variance of the magnitude of the m th complex exponential of the l th NSCE process µ l,k Mean of the l th NSCE process, vector of length k σ Variance of the white Gaussian noise added to the l th NSCE process θ l,m Normalized angular frequency of the m th complex exponential of the l th NSCE process {a i,l } i=0,...,p AR parameters of the l th ARMA process {b i,l } i=0,...,p MA parameters of the l th ARMA process {p i,l } i=0,...,p Poles of the l th ARMA process {z i,l } i=0,...,q Zeros of the l th ARMA process A m Magnitude of the m th complex exponential of the NSCE process H l (z) Transfer function of the l th random process I k Identity matrix of size k JD k Jeffreys divergence between the joint distributions of two random vectors of length k k Number of samples of a random process KL k Kullback-Leibler divergence between the joint distributions of two random vectors of length k l index l referring to the l th random process with l = 1, 2 M l Number of complex exponentials P l Diagonal matrix of the l th random process p l (x 1:k ) Joint distribution of k successive values of the l th random process Q l,k k × k covariance matrix of the l th random process r ARM A,l,τ (z) Correlation function of the l th ARMA process where τ denotes the lag u n,l n th sample of the driving process of the l th ARMA process x k Observation of the random process at time k x k 1 :k 2 Collection of samples of the process x from time k 1 to k 2 Tr Trace operator

Fig. 1 . 1

 11 Fig. 1.1 Block diagram of a monostatic radar

23 ) 2 n and x = z σ 2 n 2

 2322 At this stage, let us define the following variables a = 2C 2 σ

Fig. 1 . 3

 13 Fig. 1.3 Probability of detection vs. probability of false alarm, for different SNRs

Fig. 1 . 4

 14 Fig. 1.4 Instantaneous frequency of the LFM waveform with B p = 100 MHz and T p = 20 µs

  -a) + jS(-a) + C(b) + jS(b)

Fig. 1 . 5

 15 Fig. 1.5 Autocorrelation function of the Barker code of length 13

Fig. 1

 1 .6, Fig. 1.7, Fig. 1.8, and Fig. 1.9. 5 Here, the autocorrelation function of a sequence

Fig. 1 . 6 Fig. 1 . 7 Fig. 1 . 8 Fig. 1 . 9

 16171819 Fig. 1.6 Autocorrelation function of the polyphase Barker code of length 54

1. 6 . 1 . 3 . 1

 6131 Tangent-based NLFM waveform The tangent-based NLFM waveform is represented by (1.1) where:

Fig. 1 .Fig. 1 .

 11 Fig. 1.10 Instantaneous frequency of the tangent-based NLFM waveform for different values of β with B = 100 MHz and T p = 20 µs.

Fig. 1 .

 1 Fig. 1.12 Power spectrum of the tangent-based NLFM pulse, based on the DFT with B = 200 MHz, when (a) β = 0.45 (b) β = 0.95 (c) β = 1.42.

5 Fig. 1 .

 51 Fig. 1.13 Instantaneous frequency of the PW NLFM waveform

Fig. 1 .

 1 Fig. 1.14 Stepped-frequency linear frequency modulated waveform

1 (k) 2 }

 12 2N -2 for m ∈ 1, N p on one side and the sequence defined by { V pad k=0,...,2N -2 repeated N p times on the other.

Fig. 1 .

 1 Fig. 1.15 Block diagram of the IFFT algorithm

Fig. 2 . 1

 21 Fig. 2.1 Block diagram of the whole processing chain of the modified SF waveform at both transmitter and receiver sides

Fig. 2 . 2

 22 Fig. 2.2 Evolution of the difference of the computational costs D(N, N p ) with N and N p

40 )

 40 In order to find a relation between P dopad ref,m (k) and P down ref,m (k), let us recall the discrete-time Fourier transform (DTFT) and the IDFT of p down ref,m (n):

46 )

 46 This amounts to reproducing N p times P ref (f ), or equivalently a portion of the Fourier transform V • (f ), with a shift of frequency equal to k F (T x) s Np with k = 1, ..., N p -1. Since the latter is band-limited on a frequency band equal to F (T x) s Np , the different terms of the sum in (2.46) do not overlap. Hence, by taking advantage of the periodicity of the DTFT, (2.46) in the frequency band [0, F (T x) s Np ] reduces to:

  47) Given (2.47), a relation can be deduced between the DFT of {p ref,m (n)} n=0,...,N -1 and that of {p down ref,m (n)} n=0,...,N/Np-1 computed at the sampling frequency F (T x) s and F (T x) s

Fig. 2 .Fig. 2 . 3 4 0

 2234 Fig. 2.3 Real and imaginary parts of the modified SFPC pulses with B = 200 MHz, T p = 1 µs, and N p = 4

  defined in(2.33) since the reconstruction step plays a key role to deduce the HRRP. Then, some examples of the HRRP of the modified SF-NLFM waveform processed by both the TWR and the modified FD algorithms are provided. Finally, the performance of the modified SF-NLFM waveform in terms of PSLR and ISLR is analyzed. In the following, the modified SF-NLFM waveform is generated from a tangent-based NLFM pulse. The general parameters used are B = 200 MHz, T r = 40 µs, f c = 3 GHz, and T p = 1 µs. 2.3.1.1 About the relevance of the approximation done in (2.29) in the TWR algorithm In this section, we aim at investigating the relevance of the approximation done in (2.29) for different values of N p . For this purpose, denoting the right-hand side of (2.29) by s ref,m (n), let us compute the normalized error E l between {s int r,m (n)} n=0,...,N -1 and {s ref,m (n)} n=0,...,N -1 for the different pulses, i.e. for the different values of m (1, ..., N p ) equal to: E m = ||s int r,m -s ref,m || 2 ||s ref,m || 2 (2.50) where ||.|| 2 denotes the Euclidean norm, and s int r,m and s ref,m respectively denotes the vectors that store the set of samples of s int r,m (n) and s ref,m (n) with n = 0, ..., N -1.

Fig. 2 . 5 Fig. 2 . 6 50 2. 3 . 1 . 2

 252650312 Fig. 2.5 Normalized error vs. β when N p = 4

  when β = 1.21 and for different values of N p . S N LF M ref (k) is presented in Fig. 2.7 whereas

Fig. 2 . 7

 27 Fig. 2.7 Power spectrum of v N LF M (n) padded with N zeros when β = 1.21

Fig. 2 . 8 Fig. 2 . 9

 2829 Fig. 2.8 Reconstructed power spectrum of the modified SF-NLFM waveform when β = 1.21 and N p = 4 using (a) the modified FD algorithm (b) the TWR algorithm

Fig. 2 .Fig. 2 .

 22 Fig. 2.10 Spectral distance vs. β when the TWR algorithm is used for (a) N p = 4 (b) N p = 50.

Fig. 2 .Fig. 2 .

 22 Fig. 2.12 HRRP of the modified SF-NLFM waveform when N p = 4 using (a) the TWR algorithm, (b) the modified FD algorithm

Fig. 2 .

 2 Fig. 2.14 Mean value of the PSLR vs. SNR using N p portions, when the modified FD and the TWR algorithms are used. SF-NLFM case

Fig. 2 .Fig. 2 .Fig. 2 .

 222 Fig. 2.16 Power spectrum of v P C (n) padded with N zeros when M = 100

Fig. 2 .Fig. 2 .

 22 Fig. 2.19 HRRP of the modified SFPC waveform when N p = 4 using (a) the TWR algorithm, (b) the modified FD algorithm

Fig. 2 .

 2 Fig. 2.21 Mean value of the PSLR vs. SNR using N p portions, when the FD and the TWR algorithms are used. SFPC case

Fig. 2 .Fig. 2 .Fig. 2 .

 222 Fig. 2.23 Mean value of PSLR vs. SNR using N p portions of the polyphase P4 code (M = 100)

Fig. 2 .Fig. 2 .

 22 Fig. 2.26 Mean value of the PSLR vs. SNR using N p overlapping portions of the polyphase P4 code (M = 100)

Fig. 2 .

 2 Fig. 2.28 HRRP of the SFPC waveform when using the FD algorithm. The phase code used is the polyphase Barker code with M = 60

Fig. 2 .

 2 Fig. 2.29 Mean values of the PSLR and the ISLR vs. SNR for the SFPC, treated with the FD algorithm, and for the modified SFPC waveforms using a polyphase Barker code (M = 60)

Fig. 2 .

 2 Fig. 2.30 Mean values of the PSLR and the ISLR vs. SNR for the SFPC, treated with IFFT, and for the modified SFPC, treated with the FD algorithm, when using a polyphase P4 code (M = 100)

  bandwidth and the duration of the m th pulse, with Np m=1 B (m) p = B. In addition, the carrier frequency still follows (1.64) but ∆f (m) now satisfies:

p

  } m=1,...,Np and of the pulse durations {T (m) p } m=1,...,Np .

Fig. 3 . 2 2 .

 322 Fig. 3.2 Instantaneous frequency of (a) the tangent-based NLFM waveform with β = 1.21 (b) the PW-NLFM waveform

2

  is required in the rect function to focus on the interval [(m -1)T r , (m -1)T r + τ m+1 -τ m ]. It is also required in the exponential to have a spectrum centered around zero.

Fig. 3 . 3

 33 Fig. 3.3 Instantaneous frequency of (a) the train of baseband chirp pulses (b) the transmitted variable chirp rate SF-LFM waveform with center frequency f c

Fig. 3 . 4 Fig. 3 . 5

 3435 Fig. 3.4 Instantaneous frequency of (a) the train of received baseband chirp pulses (b) the train of received chirp pulses shifted in frequency

4 .

 4 Time shift the result obtained for each pulse in (3.45) by δt m defined in(3.22). When F

Fig. 3 . 6

 36 Fig. 3.6 Evolutionary process of GA

Table 3 . 3 2 .Fig. 3 . 7 Fig. 3 . 8

 3323738 Fig. 3.7 Instantaneous frequency of the tangent-based NLFM waveform and the approximated PW-NLFM waveform when L = 1, λ 1 = 0.4 and λ 2 = 0.2. τ 2 = 1.6099µs

2 Fig. 3 . 9 2 0Fig. 3 . 2 0Fig. 3 . 2 Let us look at λ 1 = 0. 1 .Fig. 3 .

 23923232113 Fig. 3.9 Value of the time instant versus λ 1 and λ 2

3. 4 . 3 . 1

 431 Waveform parameters based on the multi-objective criterion deduced by GA and corresponding performance measures: predefined weights The weights are still selected as follows: λ 1 = 0.4 and λ 2 = 0.2. The results are presented in Table.

3 . 4

 34 and Fig.3.13.

Fig. 3 .

 3 Fig. 3.13 Instantaneous frequency of the tangent-based NLFM waveform and the approximated PW-NLFM waveform when L = 2.

2 PSLRFig. 3 .

 23 Fig. 3.14 Value of the PSLR versus λ 1 and λ 2 when L = 2.

2 ISLRFig. 3 .Fig. 3 .Fig. 3 .

 2333 Fig. 3.15 Value of the ISLR versus λ 1 and λ 2 when L = 2

.

  The second one, which corresponds to the Jeffreys divergence, consists in taking the mean of KL

e

  j(k-1)θ l,1 . . . e j(k-1)θ l,M l At this stage, let us recall the following properties that the vectors {S m l,k } m=1,...M l satisfy. For any m = 1, ...M l and n = 1, ...M l with n = m, one has:

Fig. A. 1

 1 Fig. A.1 JD (N SCE 1 ,N SCE 2 ) k defined in (A.5) for random vectors of dimension (a) k = 5, (b) k = 15, (c) k = 50

Fig. A. 3 1 3 ) θ 2 , 1 = - 1 Fig. A. 4 5 A. 5 . 4

 3132114554 Fig. A.3 Asymptotic increment vs increment with three simulations where σ 22 is modified TableA.3 Parameters of the NSCE processes under study for the 3 rd simulation protocol1 st -process NSCE parameters 2 nd -process NSCE parameters M 1 = 1 M 2 = 1 σ 2 1 = 1 σ 2 2 = 0.1 γ 1,1 = 100 γ 2,1 = 40 1 value for θ 1,1 : 3 cases for θ 2,1 : θ 1,1 = -π/5 1) θ 2,1 = -2π/52) θ 2,1 = -1.5π/5 3) θ 2,1 = -1.1π/5

u,l , 1 σ l 2 + σ l 2 σ 2 u,l , 1 ) (A. 106 )

 1221106 It corresponds to the JD increment between white noises with variance σ 2 u,l ,1

σ 2 u, 1 , 1 1-|a 1 , 1 | 2 .A. 5 . 5 . 1

 211112551 Influence of the AR-parameter argumentThe parameters of the processes under study are given in TableA.5.Table A.5 Parameters of the AR and NSCE processes under study NSCE process parametersAR process parametersM 2 = 1 |a 1,1 | = 0.95 σ 2 2 = 1 σ u,1,1 = 1 γ 2,1 = 10 argument of a 1,1 varies θ 2,1 = π/2 in the interval ] -π, 0[ Based on Fig. A.8

2 Fig. A. 8 1 A. 5 . 5 . 2

 281552 Fig. A.8 Evolution of the asymptotic JD increment as a function of the argument of a 1,1

Fig. A. 9

 9 Fig. A.9 JD derivative vs number of variates, second simulation with six cases where the modulus of a 1 varies

Fig. A. 10

 10 Fig. A.10 Evolution of the asymptotic JD increment as a function of a 1,1 where the dotted points corresponds to the cases addressed in Fig. A.9.

=

  1,l | = 1. It was equal to:∆JD (M A 1 ,M A 2 ) k

2 |1-b 2 1, 2 | 2 if |b 1 , 2 | < 1 σ 2 u, 1 , 1 σ 2 u, 2 , 1 ( 2 b 1 ,1 b 1 , 2 )(b 1 , 2 ) 2 - 1 if |b 1 , 2 | > 1 (

 222212121122121121221121 can be re-expressed as follows: 1+(b 1,1 ) 2 -A.125)

5 d 2 ∈

 52 fer function (1 -z -1 ) d 2 -d 1 . Depending on the values of d 1 and d 2 , different situations occur. See table A.9. Table A.9 Different cases under study d 1 ∈ (-0.5, 0) d 1 ∈ (0, 0.5) d 2 ∈ (-0.5, 0) As |d 1 -d 2 | < 0.5, the filter output is a F I white noise |d 1 -d 2 | can be smaller or larger than 0.(0, 0.5) |d 1 -d 2 | can be smaller or larger than 0.5 As |d 1 -d 2 | < 0.5, the filter output is a F I white noise

d 2 = 2 u, 1 σ 2 u, 2

 22122 Fig. A.16a and Fig. A.16b

( a )

 a Fig. A.18 Evolutions of JDs and JD derivatives between long-memory F I and shortmemory F I white noises vs number of variates k

Fig. A. 19

 19 Fig. A.19 Asymptotic increment of the JD between F I white noises as a function of ∆d = d 1 -d 2 and ρ = σ 1 2

5 d 2 ∈

 52 62)-(A.64) applied to the zeros of the second process. Depending on the values of d 1 and d 2 , different situations occur and are summarized in table A.11. Table A.11 Different cases under studyd 1 ∈ (-0.5, 0) d 1 ∈ (0, 0.5) d 2 ∈ (-0.5, 0) As |d 1 -d 2 | < 0.5, the filter output is a ARF IM A process |d 1 -d 2 | can be smaller or larger than 0.(0, 0.5) |d 1 -d 2 | can be smaller or larger than 0.5 As |d 1 -d 2 | < 0.5, the filter output is a ARF IM A process 1. Let us look at the outputs of the two inverse filters when (d 1 , d 2 ) ∈ (-0.5, 0)

.Fig. A. 23

 23 Fig. A.23 Evolutions of JDs and JD derivatives between two short-memory ARF IM A processes vs number of variates k

  M F (t) denotes the output of the MF. When the latter is tested at a certain time t te , one of two hypothesis can be assumed: either the measurement s M F (t te ) is the result of disturbance only and this hypothesis is denoted as H 0 or s M F (t te ) is the result of the disturbance and echoes from targets. This hypothesis is denoted as H 1 . Radar designers are interested in statistically describing the data under each hypothesis to design good detection algorithms and analyze the radar performance. Hence, two probability density functions (PDFs) are required: 1. p s M F (t te )|H 1 denotes the PDF of s M F (t te ) under the hypothesis H 1 , i.e. when a target is present, 2. p s M F (t te )|H 0 denotes the PDF of s M F (t te ) under the hypothesis H 0 , i.e. when a target is not present.

	1.4.3 Optimum detector for nonfluctuating radar signals
	Let us assume that s

Table 1 . 1

 11 Polyphase Barker codes

	Code length (M )	P	φ i , (i = 1, . . . , M )
	54	200	0, 0, 23, 43, 16, 9, 40, 51, 20, 7, 67, 126, 178,
			180, 71, 120, 144, 151, 61, 25, 45, 100, 86, 9,
			172, 161, 142, 22, 85, 8, 96, 128, 81, 1, 18,
			137, 0, 95, 132, 59, 44, 155, 16, 129, 157, 98,
			47, 174, 73, 18, 145, 65, 170, 100
	60	210	0, 0, 16, 208, 180, 153, 126, 161, 135, 78, 83,
			98, 143, 127, 162, 153, 183, 141, 72, 207, 149,
			167, 15, 13, 146, 58, 23, 109, 169, 208, 75,
			143, 173, 199, 51, 50, 31, 142, 152, 84, 74, 6,
			147, 205, 151, 66, 51, 151, 27, 101, 170, 75,
			172, 91, 20, 131, 1, 78, 166, 68

Table 1 . 2

 12 Relation between the bandwidths of different waveforms

Table. 1.2 summarizes the relationship between B b , B and B ef f for various waveforms.

  • m (t)} m=1,...,Np by using a DAC, as shown in Fig.2.1.6. Multiply the latter pulses by exp[j2πf c t] to frequency translate each portion bythe central carrier frequency f c similar to that introduced in (1.64). The resulting signal is hence given by:

• s,m (n)} m=1,...,Np to the continuous-time domain leading to the signals { 1 αm p

  For this purpose, N p reference pulses {p ref,m (n)} m=1,...,Np have to be prepared.

							They respectively
	correspond	to	the	non-zero	components	of	the	spectra
	{P •

m (k)} m=1,...,Np . Thus, for m ∈ 1, N p , the reference signals are given by:

  .28) 2. Interpolate s r,m (n) so that the new sampling frequency becomes F

	(int) s	= N p F	(Rx) s	= B. This can be done either in the time domain or in the
	frequency domain 2 . Sampling at F	(Rx) s

  2π N/Np . Otherwise, intermediate values are obtained. The DFT of the padded signal p dopad ref,m (n) is given by evaluating P down ref,m (θ) at the normalized angular frequency θ = k 2π 2N/Np-1 where k ∈ 0, 2N Np -2 . Hence, P dopad ref,m (k) is given by:

N

  Np samples. Table.2.2 summarizes the computational cost of each step for both the

Table 2 . 2

 22 Computational costs of the modifed SF and the standard SF waveforms at the transmitter and receiver sides. The symbol / is used when it is not applicable

	Steps	SFPC	Transmitter modified SFPC	SFPC	Receiver modified SFPC
	1

Table 2

 2

	.3 General parameters used in the simulation section
	Parameter	Value
	Pulse repetition frequency (P RF )	250 Hz
	Carrier frequency (f c )	3 GHz
	Pulse width (T p )	1 µs
	consists of a single carrier whereas the steps given in section 2.2.2.2 are done for the
	(Rx) s modified SFPC waveform. Therefore, as F

  Table. 2.4. In every row of this table, the value of N p for the overlapping and the corresponding nonoverlapping cases are carefully chosen. For N p = 9 in the overlapping case which requires F MHz, we choose two corresponding values of N p for the nonoverlapping case: N p = 5 which is almost twice smaller but used with the same sampling frequency at the receiver and N p = 10 which requires almost the same number of portions but used with the sampling frequency which is twice smaller. Similar comments can be drawn for the other three cases presented in the table. Given the results presented in table 2.4, the overlapping scenario outperforms the non-overlapping one, in terms of PSLR and ISLR. The tax to be paid is either a higher F

	(Rx) s	= 40 (Rx) s	or a larger N p .

Table 2 .

 2 4 PSLR and ISLR of the modified SFPC waveform for overlapping vs. corresponding non-overlapping cases, using the polyphase P4 code (M = 100) in a noiseless scenario

	N p F	(Rx) s	(MHz) PSLR (dB)	ISLR (dB)	50% Overlap
	5	40	-26.46	-12.1	No
	9	40	-27.04	-15.5	Yes
	10	20	-24.15	-12.8	No
	10	20	-24.15	-12.8	No
	19	20	-27.52	-15.76	Yes
	20	10	-23.7	-7.9	No
	20	10	-23.7	-7.9	No
	39	10	-29.63	-13.54	Yes
	40		5	-22.62	-8.65	No
	50		4	-20.5	-7.8	No
	99		4	-20.96	-11.86	Yes
	100		2	-12	-5.94	No
	2.3.2.5 Modified SFPC waveform vs. standard SFPC waveform

Table 2 . 6

 26 Parameters for contrasting the modified SFPC and SFPC waveforms when using the IFFT algorithm. The polyphase P4 code is used as an intra-pulse modulation PC and the SFPC waveforms. From one side, both the PC and the modified SFPC waveforms exhibit same range resolution and minimum range. Nevertheless, F

	Parameter	Modified SFPC ∀ N p	SFPC N p = 10 N p = 20 N p = 40
		T p (µs)	1	10	20	40
	B b (MHz)	B 2Np	10	5	2.5
	F	(Rx) s	(MHz)	B Np	20	10	5
	∆f (MHz)	B Np	20	10	5
		B (MHz)	200	200	200	200
	B ef f (MHz)	100	200	200	200
		R res (m)	1.5	0.75	0.75	0.75
		R min (m)	150	1500	3000	6000

Summary about modified SFPC vs. both PC and SFPC The various waveforms that have been examined in this section are listed in Table

.

2.7. The values of the PSLR and the ISLR are measured for a P4 code in a noiseless environment. This table illustrates how the modified SFPC waveform processed with the FD algorithm outperforms both the

Table 2 . 7

 27 Summary of the performance of various waveforms

	Waveform	PC	SFPC	SFPC	Modified SFPC
	Algorithm	MF	FD	IFFT	FD
	F s (Rx)	(M Hz)	B	B/N p	B/N p	B/N p
	R res	c/B	c/B	c/2B	c/B
	No. of pulses	1	N p	N p	N p
	R min	cT p /2	cT p N p /2 cT p N p /2	cT p /2
	P SLR(dB)	-26.32	-13.2	-12	[-21; -29.3]
	ISLR(dB)	-13.88	-6.67	-10.4	[-7.8; -13.7]
	Artifacts	No	No	Yes	No

Table 3 . 1

 31 Reference measures taken into account for the optimization issue based on GA.

	PSLR (dB)	ISLR (dB)	R res (m)
	-31.2	-25	1.37
	3.4.2 Simulation results and comments, L = 1
	3.4.2.1 Waveform parameters based on a priori selection and corresponding perfor-
	mance measures

Table 3 . 2

 32 Performance measures of the approximated PW-NLFM using one time instant based on a priori selection.

	Method	Time instants (µs)	PSLR	ISLR	R res (m)
			(dB)	(dB)
	1 st meth. based	τ 2 =3.1186	-16.3432	-11.8106	1.2377
	on (3.58)
	2 nd meth. based	τ 2 =1.1612	-16.6322	-12.8179	1.1360
	on (3.60)
	3.4.2.2 Waveform parameters based on the multi-objective criterion deduced by GA
	and corresponding performance measures: predefined values of the weights

Table 3 . 4

 34 Performance measures and the values of the time instants of the approximated PW-NLFM when L = 2.

	Time instants (µs)	PSLR (dB) ISLR (dB) R res (m)
	τ 2 =2.2970; τ 3 =5.0211	-24.7427	-16.0011	1.3107

  ,θ 2,n with δ θ 1,m ,θ 2,n equal to 1 when θ 1,m = θ 2,n and zero otherwise. In other words, δ m 1,2 = 1 if both processes share the normalized angular frequency θ 1,m . Otherwise it is equal to 0.

	M 2 n=1 δ θ 1,m Combining (A.72) and (A.77), this leads to: where δ m 1,2 =
	A.77)

  By replacing in (A.123), r AR,2,τ with τ = -1, 0, 1 by their expressions recalled in (A.23), we retrieve R u 1-2a 1,1 a 1,2 +a 2

	1,1	r AR,2,0 +	a 1,1 σ 2 u,1,1	r AR,2,1 +	a 1,1 u,1,1 σ 2	r AR,2,-1	(A.123)
		1,1
	1-a 2 1,2

  .10.Table A.10 Parameters of the FI white noises under study

	1 st -process parameters	2 nd -process parameters	Figures
	Two short-

Stationary target is a target whose relative velocity is equal to zero.

in the absence of noise.

The time duration of the N p pulses processed coherently by using coherent integration or fast Fourier transform processing is called the CPI.

Having two sequences of length L and N respectively, before applying MF in the frequency domain, each sequence should be padded by zeros so that its length is at least L+N -1.

In these days, an order of magnitude could be 100MHz.

In the time domain, the signal is upsampled by a factor equal to N p and then is interpolated by using a low-pass finite-impulse response (FIR) filter. Polyphase filters are often considered. As for the frequency domain, the DFT of s r,m (n) is decomposed into two halves. Zeros are introduced in between to obtain a sequence of length N . Then, the inverse DFT is computed[2]. In our case the latter method is used.

It is the time that an antenna beam spends on a target. It can correspond to the duration of one CPI or more CPIs

In the time domain, the signal is upsampled by a factor equal to η and then interpolated by using a low-pass finite-impulse response (FIR) filter. Polyphase filters are often considered. As for the frequency domain, the discrete Fourier transform (DFT) of s rx,bb,m is decomposed into two parts. Zeros are then introduced in between to obtain a sequence of length ητ m . Then the real part of the inverse DFT is computed.[2]

It is the time that an antenna beam spends on a target. It can correspond to the duration of one CPI or more CPIs.

A long-memory process in the covariance sense is characterized by a a normalized covariance function which decays slowly so that the sum of these terms goes to infinity.

In the complex case, T is replaced by H and 1 2 disappears in (A.2). In the following, ∝ is used instead of =.

Given the matrices A, U , C and V where A and C are assumed to be invertible, one has:(A + U CV ) -1 = A -1 -A -1 U C -1 + V A -1 U -1 V A -1 .

It should be noted that the squares of the the partial correlation coefficients correspond to the squares of the reflection coefficients.

In the equation below, the notation we use is lim k→+∞ . This means that we always consider correlation matrices of finite size k, where k becomes larger and larger.

Indeed, one has: r AR,l ,0 ∝ S AR,l ,p (θ) dθ

(m,n) k , defined in Fig. A.21, satisfies:

(a) Evolution of the JDs vs the variate number k (b) Convergence of the JD derivative towards ∆JD (ARF IM A 1 ,ARF IM A 2 )

Modified SFPC vs. SFPC using IFFT algorithm

In Fig. 2.30a and Fig. 2.30b, the evolutions of the mean value of the PSLR and ISLR of both waveforms for various N p are presented respectively. Fig. 2.30a shows clearly the enhancement of the PSLR of the modified SFPC waveform compared to that of the SFPC waveform. As for the ISLR, the results are highly dependent on N p . Hence, the modified SFPC waveform outperforms the SFPC for certain values of N p , and the converse occurs for other values. Looking at both waveforms from another viewpoint, the range resolution for the SFPC waveform is twice better than that of the modified SFPC. However, the minimum range of the modified SFPC waveform is kept constant for any value of N p whereas that of the SFPC waveform varies with the selected N p . The larger N p is, the longer R min is. See Table . 2.6.

For this reason, in the following section, we suggest optimizing the waveform parameters when L = 10.

Simulation results and comments, L = 10

In this section, the waveform parameters when L = 10 are optimized by using GA for λ 1 = 0.4 and λ 1 = 0.2. In this case, Q p is equal to 1000. The results are presented in Table . 3.5 and Fig. 3.18. Based on the time instants provided in Table . 3.5, the sampling frequency that should be used is equal to 33.3333 MHz. Given Table . 3.5, we can observe that the normalized differences on the PSLR and the range resolution are approximately equal to zero. As for the normalized difference on the ISLR, it is equal to 0.1171.

General comments on the results

In this section, we aim at giving some comments on the number of time instants, i.e., the number of pulses within the CPI, that could be selected by the practitioner. According to the obtained results, two conclusions can be drawn:

JEFFREYS DIVERGENCE FOR PROCESS COMPARISON:

Properties, asymptotic analysis, physical interpretation and way to use it in practical cases At the very beginning of the collaboration between the University of Bordeaux and the University of Lebanon, presentations of the research activities were organized. The selection and comparison of models, as well as the design of parametric signal processing approaches based on an a priori model drew the attention of both research teams. In addition, applications in mobile communication systems or radar processing were among some common applications of interest. Therefore, different works shared by both teams started to be addressed. Among them, properly reconstructing the high-resolution range profile of stepped frequency waveform in the presence of missed data, optimizing the selection of the waveform, modeling the clutter and comparing different types of clutter by comparing their statistics. As for the last issue, the idea was to use divergences. Divergences had been already exploited by the French team to compare motion models and select two or three motion models in order to design target tracking based on multiple model approaches [START_REF] Magnant | Jeffrey's divergence for state-space model comparison[END_REF]. The French team also focused its attention on the asymptotic behavior of the divergences. In information theory, this corresponds to analyze what is called the divergence rate [START_REF] Liese | On divergences and informations in statistics and information theory[END_REF]. Various works were conducted on the analysis of Jeffreys divergence between 1 st -order moving average (MA) processes and 1 st -order autoregressive (AR) processes [START_REF] Legrand | Jeffrey's divergence between moving-average and autoregressive models[END_REF]. Although these results were of interest, they were dedicated to very specific signals. Therefore, we decided to address more general cases, i.e. sum of complex exponentials disturbed by an additive white noise, autoregressive moving average (ARMA) processes and finally long-memory processes like autoregressive fractionally moving average (ARFIMA) processes. These different studies were conducted at the very beginning of the PhD. All these studies were finally useful since it enables us to provide a physical interpretation of the JD. The purpose of this appendix is to present the complementary results that were obtained during this PhD. Using (A.3) and (A.4) leads to the following expression of the JD between zero-mean Gaussian processes:

) no longer involves a logarithm but depends on the k × k covariance matrices of the processes. Depending on the applications, the signals we can compare can have a discrete or a continuous spectrum. They can be short or long-memory processes. For this reason, the next section deals with the properties of NSCE processes as well as ARMA and ARFIMA processes. More particularly, some properties on the covariance matrices are presented as they will be useful for the computation of the JD and its analysis.

A.3 Presentation of the processes under study

This section is organized as follows. The NSCE processes are first presented. Then, the properties of the ARMA processes are detailed. Finally, we focus our attention on ARFIMA processes. As our purpose is to compare two zero-mean processes by means of the JD, a subscript l is introduced in the notations to refer the l th process under study, with l = 1, 2.

A.3.1 About the sum of complex exponentials (SCE) disturbed by an additive noise A.3.1.1 Definition and spectral properties of the NSCE processes

The sum of complex exponentials (SCE) disturbed by an additive noise is widely used in high-resolution spectrum analysis such as Pisarenko's method or its generalization called MUSIC. In this case, the n th sample of l th process is defined as follows:

where M l is the number of complex exponentials and {θ l,m } m=1,...,M l are the normalized angular frequencies. For each process, these latter are in the interval [-π, π[ and are assumed to be different from each other. However, both processes can have common normalized angular frequencies. The magnitudes {A m } m=0,...M l are Gaussian, zeromean with variances {γ l,m } m=1,...,M l . Finally η n,l is the n th sample of a zero-mean Gaussian white noise with variance σ 2 l , uncorrelated with the exponential magnitudes. This process is characterized by a power spectrum which has a discrete part due to the complex exponential functions and a continuous part equal to σ 2 l due to the additive white noise.

A.3.2 About ARMA processes

A.3.2.1 Definitions, poles and zeros and PSD expression of the ARMA processes

Let us consider the l th w.s.s. ARMA(p,q) process [START_REF] Box | Time Series Analysis, Forecasting, and Control[END_REF]. Its n th sample, denoted as x n,l , is defined as follows:

where u n,l is the n th sample of the driving process, assumed to be white, Gaussian, zero-mean with variance σ 2 u,l . In the remainder of this appendix, as ARMA and NSCE processes can be compared, the driving process is assumed to be uncorrelated with the additive noises of the NSCE processes. By introducing a 0,l = 1, {a i,l } i=0,...,p are the AR parameters of the l th ARMA process. If b 0,l = 1, {b j,l } j=0,...,q denote the MA parameters. Given (A.18), the ARMA processes can be seen as the outputs of filters whose inputs are zero-mean white sequences with unit-variance and whose transfer functions H l (z) are defined by their poles {p i,l } i=1,...,p and their zeros {z i,l } i=1,...,q :

When {a i,l } i=1,...,p = 0, the ARMA process which reduces to a MA process is also called an all-zero process. Similarly, if {b j,l } j=1,...,q = 0, the process, which is only AR, is also called an all-pole process. They belong to the class of sourcefilter models. The PSDs, evaluated at the normalized angular frequency θ, are denoted as S ARM A,l (θ) and satisfy:

Therefore, when the AR poles are close to the unit circle in the z-plane, the PSD exhibits resonances which are located at the normalized angular frequencies around the pole arguments whereas zeros located on the unit-circle in the z-plane lead to a PSD which is null at the normalized angular frequencies corresponding to arguments of the zeros.

A.3.2.2 Correlation properties of the ARMA processes

Let us now focus our attention on the mean and the autocorrelation function of the ARMA processes. As the driving process is zero-mean, the mean of the ARMA process is null. Therefore, the autocorrelation function (resp. matrix) and the covariance function (resp. matrix) are equal. As the ARMA parameters and the variance of the driving process do not vary over time, the ARMA process is w.s.s. and can be characterized by the autocorrelation function r ARM A,l,τ , where τ is the lag. When p = 0, i.e. when the ARMA process reduces to a q th -order MA process, the number of non-zero autocorrelation-function coefficients is finite: to the maximum, 2q + 1 coefficients are non zero. For instance, for a real 1 st -order MA process with MA parameter b 1,l and driving process variance σ 2 u,l , one has:

More generally, for any real q th -order MA process, one has:

Therefore, for any order q, r M A,l,τ is absolutely summable. Concerning a 1 st -order AR process with AR parameter a 1,l and driving-process variance σ 2 u,l , the autocorrelation function satisfies:

By using D'alembert criterion for series convergence [START_REF] Polyanin | Handbook of Mathematics for Engineers and Scientists[END_REF], one can easily show that r AR,l,τ is absolutely summable.

For a p th -order AR process, the autocorrelation function satisfies the following p th -order difference equation for τ = 0:

Given (A.24), it can be shown that r AR,l,τ can be expressed as a linear combination of the poles {p τ i,l } i=1,...,p . Therefore, one can deduce that the autocorrelation function exponentially decays as the lag τ increases. This property also holds for ARMA(p,q) processes, with p = 0. Therefore, r AR,l,τ and r ARM A,l,τ are absolutely summable. A.3.3 About ARF IM A(p, d, q) processes A. 3

.3.1 Preamble

In time series analysis, an autoregressive integrated moving average process ARIM A(p, d, q), where d is an integer, is a generalization of the ARMA process. In this case, the ARIMA process has d poles with modulus equal to 1. An ARIM A(0, 0, 0) process is a white noise whereas an ARIM A(0, 1, 0) process is a random walk. In addition, an ARIM A(0, 1, 1) process is an exponential smoothing process [START_REF] Box | Time Series Analysis, Forecasting, and Control[END_REF]. They are all non-stationary. The class of ARF IM A(p, d, q) processes is then a new level of generalization, which allows the degree of differencing d to take any real value. Before presenting the properties of ARFIMA processes, let us introduce some mathematical definitions that will be useful in the following.

1. The gamma function, denoted as Γ (.), is defined by:

When x is an integer larger or equal to 1, Γ (x) = (x -1)!.

2. The Γ function satisfies Euler's reflection formula, which is given by:

3. The binomial expansion, that will be used in the following, can be expressed as follows:

(

In the above equation, according to [START_REF] Laforgia | On the asymptotic expansion of a ratio of gamma functions[END_REF], for large value of i, one has:

At this stage, using the standard binomial expansion (A.38) of (1 -z -1 ) g for g = -d where d is an integer value, let us study the properties of a process xn defined as follows:

where u n is a zero-mean white noise.

It should be noted that the variance of xn is proportional to 1 + 1) . This infinite sum converges if 2(d -1) < -1 or equivalently if d < 1 2 . In the next subsection, let us present a specific case of ARFIMA processes, namely the fractionnally integrated white noise before addressing the general case.

A.3.3.2 Definitions and properties of fractionally integrated F I(d) white noise

The n th sample of the l th fractionally integrated white noise, denoted as x n,l , can be defined by looking at the relationship between X l (z) the z-transform of the F I(d) white noise and U l (z) the z-transform of the driving process u n :

This amounts to saying that the process whose z-transform is X l (z)(1 -z -1 ) d is a white noise. Using (A.41) and the link between the z-transform and the Fourier transform, the PSD of the l th FI white noise, denoted as S F I,l (θ), can be expressed as follows :

Therefore, the PSD tends to infinity when θ tends to 0 if d > 0 whereas it tends to 0 when θ tends to 0 if d < 0. It also means that the process has an infinite power when |d| > 1 2 . At this stage, using the inverse Fourier transform of (A.42) and taking advantage of the results of some integrals 5 recalled by Gradshteyn and Ryzhik in [START_REF] Gradshteyn | Table of Integrals, Series, and Products[END_REF], it can be shown that the autocorrelation function of the l th F I(d) white noise satisfies when

Given (A.43), the ratio between two consecutive values of the autocorrelation function is equal to:

) , with Re(ν) > 0 and B(x, y) is the Beta function which satisfies B(x, y) = Γ (x)Γ (y) Γ (x+y) .

Therefore, |ρ τ,l | are necessarily strictly smaller than 1 and the finite-size correlation matrix Q F I,l,k is invertible.

A.3.3.3 Definitions and properties of ARFIMA processes

As done for the F I(d) processes, let us define the n th sample of an ARFIMA process from its z-transform as follows:

where {p i,l } i=1,...,p are the poles and {z i,l } i=1,...,q are the non-unit zeros. In addition, d is chosen to describe the high-lag correlation structure, whereas the ARMA parameters describe the low-lag correlation structure of the time series.

The corresponding transfer function between the ARFIMA process x n,l and the driving process u n,l is hence defined as follows:

Under the assumption that the poles and the zeros are different and that all poles lie inside the unit circle in the z-plane, the ARF IM A(p, d, q) time series defined in (A.49) can be expressed in a unique causal infinite-order MA form [START_REF] Kozoszka | Fractional ARIMA for stable innovations[END_REF]:

where the z-transform of {c i,l } i=0,...,+∞ , i.e.

shown in [START_REF] Kozoszka | Fractional ARIMA for stable innovations[END_REF] that the IIR filter coefficients c i,l are asymptotically proportional to i d-1 . Hence, the corresponding IIR filter is BIBO stable.

1. When d > 1 2 , the process is non-stationary and it possesses infinite variance.

2. When 0 < d < 1 2 , the process exhibits long memory or persistence, whereas it has intermediate memory, also called anti-persistence when -1 2 < d < 0.

134

The PSD, S ARF IM A,l (θ), can be expressed as follows :

Concerning the autocorrelation function, when 0 < d < 1 2 , r ARF IM A,l,τ is proportional to τ 2d-1 when τ tends to infinity. It hyperbolically decays to 0 and can be expressed as follows [START_REF] Palma | Long-Memory Time Series, Theory and Methods[END_REF]:

where:

Given (A.54), φ(i) corresponds to the values of the autocorrelation function of the MA part defined from the zeros of the ARF IM A(p, d, q) process. Moreover, one has:

with β(τ, p j,l ) = F (d+τ, 1, 1-d+τ, p j,l ) and F the Gaussian hypergeometric function 7 .

A recursive relation for the latter function exists [START_REF] Sowell | Maximum likelihood estimation of stationary univariate fractionally integrated time series models[END_REF] and is given by:

n! where (a) n is the Pochhammer symbol that satisfies:

Thus, given (A.56) and (A.57), it can be shown that:

Note that this relation will be useful in the simulation part to reduce the computational cost and to avoid numerical issues.

Let us now study whether the correlation matrix Q ARF IM A,l,k is singular or not, for any finite size k. As recalled in (A.48), the correlation matrix is invertible if the moduli of the partial correlation coefficients are not equal to 1. However, there is no analytical expression of these coefficients for ARFIMA processes. Therefore, let us look at the reflection coefficients of the ARFIMA process, since they correspond up to a multiplicative value equal to -1 to the partial correlation coefficients when the process is w.s.s.. They can be obtained with the Durbin-Levinson algorithm when the zeros are inside the unit circle in the z-plane [START_REF] Najim | Modeling, estimation and optimal filtering in signal processing[END_REF]. Using a reductio ad absurdum, let us show that the correlation coefficient for an order ν is not equal to 1. Thus, if the modulus of the reflection coefficient was equal to 1, the variance of the driving process of the corresponding ν th -order AR process defined by the AR parameters {a ν i,l } i=0,...,ν would be equal to 0, meaning that for any lag τ (even τ = 0):

The correlation matrix of size ν × ν would be rank deficient. The spectrum of the process under study would be defined by less than ν frequencies, which is not coherent with the continuous feature of (A.52). When ν becomes very large and tends to infinity, we could wonder whether this justification could be still considered. In this case, the partial correlation coefficients of a w.s.s. ARFIMA process with 0 < d < 1 2 satisfies: |ρ ν,l | ∼ d ν when ν tends to infinity [START_REF] Inoue | Asymptic behaviour for partial autocorrelation functions of fractionnal arima processes[END_REF]. Therefore, the finite-size correlation matrix

2 . To end up this section, it should be noted there are various approaches for the estimations of the ARFIMA parameters based on the maximum likelihood criterion. Among them, d can be estimated from different variants of the log periodogram regression estimator. The reader can refer to [START_REF] Geweke | The estimation and application of long-memory time series models[END_REF] and [START_REF] Cheung | Long memory in foreign-exchange rates[END_REF] for other existing estimation methods.

A.3.4 Inverse filter associated to ARFIMA processes

Given section A.3.3.3, the l th ARFIMA process can be seen as a BIBO-stable IIR filtering of a white noise, whose transfer function H l (z) is given in (A.50) and can be re-expressed as follows:

Let us now define the corresponding "stable" inverse filtering. It satisfies:

Depending on the position of the zero z i,l , the definition of H -1 i,l (z) differs to get a stable filter:

1. If the zero z i,l is inside the unit-circle in the z-plane, H -1 i,l (z) is defined by:

2. When |z i,l | > 1 and as done for ARMA processes, the following relationship must be taken into account:

where

1-z i,l z -1 is a Blaschke product [START_REF] Colwell | Blaschke products: bounded analytical functions[END_REF] that corresponds to the transfer function of an all-pass filter. In this case, H -1 i,l (z) is defined by:

Whatever the positions of the zeros, the resulting inverse-filtering transfer function H -1 l (z) corresponds to an IIR filter that is stable in the BIBO sense provided that

Given the statistical properties of the different processes we presented in this section, let us now compare them by using the JD. In the next section, we focus our attention on the JD between sums of complex exponentials disturbed by additive noise. Our purpose is to analyze how the JD evolves when the number of variates k increases.

A.4 Jeffreys divergence between sums of complex exponentials disturbed by additive noises

In this section, using the definitions (A.7), and (A.12), as well as the properties of the JD and of the processes under study, we propose to study how Tr(Q

Given (A.7) and (A.12), after some mathematical developments, the trace

) can be written as follows:

As mentioned in the above equation, the trace corresponds to the sum of four terms denoted as A (2,1) , B (2,1) , C (2,1) and D (2,1) . The order of the upperscripts in these notations corresponds to the order of the matrices in the trace to be computed. Let us start expressing the first one. Since Tr(F G) = Tr(GF ) where F and G are non square matrices but F G and GF are square matrices, one has:

Tr(P 1 ) =

Then, one has:

In (A.68), let us focus our attention on the matrix

. By applying the inversion matrix lemma, one has:

Using (A.68) and (A.69) leads to:

When k increases, due to the asymptotic properties (A.11) of "orthogonality", the sec-and

In this case, (A.81) becomes:

Given the above analysis, it is now possible to analyze how the JD between two NSCE processes evolves when k increases and tends to infinity. This is the goal of the next subsection.

A.4.2 Analytic expression of the Jeffreys divergence

As the expression of Tr(Q

) when k becomes high is similarly defined when the indices 1 and 2 are switched. Given (A.5) and (A.81), the JD can be approximated when k becomes large as follows:

The JD depends on the parameters of the processes, namely the normalized angular frequencies of each process, the variances of the magnitudes of each component as well as the variances of the additive noises. The first term of the expression includes the JD between two zero-mean white noises with variances σ 2 1 and σ 2 2 . Indeed, when there is no complex exponentials in both processes, the processes under study correspond to white noises and (A.85) reduces to:

Note that the two other terms in (A.85) make it possible to point out the differences between the sets of complex exponentials.

In the following section, let us analyze how the JD evolves when k is incremented.

A.4.3 Analysis of the increment of the Jeffreys divergence

Given the expression (A.85) of the JD we obtained, let us now deduce the asymptotic increment, i.e. the increment of the JD when k becomes large 8 :

Therefore, one can easily deduce that:

The ratio of the additive-white-noise variances has an influence on the slope of the JD. In addition, when there are common normalized angular frequencies between two processes, ∆JD is smaller. After this theoretical analysis, let us give some examples based on synthetic data and let us comment the results we obtain. Table A.1 Parameters of the NSCE processes under study for the 1 st simulation protocol

with a step equal to 0.02

As illustrated by Fig. A.3, the asymptotic increment is modified according to (A.88). It always depends on σ 1 2 σ 2 2 + σ 2 2 σ 1 2 and may depend on the set of ratios As presented in (A.88) in the theoretical analysis in section A.4.3, we can see that the asymptotic increment is here the same for all the simulations. The only difference stands in the convergence speed towards the stationary regime. Indeed, given Fig. A.5, the convergence of the JD derivative towards the asymptotic increment is faster when the difference between the normalized angular frequencies is large. The fluctuations of the derivatives around the asymptotic increment are mainly due to the products such as S H 1,k S 2,k and S H 2,k S 1,k that must be computed. They correspond to values of sin(

whose square is periodic with respect to k with period equal to Table A.4 Parameters of the NSCE processes under study for the 4 th simulation protocol

3 cases for θ 2,2 : , we present the evolution of the JD and the increment over time obtained from (A.5) respectively. We compare the increment (i.e. the derivative of the JD with respect to the number of variates k) with the asymptotic increment obtained in (A.88). It confirms our theoretical analysis. In practical cases, the covariance matrices can be estimated in the maximum likelihood sense from the data. Then, the JD is deduced for various sizes k of the covariance matrices. In order to make the interpretation easier, the derivative of the JD with respect to k must be computed and plotted to analyze the convergence speed towards the stationary regime. In various signal processing applications such as speech processing and mobile communication, the spectrum of the signal may exhibit some resonances. In this case, the signals can be modeled by an AR process. When comparing two processes using the JD, the less dissimilar the pdf are, the smaller the divergences are. As a consequence, we could a priori guess that the JD between a NSCE process, defined by the normalized angular frequencies of its complex exponentials, and an AR process which has sharper and sharper resonances in its spectrum at the same normalized angular frequencies becomes smaller and smaller. Nevertheless, in practice, the results that can be obtained do not necessarily confirm this statement. For this purpose, we propose to analyze the JD between an AR process and a NSCE process in the next section.

A.5 Jeffreys divergence between an autoregressive process and a sum of complex exponential process

In this section, using the definitions and properties of the JD and the processes under study, we first propose to express Tr(Q -1 N SCE,l,k Q AR,l ,k ) and Tr(Q -1 AR,l ,k Q N SCE,l,k ), with l and l defining the labels of the processes to be compared. We will see if one can deduce an analytic expression of these traces when the number of variates k increases and tends to infinity. As done in the previous section, this step will be useful to deduce the asymptotic properties of the JD between an AR process and a NSCE process.

A.5.1 Expression of the trace Tr

Given (A.7) and (A.17), after developing, one can show that Tr(Q -1 N SCE,l,k Q AR,l ,k ) can be expressed as the sum of three terms denoted as Ω, Ψ and Υ :

In (A.90) and (A.91), S H l,k Q AR,l ,k S l,k has to be computed. Given the definition of the matrix S l,k in (A.9), let us study what the value of S m l,k

r AR,l ,τ e -jτ θ l,m . . .

r AR,l ,τ e -j(τ +k-1)θ l,m 1 . . . e j(k-1)θ l,m T By rearranging the terms, (A.92) can be rewritten as follows:

Given (A.89) and (A.95), Tr(Q -1 N SCE,l,k Q AR,l ,k ) can be expressed as follows when k becomes high:

Given (A.7) and (A.27), Tr(Q -1 AR,l ,k Q N SCE,l,k ) can be expressed as the sum of two traces, denoted as ∆ and Θ:

where the inverses of L and D have been defined in (A.31) and (A.30). Using (A.31), this leads to:

with a l j 2 the 2-norm of the j th -order AR-parameter vector.

Remark: When considering a 1 st -order AR process, given (A.27), Q -1 AR,l ,k has the following simple expression:

where e k,l denotes the column vector of size k with zeros except at the l th row where the element is equal to 1. Therefore, Tr(σ

. This is also confirmed by (A.98).

Then, let us give some details about the second trace. Taking into account the expression of the PSD of the AR process for any order i = 0, ..., p evaluated at normalized angular frequencies θ l,m with m = 1, ..., M , one has:

It should be noted that in the above equation (A.100), we can organize the terms as follows:

Indeed, the pseudo-spectrum S capon that can be deduced using Capon's method [START_REF] Therrien | Discrete random signals and statistical signal processing[END_REF] can be expressed by:

. . e -jpθ l,m ] H Given the above analysis, let us deduce an expression of the JD when k increases in the next subsection. This step will be useful to deduce the asymptotic increment of the JD.

A.5.3 Analytic expression of the Jeffreys divergence

Combining (A.96), (A.98) and (A.100), the JD can be approximated when k becomes large as follows:

Influence of the additive-noise variance

The parameters of the processes under study are given in A.7

2 varies between 0. 2 tends to zero. This illustrates the main difference between the spectrum properties of both processes: one tends to be discrete whereas the other is continuous. In the next section, let us give a physical interpretation of the asymptotic JD increment for ergodic w.s.s. ARMA processes.

A.6 Interpreting the asymptotic increment of Jeffreys Divergence

between some random processes A.6.1 Inverse-filtering interpretation

In this section, we suggest giving an interpretation of the asymptotic JD increment in order to better understand the influence of each process parameter on the JD and to generalize the results obtained to ARMA(p,q) processes. For this reason, our purpose is to provide a new way to derive the asymptotic JD increment between zero-mean w.s.s. ARMA processes.

Let us rewrite the definition of the asymptotic JD increment as follows:

where the asymptotic increments of the traces ∆T (ARM A l ,ARM A l ) satisfy with (l, l ) equal to (1, 2) or (2, 1) :

In the above equation, the covariance matrices are of finite sizes k and k -1. Then, we study how the increment of the matrix traces evolves when k increases. If infinite-size correlation matrices are considered, the expressions of the KL (A.2) and of the JD (A.5) no longer hold when the corresponding transfer functions of the ARMA processes have unit roots since the covariance matrices are no longer invertible. As we will see later, it is not a hazard if these cases lead to specific behaviors of the JD increment in our analysis.

To evaluate ∆T (ARM A l ,ARM A l ) , we propose a particular way for interpreting the asymptotic JD increment. This basically avoids long mathematical developments. The covariance matrix Q ARM A,l,k can be decomposed by means of an eigenvalue decomposition:

where E l denotes the unitary matrix storing the eigenvectors of Q ARM A,l,k and D l is the diagonal matrix storing the non-null real positive eigenvalues.

Then, one has:

Our interpretation is the following: Let X l,k be the zero-mean random vector storing k consecutive samples of the l th zero-mean process ARMA process associated with covariance matrix

l amounts to whitening the process X l,k and making its power equal to unity. When k tends to infinity, it amounts to filtering all the samples stored in X l,k by an inverse filter whose transfer function is H -1 l (z). Similarly, pre-multiplying X l ,k by D -1/2 l P H l is equivalent, when k tends to infinity, to filter the data stored in the vector X l ,k by the inverse filter whose transfer function is equal to H -1 l (z). As a consequence, the asymptotic increment ∆T (ARM A l ,ARM A l ) corresponds to the power of process whose samples are stored in X l ,k and then filtered by H -1 l (z).

Based on the inverse filtering interpretation, the asymptotic JD increment amounts to computing the power P (ARM A l ,ARM A l ) of the l th random process filtered by the inverse filter corresponding to the l th process, with (l, l ) = (1, 2) or (2, 1):

Applications

In this section, we recall the results obtained in previous works [START_REF] Magnant | On computing Jeffrey's divergence between time-varying autoregressive models[END_REF], [START_REF] Legrand | Jeffrey's divergence between moving-average models that are real or complex, noise-free or disturbed by additive white noises[END_REF] and [START_REF] Legrand | Jeffrey's divergence between moving-average and autoregressive models[END_REF] and propose to retrieve the expressions of the asymptotic JD increments by using the inverse filtering interpretation in the following cases:

1. the JD between two white noises, 2. the JD between a 1 st -order AR process and a white noise, 3. the JD between two real 1 st -order MA processes 4. the JD between 1 st -order AR and MA processes.

Then, three new cases that were not addressed before can be now presented thanks to the results of section A.6.1:

1. the JD between q th -order MA processes, 2. the JD between NSCE processes, 3. the JD between an NSCE process and a p th -order AR process.

In these latter cases, we propose to illustrate the influences of the parameters of the processes on the asymptotic JD increment and consequently on the JD.

A.6.2.1 JD between two white noises

Even if this is a trivial case, by starting from the definition (A.5), our proposed interpretation in section A.6.1 makes it possible to deduce the asymptotic increment of the JD between zero-mean Gaussian white noises (WN) with variances σ 2 u,1 and σ 2 u,2 . It can be viewed as a toy example:

Indeed, the term

can be explained as follows: the first white noise with variance σ u,1 is filtered by an all-pass filter with a transfer function equal to 1 σ u,2 . The power of the filter output is hence equal to

can be explained similarly. By introducing

the above equation (A.112) becomes:

It should be noted that (A.114) can be rewritten by expressing σ 2 u,2 as σ 2 u,1 + δσ 2 u and by introducing the relative difference between the noise-variances ∆σ

. In this case, one has:

and A.6.2.2 JD between a 1 st -order AR process and a white noise By taking advantage of the inverse of the correlation matrix of a 1 st -order AR process [START_REF] Cernuschi-Frias | A derivation of the gohberg-semencul relation [signal analysis[END_REF], the asymptotic JD increment can be expressed as follows:

By using the inverse filtering interpretation presented in the section A.6.1, the term ∆T (W N,AR) which is equal to

(1-(a 1,1 ) 2 ) corresponds to the power of the 1 st -order AR process filtered by an all-pass filter with a transfer function equal to 1 σ u,2,1 . Meanwhile, the second term ∆T (AR,W N ) , which is equal to

(1 + (a 1,1 ) 2 ), corresponds to the power of the zero-mean white noise filtered by the inverse AR filter whose transfer function is equal to

). Once again, (A.117) can be rewritten as follows:

In Fig. A.13, the JD between an AR process and a white noise is presented as a function of the AR parameter a 1,1 and the relative difference between the noise variances ∆σ 2 u . This latter varies in the interval (-1, 20] with a small step equal to 0.03. Only positive values of the AR parameter a 1,1 are considered because the JD is an even function with respect to the AR parameter in this case. The AR parameter varies in the interval [0, 1) with a step equal to 0.01. Thus, this illustration makes it possible to present a large set of situations that could happen. It should be noted that when a 1,1 = 0, we retrieve the results given in (A.116). When the white-noise variance and the driving process variance are equal (i.e. when the relative difference between the noise variance is equal to zero) and if the AR parameter is equal to zero, the JD is of course null. Then, for any value of the relative difference between the noise variance, the JD between the AR process and the white noise is all the higher as the PSD of the AR process is spiky, i.e. the modulus of the pole tends to 1. This phenomenon is clearly noticeable when the variances of the white noise and of the AR driving process are the same, i.e. the relative difference between the noise variance is equal to zero. In this case, according to Fig. A.13, the JD can be higher than 120 when the AR parameters tend to 1. If the step equal to 0.01 in this simulation had been chosen smaller, we could have seen that the JD tends to infinity when the AR parameter tends to 1. When the variance of the white noise increases, the phenomenon is the same but the sensitivity of the JD with respect to the AR parameter is less pronounced. This is confirmed by calculating the derivative of ∆JD k (AR,W N ) with respect to the AR parameter.

One can also calculate the derivative of ∆JD k (AR,W N ) with respect to the relative difference between the noise variances ∆σ 2 u . In this case, one can notice that the minimum value of the asymptotic JD increment is obtained when ∆σ In [START_REF] Magnant | On computing Jeffrey's divergence between time-varying autoregressive models[END_REF], a recursive way was suggested to deduce the JD between two 1 st -order AR processes:

However, by reorganizing the terms in the above equation, one has:

If |b 1,2 | < 1, it corresponds to the power of the 1 st -order MA process filtered by the inverse filter associated to the second 1 st -order MA process. In this case, the transfer function is defined by

) . Then, one just has to compute the autocorrelation function of the filtered process for the lag equal to 0. This leads to:

By replacing in (A.126) r M A,1,τ with τ = 0, |1| by their expressions recalled in (A.21), we retrieve

. Similar reasoning can be done for |b 1,2 | > 1. When the zero associated to the 1 st -order MA process is on the unit circle in the z-plane, the inverse filter has its pole on the unit circle also. Therefore, the output of the inverse filter tends to infinity, except when the MA processes share the same zeros. This explains the atypical results identified for |b

The same reasoning is applied for ∆T (M A 1 ,M A 2 ) . A.6.2.5 JD between AR and MA processes In [START_REF] Legrand | Jeffrey's divergence between moving-average and autoregressive models[END_REF], the JD between a 1 st -order MA process and a 1 st -order AR process with the variances of driving processes equal to σ 2 u,1,1 and σ 2 u,2,1 was analyzed. The increment satisfied:

In (A.127), it can be shown that for k > 2:

whereas one has:

When |b 1,2 | = 1, one has:

Note that in (A.128), ∆T

(AR,M A) k corresponds to the autocorrelation function of the MA process filtered by the inverse AR filter, for a lag equal to 0. In this case, the transfer function is defined by (1+a 1,1 z -1 )

. Then, one just has to compute the autocorrelation function of the filtered process for the lag equal to 0. This leads to:

By replacing in (A.132), r M A,1,τ with τ = 0, |1| by their expressions recalled in (A.21) in the subsection A.3.2, we retrieve

Similarly in (A.129), lim k→+∞ ∆T (M A,AR) k corresponds to the autocorrelation function of the AR process filtered by the inverse MA filter, for a lag equal to 0. Like the previous case, the proof consists in computing the autocorrelation function of the filtered process for the lag equal to 0. Once again, our interpretation makes it possible to explain the different behaviors of the JD regarding the value of the MA parameter. Finally, the asymptotic JD increment clearly points out the fact that the 1 st -order MA process which has a zero with a unit modulus or equivalently whose PSD is null at a specific frequency and a 1 st -order AR process whose PSD can exhibit a resonance are totally dissimilar with respect to the JD increment.

A.6.2.6 JD between q th -order MA processes Let us now analyze the asymptotic increment of the JD between two q th -order MA processes. It is not necessarily possible to generalize to q th -order MA processes the approach we followed for 1 st -order MA process in [START_REF] Legrand | Jeffrey's divergence between moving-average models that are real or complex, noise-free or disturbed by additive white noises[END_REF]. This would require the expressions of the inverses of matrices with 2q + 1 main and sub-diagonals. It is exactly the same problem for ARMA processes. For all these processes, obtaining an analytical expression of the JD, its increment, and its asymptotic increment when it is finite is not straightforward. Thanks to the interpretation we propose, it is now easier to deduce the asymptotic JD increment. For this purpose we propose to analyze the two following cases:

Firstly, q = 5. The zeros of the MA processes are set at values recalled in Table A In this example, as the zeros are preliminarily defined, the MA parameters can be easily deduced, the autocorrelation function of both MA processes can be obtained by using (A.22) and the Toeplitz covariance matrices can also be built. As a consequence, we can compute the theoretical JD derivative for different values of k, as well as the asymptotic JD increment. They are given in Using the interpretation we have presented, it is now easy to deduce the asymptotic JD increment. We have generated a set of data (10000 samples) for both MA processes. Then each set of data is filtered by the inverse filter associated with the other process using the true parameters defined at the beginning of this section. The powers of the filter outputs are computed and the asymptotic JD increment is deduced. Finally, the average value of the asymptotic JD increment is computed over 200 realizations. It is compared with the theoretical one obtained in Fig. A.15. The percentage of the normalized error 11 E is 0.25%. Secondly, to address more general cases, Monte Carlo simulations were carried out. 1000 situations were simulated where the zeros and the variance of the driving process of each real 5 th -order MA process were uniformly randomly drawn. Note that the zeros were necessarily complex conjugates or real. At the end of the simulations, the normalized error is averaged over the whole cases. The percentage of the average normalized error obtained is 2.45%. It confirms the interpretation we have proposed. All this work can be easily generalized to any AR/MA or ARMA process of any order.

ARMA processes are short-memory processes. However, in some applications, longmemory processes such as ARFIMA processes are considered. In the following, the JD between ARFIMA processes is analyzed.

2 ) and their driving process variances {σ 2 u,l } l=1,2 . As seen in section A.3.3.1, the l th F I white noise can be seen as a BIBO-stable IIR filtering of a white noise, with l = 1, 2. The corresponding inverse filters are defined by the following transfer functions:

where ∆JD represents the asymptotic JD increment that is computed by using the interpretation while ∆JD represents the theoretical asymptotic JD increment.

3. For a given ρ, by deriving (A.111) with respect to ∆d and by taking advantage of the properties of the digamma function, defined as the logarithmic derivative of the gamma function, it can also be shown that ∆JD (F I 1 ,F I 2 ) decreases when ∆d ∈ (-1 2 , 0] and increases when ∆d ∈ [0, 1 2 ). Its minimum is attained when ∆d = 0.

4. When ρ is equal to 1, ∆JD (F I 1 ,F I 2 ) is also a symmetric function with respect to ∆d. To end up this part, let us illustrate the comparison between two stationary ergodic F I white noise p rocesses when ∆d > 1 2 . In this case, the asymptotic JD increment should go to infinity since the power of a F I(∆d) process is infinite when ∆d > 1 2 . In

JD between unit-power F I white noises

As suggested in [START_REF] Grivel | Process comparison combining signal power ratio and Jeffrey's divergence between unit-power signals[END_REF], we now propose to compare two unit-power F I white noises, denoted as {y n,l } with l = (1, 2), in order to study the influences of d 1 and d 2 on the value of the asymptotic JD increment ∆JD (F I 1 ,F I 2 ) . As seen in Fig. A.21a which depicts the normalization to be done, both unit-power F I white noises {y n,l } l=1,2 no longer depend on {σ 2 u,l } l=1,2 . Then, following the processing presented in Fig. A.21b, the power P m,n of the inverse-filter output t

Given (A.111) and (A.136), the asymptotic JD increment ∆JD (F I 1 ,F I 2 ) between two unit-power F I white noises is a function of d 1 and d 2 only. In addition, one has:

When d 1 and d 2 both tend to 0, 0.5 or -0.5, P (F I 1 ,F I 2 ) and P (F I 2 ,F I 1 ) tend to 1 according to (A.136). Therefore, ∆JD (F I 1 ,F I 2 ) tends to 0. When d 1 is positive (resp. negative) and tends to 0, P (F I 1 ,F I It is given by:

u,l × (A.140) q l +p l i=-(q l +p l ) p l +q l s=1 φ l,l (i) l,l s C(d l -d l , p l + q l + i, p s )

where {p s } s=1,...p l +q l are the poles of the ARF IM A(p l + q l , d l -d l , q l + p l ) process, φ m,n (i) corresponds to the value of the autocorrelation function for lag l of the MA part defined from the zeros of the ARF IM A(p l + q l , d l -d l , q l + p l ) process, and { l,l s } s=1,...p l +q l is similarly defined as in (A.55) but adjusted to the poles {p s } s=1,...p l +q l . Then the asymptotic JD increment can be deduced by using (A.111).

2. When (d 1 , d 2 ) ∈ (0, 0.5), the processes to be compared are long-memory ARFIMA processes. The inverse-filter outputs are stationary ARFIMA processes and the same conclusions as in the previous case can be drawn.

3. When d 1 and d 2 have opposite signs, |d 1 -d 2 | can be smaller or larger than 1 2 . If the difference is smaller than 1 2 , the same comments as in the two previous cases can be drawn. If it is larger than 1 2 , one of the filter outputs should correspond to an ARFIMA process with an differencing order larger than 1 2 . It is a nonstationary process with an infinite variance.

A.7.2.2 Illustrations and comments

Let us now study the evolutions of the JD and the JD increment with respect to the variate number between two real ARF IM A(1, d l , 1) processes. For this purpose, three types of simulations were done.

1. In the first one, two short-memory ARF IM A(1, d l , 1) processes are compared.

2. In the second one, the case of two long-memory ARF IM A(1, d l , 1) processes is addressed.

3. As for the last illustration, we compare a short-memory ARF IM A(1, d 1 , 1) with a long-memory ARF IM A(1, d 2 , 1) process.

Let us study the influence of the type of the ARF IM A(1, d l , 1) processes on the convergence of the JD derivative towards the asymptotic JD increment. The parameters of the processes in the three cases are given in Table . A.12. In each illustration, d To end up this part, we will compare two stationary ergodic ARFIMA processes when ∆d > 1 2 as we did for the F I white noise processes. In

A.7.3 JD between ARFIMA and ARMA processes

In this last section, we propose to study the evolution of the JD and the JD increment versus the number of variates between a real ARF IM A(1, d l , 1) process and a real ARM A(1, 1) process. For this purpose, two types of simulations were done. In the first one, a real ARM A(1, 1) process is compared with a short-memory ARF IM A(1, d l , 1) process. As for the second one, a real ARM A process is compared with a long-memory ARF IM A(1, d l , 1) process. The parameters of processes under study are given in