Member, IEEE Rafik Zayani

Jean-Baptiste Dore

Member, IEEE Benoit Miscopein

Member, IEEE David Demmer

Member, IEEE Jean-Baptiste Doré

Local PAPR-aware precoding for OFDM based cell-free massive MIMO-OFDM systems

Keywords: Cell-free massive MIMO, Precoding schemes, OFDM, Hardware impairment, Power amplifier, PAPR reduction, Spectral-efficiency, Energy-efficiency

ouverte pluridisciplinaire HAL, est destinée au dépôt

I. INTRODUCTION

During the last decades, the exponential growth in mobile data traffic has been enabled by the densification of the network infrastructure, which can be ensured by (i) increasing the cell density known as ultra-dense network (UDN) [START_REF] Hwang | A holistic view on hyper-dense heterogeneous and small cell networks[END_REF][2] and/or (ii) the adoption of a large number of active antennas per access point (AP), referred to as massive MIMO (mMIMO) [START_REF] Marzetta | Noncooperative cellular wireless with unlimited numbers of base station antennas[END_REF] [START_REF] Björnson | Massive MIMO Networks: Spectral, Energy, and Hardware Efficiency[END_REF]. Nevertheless, both cell densification and mMIMO have fundamental limitations, R. Zayani is with ... ... are with Anonymous University. [START_REF] Price | A useful theorem for nonlinear devices having gaussian inputs[END_REF] caused by the inter-cell interference and large quality-of-service variations, making them unable to cope with the challenges of the next/sixth generation (6G) networks in terms of increasing the user-experienced data rates.

Moreover, 6G is expected to drive the future of the Industry 4.0, where private/industrial networks, applications with rapid deployment of wireless access infrastructure and low power consumption are the prominent subset.

Towards this end, a revolutionary wireless technology has recently retained substantial attention in literature [START_REF] Ngo | Cell-free massive mimo versus small cells[END_REF] that exploits the best aspects of ultra-dense cellular networks and the cellular mMIMO technology to overcome their aforementioned drawbacks, known as cell-free massive MIMO (CF-mMIMO). The main propriety of the CF-mMIMO technology is that there are many geographically distributed APs, but the coverage area is not divided into disjoint cells. Note that early embodiment include the popular concept of "network MIMO" [START_REF] Venkatesan | Network mimo: Overcoming intercell interference in indoor wireless systems[END_REF], "distributed antenna systems (DAS)" [START_REF] You | Cooperative distributed antenna systems for mobile communications [coordinated and distributed mimo[END_REF] and "distributed wireless communication systems" [START_REF] Zhou | Distributed wireless communication system: a new architecture for future public wireless access[END_REF], known to the third generation partnership project (3GPP) as "coordinated multi-point (CoMP)" [START_REF] Irmer | Coordinated multipoint: Concepts, performance, and field trial results[END_REF]. We can distinguish CF-mMIMO from the other early technologies by two main factors which are (i) the operating regime with more APs than UEs together with the PHY operation inspired by the recent advancement in the mMIMO field and (ii) every UE is served by all the surrounding APs, stressing the cell-free aspect.

The original version of the CF-mMIMO technology was unscalable, i.e., the front-haul capacity and the computational complexity grow exponentially with the number of UEs. This is caused by the fact that all APs are connected to a central processing unit (CPU) which is responsible to coordinate and process the signals of all UEs. Very recently, a new scalable version of the CF-mMIMO was introduced, in [START_REF] Björnson | Scalable cell-free massive mimo systems[END_REF], where fully distributed processing is adopted. In [START_REF] Interdonato | Local partial zero-forcing precoding for cell-free massive mimo[END_REF] and [START_REF] Zhang | Local partial zero-forcing combining for cell-free massive mimo systems[END_REF], authors introduced new scalable precoding and combining schemes together with channel estimation, power allocation, AP clustering methods, achieving very good benefits of CF-mMIMO.

On the other hand, broadband wireless channels encounter, however, large delay spread, causing frequencyselective fading in sub-6GHz bands. Encoding digital symbols on multiple orthogonal subcarriers is the best way to deal with frequency-selective channels. Here we talk about the adoption of the well-known orthogonal frequencydivision multiplexing (OFDM) scheme. Therefore, in this investigation, we combine OFDM and the scalable version of CF-mMIMO, representing a very promising combination to satisfy the ever growing demands for higher data traffic and link readability.

However, CF-mMIMO-OFDM systems carry high peak-to-average power ratio (PAPR) to transmit signals.

Moreover, CF-mMIMO-OFDM can be commercially viable, only when APs are deployed using power-efficient and inexpensive hardware. Consequently, high PAPR and energy-efficient hardware results in severe hardware impairments (HWIs), which is dominated by the non-linearity of the radio frequency (RF) power amplifiers (PAs).

Then, major degradation in the transmission quality is occurred and CF-mMIMO-OFDM loses rapidly all its potential benefits. Therefore, it is of paramount importance to reduce the PAPR of OFDM based CF-mMIMO systems to enable cost-and energy-efficient AP deployments.

A. Related works

The PAPR reduction problem has been now studied for a long time, the first method proposed for single-input single-output (SISO)-OFDM was in 1999 [START_REF] Wulich | Reduction of peak factor in orthogonal multicarrier modulation by amplitude limiting and coding[END_REF]. Then, some enhanced methods have been introduced to improve the [START_REF] Price | A useful theorem for nonlinear devices having gaussian inputs[END_REF] power-efficiency, such as tone reservation (TR) [START_REF] Li | An improved tone reservation scheme with fast convergence for papr reduction in ofdm systems[END_REF], selective mapping (SLM) [START_REF] Chaitanya Bulusu | Reduction of papr for fbmc-oqam systems using dispersive slm technique[END_REF], partial transmit sequence (PTS) [START_REF] Chen | Partial transmit sequences for peak-to-average power ratio reduction of ofdm signals with the cross-entropy method[END_REF], active constellation extension (ACE) [START_REF] Wang | Papr reduction in ofdm systems using active constellation extension and subcarrier grouping techniques[END_REF], coding [START_REF] Shu | Invertible subset qc-ldpc codes for papr reduction of ofdm signals[END_REF] and iterative clipping and filtering [START_REF] Anoh | On the optimization of iterative clipping and filtering for papr reduction in ofdm systems[END_REF]. These methods, which have been proposed for SISO-OFDM and classical MIMO-OFDM, can, unfortunately, provide moderate PAPR reduction that does not satisfy the requirements in highly power-efficiency for massive MIMO based 6G networks.

Moreover, the bottlenecks of these methods is related to their respective drawbacks like increase of the average power (i.e., signal-to-noise ratio gap), loss in SE due to the reservation of some subcarriers, high computational complexity and high latency. That is why they do not adequate with OFDM based massive MIMO systems.

In this regard, more interesting techniques have been proposed for co-located massive MIMO-OFDM systems, like fast truncation (FITRA) algorithm [START_REF] Studer | Par-aware large-scale multi-user mimo-ofdm downlink[END_REF], approximate passing (AMP)-based Bayesian method (EM-TGM-GAMP) [START_REF] Bao | An efficient bayesian papr reduction method for ofdm-based massive mimo systems[END_REF] and perturbation assisted based alternative direction method of multipliers (ADMM) technique [START_REF] Bao | An admm approach for papr reduction for large-scale mimo-ofdm systems[END_REF]. More recently, joint MU precoding and PAPR reduction schemes have been studied in [START_REF] Zayani | PAPR-Aware Massive MIMO-OFDM Downlink[END_REF] [START_REF] Zayani | Low-complexity linear precoding for low-papr massive mu-mimo-ofdm downlink systems[END_REF], referred to as MU-PP-GDm, RZF-OPNS, POLY-POLY-Horner and POLY-OPNS. They consist in gradient-iterative method-based linear precoding. Unfortunately, all of these techniques were limited to traditional/co-located massive MIMO systems. Indeed, in this latter, a high-dimensional degrees of freedom (DoFs) are offered by equipping the base station (BS) with a large number of antennas compared to the number of served users. Interestingly, these DoFs can be exploited to effectively reduce the PAPR. In centralized CF-mMIMO, i.e., adopting high capacity front-haul connections to transfer data between the APs and the CPU, it would be possible to re-use the aforementioned techniques already proposed for co-localised mMIMO. However, their provided gain in terms of energy-efficiency will be in the cost of heavy front-hauling load, leading to reduced global system energy-efficiency. In this regard, this paper aims to propose new PAPR reduction method that is performed locally at each AP, with no instantaneous information sharing (e.g. CSI), leading to a reduced front-hauling overhead. Here, we use local PAPR reduction technique to stress the local nature of the proposed strategy, which is primordial to ensure the CF-mMIMO-OFDM system scalability.

B. Motivation

The development of fully distributed and scalable precoding [START_REF] Interdonato | Local partial zero-forcing precoding for cell-free massive mimo[END_REF] and combining [START_REF] Interdonato | Downlink spectral efficiency of cell-free massive mimo with full-pilot zero-forcing[END_REF] schemes have been carried out for, respectively, DL and UL CF-mMIMO, representing extremely interesting SE performance. In this paper, we introduce new PAPR-aware precoding scheme adapted to CF-mMIMO-OFDM. Interestingly, the proposed method can be implemented in a fully distributed and scalable manner and hence do not need any instantaneous CSI exchange via the front-haul links. Moreover, it provides a good balance between PAPR reduction and maintaining good transmission quality. Indeed, the motivation behind this work is related to the following observations: 1) CF-mMIMO is a fresh research topic: a number of issues need to be tackled before rolling out it into practice; one of the most challenging issue is the HWI mitigation. 2) CF-mMIMO can be economically attractive only if its implementation is based on low-cost hardware, which are more subject to HWIs. To the best of our knowledge, there is no prior work investigating the PAPR reduction in emerging OFDM based CF-mMIMO systems. 3) Local PAPR reduction together with the latest advancements of local precoding schemes [START_REF] Interdonato | Local partial zero-forcing precoding for cell-free massive mimo[END_REF] can offer radical performance improvements, leading to practical [START_REF] Price | A useful theorem for nonlinear devices having gaussian inputs[END_REF] and energy-efficient CF-mMIMO-OFDM. 4) There is a high demand of green signal processing solutions due to concerns over making the telecommunication sector sustainable and, correspondingly, decreasing carbon dioxide (CO2) emissions, resulting in power saving and reduced environmental pollution.

C. Contributions

The main technical contributions of this paper are as follows:

• We analyze the DL SE of OFDM based CF-mMIMO running under frequency-selective channels and nonlinear power amplifiers. Contrary to works [START_REF] Jiang | Cell-free massive mimo-ofdm transmission over frequency-selective fading channels[END_REF][27] that only apply conjugate beamforming/matched filter, we consider local precoding schemes like local full-pilot zero-forcing (FZF) and local regularized zero-forcing (RZF) studied in [START_REF] Interdonato | Local partial zero-forcing precoding for cell-free massive mimo[END_REF].

• We derive analytical expressions for an achievable DL SE in CF-mmIMO-OFDM under PA nonlinearity and the assumption of independent Rayleigh fading, contrary to the work in [START_REF] Papazafeiropoulos | Scalable cell-free massive mimo systems with hardware impairments[END_REF] that focuses only on UL CF-mMIMO.

Unlike to [START_REF] Mokhtari | Sum-rate of cell free massive mimo systems with power amplifier non-linearity[END_REF], these expressions take into consideration channel estimation errors and pilot contamination.

• We develop a new algorithm, referred to as localPAPRfree to efficiently reduce the PAPR of the transmit OFDM signals in scalable CF-mMIMO-OFDM. To the best of our knowledge, it is the first solution proposed, in the open literature, that can be implemented in a fully distributed and scalable manner. Moreover, we analyze the DL SE of the considered system taking into account the PAPR reduction. This analysis explicitly reveals how PA impairments at the APs affect the SE and provide important insights into the practical deployment of CF-mMIMO systems while considering real PA models.

• We investigate the energy efficiency of our considered system with and without PAPR reduction. We consider the front-haul power consumption and the consumption of real power amplifiers operated with different input power levels.

The remainder of this paper is organized as follows. In Section II, we present the OFDM based CF-mMIMO system model. We derive the DL SE and EE in Section III. The proposed algorithm is discussed in Section IV. In Section V, numerical results and discussions are provided. Finally, the conclusion and some perspectives are given in Section VI.

D. Notations

The notations used in this paper are listed as follows:

• Lowercase boldface letters (e.g. x) stand for column vectors,

• Bold lowercase letters with a superscript (.) t (e.g. x t ) denotes row vectors,

• Bold uppercase letters (e.g. X) denotes matrices,

• The˘marks time-domain variables over the paper,

• We denote by matrix transpose, matrix conjugate transpose, matrix pseudo-inverse and trace of a matrix by X T ,X H , X † and tr(X), respectively,

• For a M × N -dimensional matrix X = {x mn }, June 20, 2022

• We use x n to designate the n-th column, and x t m to designate the m-th row,

• The N × N identity matrix and the M × N all-zeros matrix are denoted by I N and 0 M ×N , respectively,

• We use ∥x∥ 2 and ∥x∥ ∞ to denote l 2 -norm and l ∞ -norm of vector x, respectively,

• X = diag{x 1 , ..., x K } denotes a diagonal matrix with elements {x i },

• the cardinality and complement of set Ξ is |Ξ| and Ξ c , respectively,

• E [.] stands for the expectation operator and j denotes √ -1.

II. OFDM BASED CELL-FREE MASSIVE MIMO

A. System Model

We study an OFDM based cell-free massive MIMO downlink system operating in time division duplex (TDD) mode. Let there be L APs, each equipped with M antennas, serving coherently K single-antenna users, where LM ≫ K. We assume that all the users and the APs are randomly located in a geographic area, as shown by Fig.

1.

Here, an OFDM modulator is implemented at each antenna branch and all the APs are connected through an unlimited front-haul network to central processing unit (CPU). Besides, broadband communications are considered, suffering from severe frequency selectivity. Then, the small-scale fading is considered to be frequency-selective and modeled as a tap delay line with T taps. Then, the time-domain channel response between the l-th AP and the k-th UE, Hl,k ∈ C M ×T , is given by

Hl,k = hl,k,0 , ..., hl,k,T -1 (1)
and hl,k,t can be written as β l,k gl,k,t , where β l,k denotes large-scale fading (i.e., variance of the channel), which is frequency and antenna independent and gl,k,t ∈ C M ×1 , t = 0, ..., T -1, have i.i.d. circularly symmetric Gaussian distributed entries with zero mean and unit variance.

We denote by h l,k,n ∈ C M ×1 its equivalent frequency-domain channel response on the n-th subcarrier, where n = 0, ..., N -1 and N denotes the total number of OFDM subcarriers.

Note that we consider a block-fading channel, which is constant during a time-frequency interval, known as the coherence interval, and varies independently between coherence intervals. Besides, we assume that the large-scale fading coefficients vary slowly, so it is constant over many coherence intervals. Then, they are assumed to be known at each AP and they are used afterwards to estimate the frequency-domain channel responses.

B. Communication Process and Impairment Model

The TDD mode separates the downlink (DL) and uplink (UL) transmissions (see Fig. 2) with the assumption of perfect channel reciprocity which can be assured by accurate calibration methods [START_REF] Vieira | Reciprocity calibration for massive mimo: Proposal, modeling, and validation[END_REF]. Moreover, the transmission of a TDD frame in OFDM based CF-mMIMO is performed within the coherence interval and the physical resource block (RB) width is smaller than the coherence bandwidth.

To be compliant with the 5G NR standard, let us consider a radio frame whose time-frequency resource is divided into N rb resource blocks. Each RB comprises N sc = N/N rb consecutive subcarriers. We note by (t, n) l,m the resource unit (RU), which represents the smallest time-frequency resource of the n-th subcarrier of the t-th OFDM symbol corresponding to the m-th antenna of the l-th AP.

June 20, 2022 Here, the TDD frame contains N c OFDM symbols that fits the shortest coherence interval of all the users. It corresponds to the transmission of N sc N c RUs per RB, where τ p of them are used as pilots which are distributed among the UL payload transmission, as shown by Fig. 2. These pilot data will serve to estimate M K frequencydomain channels, per RB, at each AP. Consequently, we leave N D = N sc N c -τ p for payload data, in samples per RB, that will be split between DL and UL transmissions as ξN D and (1 -ξ)N D , respectively, where 0 < ξ < 1.

1) Uplink Training and Channel Estimation:

In this investigation, we adopt a block fading channel and we assume that the frequency-domain channel coefficients for all RUs within one RB is identical and none within different RBs. Consequently, a per-RB channel estimation has to be performed.

Therefore, we consider τ p mutually orthogonal τ p -length frequency-domain sequences transmitted by using the first time-domain OFDM symbols (see Fig. 2), i.e., all the UEs synchronously transmit their pilot sequences to the APs, once per coherence interval. The case of practical interest is a dense network with K > τ p so that some UEs share the same pilot, per-RB, leading to pilot contamination [START_REF] Zeng | Pilot assignment for cell free massive mimo systems using a weighted graphic framework[END_REF], where APs are not able to spatially June 20, 2022 separate linearly dependent channels. For the sake of simplicity and without loss of generality, we assume that all the considered RBs are allocated to all served UEs.

Let θ i k ∈ C τp×1 is the frequency-domain pilot sequence sent by the k-th UE, where i k ∈ {1, ..., τ p } denotes the index of the pilot used by the k-th UE. We note by Θ = θ 1 , ..., θ τp ∈ C τp×τp the pilot book matrix. Then, the UE k sends η u k θ H i k , where η u k is the UL normalized transmit power. Let P k ⊂ {1, ..., K} be the subset of UEs that use the same pilot as the k-th UE, including itself. As you can see, i k and P k do not depend on the RB index so that the pilot assignment is identical for all RBs, thanks to the orthogonality ensured by OFDM. Note that the pilot sequences are normalized such that θ H i k θ i k = τ p and 0 otherwise. The frequency-domain pilot signal received at the l-th AP on the n′-th RB is given by

Y n′,l = K k=1 h n′,l,k η u k θ H i k + B l ∈ C M ×τp , (2)
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where B l ∈ C M ×τp is a Gaussian noise matrix whose elements are i.i.d. CN (0, 1).

In order to be compliant with the recent literature, we adopt, in this work, the minimum mean square error (MMSE) channel estimation method and the pilot-to-precoder mapping strategy inspired from the work in [START_REF] Interdonato | Local partial zero-forcing precoding for cell-free massive mimo[END_REF].

Hence, the frequency-domain channel estimate between the l-th AP and the k-th UE, corresponding to the n′-th RB can be given by

ĥn′,l,k = c l,k Hn′,l e i k , (3) where
• c l,k is a frequency independent scalar, which is defined as [START_REF] Kay | Fundamentals of Statistical Signal Processing: Estimation Theory[END_REF]

c l,k ≜ η u k β l,k τ p t∈P k η u t β l,t + 1 (4)
• Hn′,l ∈ C M ×τp denotes the corresponding full-rank matrix of the frequency-domain channel estimates and is computed as

Hn′,l = Y n′,l Θ (5)
• e i k is the i k -th column of I τp , i.e., taking the i k -th column of c l,k Hn′,l .

Note that the channel estimates and estimation error, corresponding to the n′-th RB, which are denoted by ĥn′,l,k and hn′,l,k = h n,l,k -ĥn′,l,k , are independent and distributed as ĥn′,l,k

∼ CN (0, γ l,k I M ), hn′,l,k ∼ CN (0, (β l,k -γ l,k ) I M ), where γ l,k ≜ η u k τ p β 2 l,k τ p t∈P k η u t β l,t + 1 (6)
Note that γ l,k , which is the mean-square of the channel estimate between the AP l and the UE k, is antenna and frequency independent.

Remark 1 (Per-RB to per-subcarrier channel mapping) The channels at AP l are estimated per-RB and are assumed to be identical for RUs within one RB. Then, each AP can effectively construct τ p channel coefficients per-subcarrier n as

ĥn,l,k = ĥn′,l,k for n ∈ RB n ′, (7)
we can also write

Hn,l = Hn′,l for n ∈ RB n ′. (8)
2) Basic OFDM based Downlink Data Transmission: The information data to be transmitted to the K users on the n-th subcarrier, which is denoted by s n ∈ C K×1 , has independent elements with unit power, i.e., E ∥s n ∥ 2 = 1.

Typically, OFDM systems specify certain unused subcarriers, as guard-band located at both ends of the spectrum.

Hence, we divide the set of available subcarriers into two subsets: (i) Ξ, used for data transmission and (ii) its complementary Ξ c , used for guard-band. Then, s n = 0 K×1 for n ∈ Ξ c , i.e., no data is transmitted on the guard-band.

Precoding is needed to be performed at each AP in order to remove the multi-user interference (MUI) at receivers.

In CF-mMIMO, the local nature of the implemented precoders is crucial to preserve system scalability. To stress this aspect, we use the terminology local precoding.

June 20, 2022

The signal vectors s n , ∀n, can be linearly precoded, at the l-th AP, as

x l,n = W l,n P l s n , (9)
where:

• x l,n ∈ C M ×1 denotes the precoded vector that contains samples to be transmitted over the the n-th subcarrier through the M antennas of AP l,

• W l,n ∈ C M ×K is the precoding matrix corresponding to the n-th subcarrier at the l-th AP,

• P l ∈ C K×K , which is frequency independent, represents a diagonal matrix whose elements √ η l,k , k = 1, ..., K are the normalized transmit powers allocated to the K UEs.

To be compliant with the 5G standard, we consider max-min fairness power control that consists in maximizing the lowest user's downlink SE. Then, the normalized transmit power (normalized by the noise power σ 2 b ) at each AP l, denoted by ∥x l,n ∥ 2 = K k=1 η l,k , ∀n has to be constrained as

K k=1 η l,k ≤ η max l
, where

η max l = 1 σ 2 b M (V sat /G) 2 IBO , (10)
where the IBO denotes the input back-off which represents the PA input power level relative to input saturation power, V sat is PA amplitude saturation level and G is the PA linear gain.

The power control optimization problem can be formulated as follows [START_REF] Interdonato | Local partial zero-forcing precoding for cell-free massive mimo[END_REF][25]

maximize {η l,k ≥0} min k SINR k,n (11)
s.t. K k=1 η l,k ≤ η max l , ∀l (12)
where SINR k,n denotes the signal-to-interference plus noise ratio (SINR) for user k at subcarrier n.

For the sake of self-containment, we consider, in this paper, the distributed power control policy where the power control coefficients are given in [START_REF] Interdonato | Local partial zero-forcing precoding for cell-free massive mimo[END_REF] as

η l,k = γ l,k K i=1 γ l,i η max l , ∀l, ∀k (13)
Inspired by [4][11][25] that study local precoding schemes in the downlink, we will consider the following linear precoders in frequency-domain of our OFDM based CF-mMIMO: (i) local full-pilot zero-forcing (FZF) and (ii)

local regularized zero-forcing (RZF) . Here, the precoding vector employed by the l-th AP towards the k-th UE on subcarrier n, denoted by

w l,n,i k ∈ C M ×1 , is w l,n,i k =            Hl,n( HH l,n Hl,n) -1 ei k E ∥ Hl,n( HH l,n Hl,n) -1 ei k ∥ 2 = (M -τ p )θ l,k Hl,n HH l,n Hl,n -1 e i k , for local FZF Ĥl,n( ĤH l,n Ĥl,n) -1 êk E ∥ Ĥl,n( ĤH l,n Ĥl,n +P -1 l ) -1 êk ∥ 2 , for local RZF (14)
where:

• θ l,k , which is the mean square of Hl,n e i k , can be defined as θ l,k = γ l,k /c 2 l,k [START_REF] Interdonato | Local partial zero-forcing precoding for cell-free massive mimo[END_REF],

June 20, 2022

• Ĥl,n ∈ C M ×K is the matrix of the channel estimates between the AP l and the K users on the subcarrier n, which can be collected as Ĥl,n = ĥl,n,1 , ..., ĥl,n,K ,

• P l ∈ R K×K is a diagonal regularization matrix,

• e i k is the k-th column of I K , i.e., taking the k-th column of Ĥl,n ĤH l,n Ĥl,n -1

.

Remark 2: It is worth mentioning that any AP can design the different aforementioned precoders by only using its local CSI. Exceptionally for the local FZF, it works only when M ≥ τ p + 1 as explained in [START_REF] Interdonato | Local partial zero-forcing precoding for cell-free massive mimo[END_REF] and [START_REF] Interdonato | Downlink spectral efficiency of cell-free massive mimo with full-pilot zero-forcing[END_REF].

In order to perform the OFDM modulation at each antenna of an AP l, one need to reorder the M -dimensional precoded vectors {x l,n , ∀n} to M transmit antennas (see Fig. 1) according to the following one-to-one mapping

x t l,1 , ..., x t l,M T = [x l,1 , ..., x l,N ] , (15)
where x t l,m ∈ C 1×N denotes the frequency-domain vector to be transmitted through the m-th antenna. Then, the OFDM modulated (time-domain) signals, performed at AP l, {ȃ t l,m , ∀m}, are obtained by applying an inverse fast Fourier transform (IFFT) to {x t l,m , ∀m}. Then, a cyclic prefix (CP) is added, at each antenna, to the time-domain samples, avoiding inter-symbol interference (ISI) to be occurred. It is clear that when the CP is larger than the channel length (T ), the input-output relation of wireless channel can be specified in frequency-domain only.

One of the major drawbacks of OFDM is related to the fact that its time-domain signal has very high amplitude fluctuations, making it power-hungry as well as very sensitive to hardware impairments. These fluctuations are characterised by the peak-to-average power ratio (PAPR), which is defined as the ratio of the highest signal peak power and its average power value. Then, the PAPR, corresponding to the signal at antenna m of AP l, can be given by

P AP R(ȃ t l,m ) = max 0≤t≤LN -1 |ȃ l,m (t)| 2 E {|ȃ l,m (t)| 2 } , (16)
where ȃt l,m = [ȃ l,m (0), ..., ȃl,m (LN -1)] and L denotes the oversampling factor. 3) Power amplifier model: In the case of NL OFDM based CF-mMIMO, the modulated signals {a t l,m , ∀m} are fed, towards each AP antennas, through M transmit chains with power amplifiers (PAs). The resulting amplified signal at antenna m of AP l cab be written as

zt l,m = f ȃt l,m (17)
where f (.) represents the nonlinear amplification operation which is assumed to be identical over all antennas of all APs, for the sake of simplicity. Specifically, the nonlinear behavior of the PA can be modeled by the following full-rank polynomial

zl,m (n) = I i=1 λ i ȃl,m (n)|ȃ l,m (n)| i-1 , (18)
where λ 1 , ..., λ I are complex-valued model parameters that capture both amplitude-to-amplitude (AM/AM) and amplitude-to-phase (AM/PM) conversions. We have neglected the memory effects because we are primarily looking for how the basic nonlinear distortions are processed and tackled within highly-efficient OFDM based CF-mMIMO.

June 20, 2022

According to the Busgang theorem [START_REF] Price | A useful theorem for nonlinear devices having gaussian inputs[END_REF], the time-domain OFDM signal at the output of the NL PA can be expressed as

zt l,m = K 0 ȃt l,m + dt l,m (19)
where K 0 is a complex gain which is frequency independent and dt l,m is a zero mean noise which is uncorrelated with ȃt l,m . This latter is not Gaussian, but at the receiver side, after the OFDM demodulation, it becomes Gaussian. Note that these nonlinear distortion (NLD) parameters can be computed analytically, letting us to be able to develop closed-form expression to evaluate the in-band performance of CF-mMIMO-OFDM systems. For the sake of self-containment, we herein present, in equations ( 20) and ( 21), the closed-form expressions for the NLD parameters in terms of the model PA parameters {λ i }. Such expressions were already given in [START_REF] Dakhli | Theoretical analysis and compensation for the joint effects of hpa nonlinearity and rf crosstalk in vblast mimo-ofdm systems over rayleigh fading channel[END_REF] and [START_REF] Shaiek | Analytical analysis of ser for beyond 5g post-ofdm waveforms in presence of high power amplifiers[END_REF].

K 0 = λ 1 + π 8 I i=2, i even (i + 1)λ i σ i-1 i-2 2 i′=0 (2i′ + 1) + 1 2 I i=3, i odd (i + 1)λ i ( √ 2σ) i-1 ( i -1 2 )! (20)
and

σ 2 d = I i=1 |λ i | 2 2 i σ 2i i! -2 |K 0 | 2 σ 2 + 4π 2 I i,l=1,i̸ =l, (i+l) odd ℜ[λ i λ * l ]σ i+l i+l-1 2 i′=0 (2i′ + 1) + 2 I i,l=1,i̸ =l, (i+l) even ℜ[λ i λ * l ]( √ 2σ) i+l ( i + l 2 )! (21)
where σ is the standard deviation of the PA input signal.

The frequency-domain received signal at UE k and subcarrier n can be given by

y k,n = L l=1 h H l,k,n z l,n + b k,n (22)
= L l=1 h H l,k,n (K 0 x l,n + d l,n ) + b k,n = L l=1 h H l,k,n K 0 x l,n + L l=1 h H l,k,n d l,n + b k,n
where z l,n ∈ C M ×1 denotes the frequency-domain amplified signal transmitted by AP l on subcarrier n, d l,n ∈ C M ×1 is the frequency-domain version of NLD noise, K 0 is the M × M diagonal matrix whose elements are equal to K 0 and b k,n ∼ CN (0, 1) is an i.i.d. Gaussian noise.

June 20, 2022 III. PERFORMANCE ANALYSIS By plugging ( 9) in [START_REF] Bao | An admm approach for papr reduction for large-scale mimo-ofdm systems[END_REF], y k,n can be expanded as

y k,n = L l=1 h H l,k,n K 0 W l,n P l s n + L l=1 h H l,k,n d l,n + b k,n (23)
= L l=1 √ η l,k h H l,k,n K 0 w l,n,i k s k,n Desired signal + L l=1 K t̸ =k √ η l,t h H l,k,n K 0 w l,n,it s t,n
Multi-user interference

+ L l=1 h H l,k,n d l,n PA impairments + b k,n

Noise

One can note that y k,n consists of four terms: (i) the desired signal for the UE k, (ii) the multi-user interference, (iii) the NLD noise caused by the different PAs in the L APs and (iv) the noise.

It is the third term (PA impairments) together with the adoption of OFDM and several local precoders that makes the analysis in this paper different from existing literature. In the following, we will use [START_REF] Zayani | PAPR-Aware Massive MIMO-OFDM Downlink[END_REF]

+ b k,n Noise ( 24
)
where the coherent precoding CP k,n is given by

CP k,n = L l=1 √ η l,k E h H l,k,n K 0 w l,n,i k , (25)
the precoding gain uncertainly PGU k,n can be written as

PGU k,n = L l=1 √ η l,k h H l,k,n K 0 w l,n,i k -E h H l,k,n K 0 w l,n,i k , (26)
the multi-user interference MUI kt,n can be expressed as

MUI kt,n = L l=1 √ η l,t h H l,k,n K 0 w l,n,it , (27)
and the PA impairments PAI k,n is given by

PAI k,n = L l=1 h H l,k,n d l,n (28)
We recall that d l,n is uncorrelated with s k,n , i.e., the first term is uncorrelated with the fourth term. Moreover, it is clear that the first term is also uncorrelated with the second and third terms since, respectively, s k,n is independent of PGU k,n and s k,n and s t,n are uncorrelated ∀ t ̸ = k. Therefore, the sum of the second, third, fourth and fifth terms in [START_REF] Zayani | Low-complexity linear precoding for low-papr massive mu-mimo-ofdm downlink systems[END_REF] can be seen as an uncorrelated effective noise, when assuming that the fifth term (noise) is assumed to be independent of the first term. Hence, the SINR can be given as stated in Corollary 1.

June 20, 2022

Corollary 1: The effective SINR of UE k at subcarrier n is given by

SINR k,n = |CP k,n | 2 E {|PGU k,n | 2 } + K t̸ =k E {|MUI kt,n | 2 } + E {|PAI k,n | 2 } + 1 (29) = L l=1 √ η l,k E h H l,k,n K 0 w l,n,i k 2 K t=1 E L l=1 √ η l,t h H l,k,n K 0 w l,n,it 2 - L l=1 √ η l,k E h H l,k,n K 0 w l,n,i k 2 + E L l=1 h H l,k,n d l,n 2 + 1
Then, the per-RB SE of a user k is lower bounded as stated by Corollary 2.

Corollary 2: The per-RB and per-user SE in the downlink is given by

SE k = ξ 1 - τ p N sc N c N sc ∆f log 2 (1 + SINR k,n ) , (30)
where ∆f denotes the subcarrier spacing.

Note that this SE expression is valid for all precoding schemes presented in [START_REF] Li | An improved tone reservation scheme with fast convergence for papr reduction in ofdm systems[END_REF] and analytical expressions regarding the FZF scheme are stated by Corollary 3. Note that closed-form expression for the local RZF is intractable and only results, performed by Monte-Carlo simulations, are presented regarding this latter precoding scheme.

Corollary 3: The per-user downlink SE (bit/s) is obtained for FZF precoding as in Corollary 2, where

SINR k,n = (M -τ p )|K 0 | 2 L l=1 √ η l,k γ l,k 2 (M -τ p )|K 0 | 2 t∈P k \{k} L l=1 √ η l,t γ l,k 2
+ |K 0 | 2 L l=1 K t=1 η l,t (β l,k -γ l,k ) + Ψ k,n + 1 , (31)
where Ψ k,n denotes the NLD variance which can be expressed as

Ψ k,n = E    L l=1 h H l,k,n d l,n 2    (32) = L l=1 L l′=1 M m=1 M m′=1 E h * l,k,n,m d l,n,m d * l′,n,m′ h l′,k,n,m′
Proof: see Appendix.

Note that Ψ k,n depends on both intra-user and inter-user nonlinear distortion, which can not be eliminated by using conventional transmit precoding techniques. Moreover, the behavior of this NLD depends on many parameters, such as number of users, number of pilots, number of APs, number antennas per-AP, the precoding scheme, channel estimation, pilot contamination and essentially the adopted PA and its operating point.

In order to derive closed-from expressions for the SINR of the k-th user at the n-th subcarrier, it is remaining to find the terms in [START_REF] Price | A useful theorem for nonlinear devices having gaussian inputs[END_REF]. Note that due to the possibility of NLD correlation, these latter are not necessary zero even when l ̸ = l′ and m ̸ = m′. But, it is still possible to have closed-form expressions, for Ψ k,n , corresponding to some scenarios as stated in Lemma 1, corresponding to FZF precoding.

Lemma 1 (NL OFDM based CF-mMIMO with local FZF precoding): With FZF precoding, when τ p → M , Ψ k can be expressed as

Ψ F ZF k = M σ 2 b L l=1 β l,k σ 2 d , (33)
June 20, 2022 and achievable SE for the k-th UE can be defined as stated in Theorem 1.

Theorem 1: In OFDM based CF-mMIMO adopting nonlinear PA and FZF precoding (τ p → M ), the downlink capacity of the k-th UE is lower bounded by

SE k = χN sc ∆f log 2   1 + (M -τ p )|K 0 | 2 L l=1 √ η l,k γ l,k 2 (M -τ p )|K 0 | 2 t∈P k \{k} L l=1 √ η l,t γ l,k 2
+ |K 0 | 2 L l=1 K t=1 η l,t (β l,k -γ l,k ) + Ψ F ZF k + 1    , (34)
where χ, which denotes the percentage of DL payload data, is expressed as χ = ξ 1 -τp NscNc and we recall that

σ 2
d is given in closed-form in [START_REF] Bao | An efficient bayesian papr reduction method for ofdm-based massive mimo systems[END_REF].

Proof: see Appendix.

B. Downlink Energy Efficiency

In this subsection, we derive the EE of OFDM based CF-mMIMO systems to investigate how it is affected by the PA's IBO and the number of APs. Inspired by [START_REF] Ngo | On the total energy efficiency of cell-free massive mimo[END_REF][37] [START_REF] Björnson | Optimal design of energy-efficient multi-user mimo systems: Is massive mimo the answer?[END_REF], we consider the per-RB EE as the ratio of the sum rate (bits/s) to the total system power consumption (Watt). The per-RB EE is defined as

EE = K k=1 SE k P total , (35)
where P total can be defined as [START_REF] Ngo | On the total energy efficiency of cell-free massive mimo[END_REF]]

P total = L l=1 P l + L l=1 P f h,l , (36)
where P l denote the power consumption at the l-th AP due to the RF and digital processing, i.e., the PA and the circuit power consumption, and P f h,l is the power consumed by the front-haul link connecting the l-th AP with the CPU.

The P l can be modeled as

P l = σ 2 b M µ l K k=1 η l,k γ l,k + M P c,l , (37)
where 0 < µ l ≤ 1 is the power amplifier efficiency, which can be expressed in terms of IBO as

µ l = µ max / √ IBO,
where µ max is the maximum power efficiency; and P c,l is the consumed power related to each antenna to run the circuit components.

In Downlink CF-mMIMO, the front-hauling cost refers to the amount of data to be transferred between the CPU and the APs to satisfy coherent transmission and other centralized network operations. Since all processing related to the precoding and PAPR reduction are fully distributed (i.e. they do not need any exchange of information), the CPU need to send, within a coherence block, |Ξ|τ D KL complex samples (i.e., {s k,n }). Inspired by [START_REF] Ngo | On the total energy efficiency of cell-free massive mimo[END_REF], the front-haul power consumption is proportional to the sum spectral efficiency, as follows

P f h,l = P 0,l + P bt,l K k=1 SE k , (38)
where P 0,l is a fixed power consumption which depends on link between the AP l and the CPU, and P bt,l is the traffic-dependent power.

June 20, 2022

For the sake of clarity and without loss of generality, we assume that all APs have the same transceiver model (i.e., µ l = µ ∀l and P c,l = P c ∀l); and P 0,l and P bt,l are AP-independent, such that P 0,l = P 0 ∀l and P bt,l = P bt ∀l.

To this end, the per-RB EE can be written as

EE = K k=1 SE k M σ 2 b µ L l=1 K k=1 η l,k γ l,k + LM P c + LP 0 + LP bt K k=1 SE k , (39)
IV. PROPOSED LOCAL PAPR REDUCTION ALGORITHM

A. Discussion

In classical (co-located) massive MIMO systems, a high-dimensional degrees of freedom (DoFs) are offered by equipping the BS with a large number of antennas compared to the number of served users. Interestingly, these DoFs can be exploited to effectively reduce the PAPR of transmit signals [START_REF] Studer | Par-aware large-scale multi-user mimo-ofdm downlink[END_REF][21] [START_REF] Bao | An admm approach for papr reduction for large-scale mimo-ofdm systems[END_REF][23] [START_REF] Zayani | Low-complexity linear precoding for low-papr massive mu-mimo-ofdm downlink systems[END_REF]. In centralized CF-mMIMO, i.e., adopting high capacity front-haul connections to transfer data between the APs and the CPU, it would be possible to re-use the PAPR reduction methods already proposed for co-localised mMIMO. However, the PAPR reduction and its associated energy-efficiency improvement will be in the cost of heavy front-hauling load, leading to reduced global system energy-efficiency.

In this regard, we propose new PAPR reduction technique that is performed locally at each AP, with no instantaneous information sharing (e.g. CSI), leading to a reduced front-hauling overhead. Consequently, it becomes very challenging to find a solution that is able to provide considerable PAPR reduction while it is constrained to i) be implementable with APs adopting few antennas ii) do not need additional front-hauling overhead and iii) do not damaging the transmission quality.

The principle behind our proposed solution (see Fig. 1) is that each AP

= argmin

{r t m } max ∥IDF T x t 1 + r t 1 ∥ ∞ , ..., ∥IDF T x t M + r t M ∥ ∞
The key idea of the proposed solution is to anticipate the distortion, so that instead of transmitting high-PAPR signals through NL PAs that cause severe distortions (not controllable at all), we transmit low-PAPR signals (avoiding such severe PA-related distortion) by generating the PCSs {r t m , ∀m} which can add distortion to the transmitted signals but, interestingly, we can control it. Indeed, PA-related distortion is proportional to the amount of the power allocated to the desired signal, making the strongest users, i.e. having the largest channel gains, to be the most affected and, without interest, the weakest users to be less affected.

Towards this end, we optimize the PCSs that satisfy the Statement 1.

June 20, 2022

Statement 1: Frequency-domain peak-cancelling signals are optimized such that distortion caused to the strongest users is negligible or even null while distortion caused to the weakest users are tolerable. Specifically, the PCSrelated distortions are constrained to be beam-formed only towards the weakest users in a flexible way, so that we continue to provide service for these latter.

More specifically, each AP l divides the set of active users into two disjoint subsets: strong users, Ω l ⊂ {1, ..., K}, and weak users, Ω c l ⊂ {1, ..., K}, where

Ω l ∩ Ω c l = Ø and |Ω l | + |Ω c l | = K.
The user grouping is based on the user path-loss: the τ s users having the highest path-losses belong to Ω l , while τ p -τ s users belong to Ω c l . Note that if UE k ∈ Ω l , any UE t ∈ P k belongs to Ω l .

B. Computing frequency-domain peak-canceling signals (PCSs)

We discuss, in this subsection, how to design the optimal frequency-domain PCSs {r n , ∀n}, at each AP l,

to
ām (t) =    ȃm (t), if |ȃ m (t)| < δ δe jϕ(t) , if |ȃ m (t)| > δ , (41)
where ȃm (t) = |ȃ m (t)|e jϕ(t) , ϕ(t) is the phase of ȃm (t) and δ denotes the clipping threshold.

In order to obtain the best PAPR reduction, δ should be related to the OFDM signal mean power and the percentage of the reserved users τ p /τ s . Then, it is chosen to be

δ = σ 2 τ p τ s , ( 42
)
where σ 2 denotes the variance of the modulated signal at each antenna. It is worth to mention that δ has to satisfy the out-of-band constraint and should be δ ≥ σ 2 ln N Na [START_REF] Zayani | PAPR-Aware Massive MIMO-OFDM Downlink[END_REF], where N a = |Ξ| denotes the number of activated subcarriers.

Then, the original frequency-domain clipping-noise signal associated to the m-th transmit antenna is

ϵ t m = DF T ām -ȃm . ( 43
)
Here, the vectors {ϵ n , ∀n} are collected from the M vectors {ϵ t 1 , ϵ t 2 , ..., ϵ t M }. Plugging (43) into (40), we have

{r t,opt 1 , .., r t,opt M } = argmin {r t m } max ∥IDF T x t 1 + r t 1 -ϵ t 1 ∥ ∞ , ..., ∥IDF T x t M + r t M -ϵ t M ∥ ∞ (44)
According to (44), it is clear that the best PAPR is obtained when the PCSs, {r t m }, equal the original clippingnoise signals {ϵ t m }. However, the PCSs hardly equal the original peak-cancelling signals because the have to satisfy the necessary constraints. Indeed, the frequency-domain PCSs are obtained through a projection of the original frequency-domain clipping-noise signals, {ϵ t m }, such that they satisfy the Statement 1. In addition, they have to [START_REF] Price | A useful theorem for nonlinear devices having gaussian inputs[END_REF] respect the out-of-band constraint such that they have to be set to zero on the guard-band. Then, frequency-domain PCSs can be defined as

r n = ν l V l,n ϵ n , ∀n ∈ Ξ (45) r n = 0 Mt×1 , ∀n ∈ Ξ c
where :

• {r n } are collected from the M vectors {r t 1 , r t 2 , ..., r t M }

• V l,n ∈ C M ×M represents the projection space corresponding to AP l, i.e., projection matrix (please refer to Section IV-C) and ν l is a regularization factor (discussed in Section IV-D).

To efficiently reduce the PAPR, we propose an iterative method that aims at fitting the PCSs to the original clipping-noise.

C. Design of flexible projection space V l,n

To significantly protect the strong users, i.e., UEs in Ω l , against the distortion caused by adding PCSs, we force the distortion to take place in the orthogonal complement of {h l,k,n ∀l, ∀n and ∀k ∈ Ω l }. To do that, let

V l,n = I M -Hl,n HH l,n Hl,n + Γ l -1 HH l,n (46)
denote the projection matrix, corresponding to AP l and subcarrier n, where Γ l ∈ R τp×τp is a regularized diagonal matrix that enables flexibility in how we project the distortion related the to PAPR reduction (i.e., adding PCSs).

Indeed, by using this projection matrix, we have ν l ĥl,n,k V l,n ϵ l,n = 0, only if Γ l [k, k] = 0, meaning that distortion caused to the corresponding user k almost vanishes (only a small contribution can survive when channel estimation errors are occurred). Thus,

Γ l [k, k] = 0 ∀k ∈ Ω l .
Conversely, ν l ĥl,n,k′ V l,n ϵ l,n ̸ = 0 when Γ l [k′, k′] ̸ = 0, meaning that there is distortion caused to UEs k′. In order to make this distortion tolerable, it would be scaled inversely proportional to the power allocated to user k′.

Then, we define Γ l [k′, k′] = 1/η l,k′ ∀k′ ∈ Ω c l . To summarize, our approach guarantees full distortion protection to UEs with high channel gains, except for channel estimation errors, while still providing service to UEs with bad channel gains. It offers a good balance between PAPR reduction and cancelling distortion (adjustable by properly choosing τ s ). That is clear now the philosophy of our approach -Reduce the PAPR and protect -.

D. Computing the optimal value of ν l

Due to the reconstruction of the precoded PCSs from their projections onto the projection matrix and the available data subcarriers in Ξ, it is obvious that the considered PCSs, at each iteration, can be smaller than their associated clipping noises when the traditional clipping and control (CC) method is employed at each AP.

Hence, a regularization factor is needed in order to generate the optimal PCSs, leading then to a fast convergence toward the optimal solution. Then, ν l is optimized such that ν l V l,n ϵ l,n is similar to ϵ l,n . It can be calculated using least-square approximation (LSA), as explained in [START_REF] Zayani | PAPR-Aware Massive MIMO-OFDM Downlink[END_REF] as

ν l = E n m |V l,n ϵ l,n ||ϵ n | m |V l,n ϵ l,n | 2 . ( 47
)
June 20, 2022

Using such regularization factor, the amplitudes of PCSs ν l V l,n ϵ l,n generated by LSA, almost equal to those of the original clipping noise ϵ n . Then, we transmit ν l V l,n ϵ l,n instead of V n ϵ n . This operation reduces the number of iterations to achieve the optimal PAPR reduction.

E. Algorithm Summary

In this subsection, we present our developed algorithm applying the above-explained PAPR reduction approach.

The PAPR reduction based CF-mMIMO-OFDM DL tranmission is achieved by alternately repeating the PAPR reduction process using the clip and control (CC) appraoch and restoring the restrictions on the clipping-noise components using the proposed flexible projection space.

The details of the proposed algorithm is summarized in -the two groups of users (stong and weak),

-the corresponding regularization diagonal matrix Γ l (subsection IV-C) 2: Compute:

-the projection matrices {V l,n }, as given by equation (

-the precoded data vectors {x l,n }, as given by equation ( 9)

3: Set the maximal iteration number N iter 4: Initialize:

-{r

l,n = 0 M ×1 } -{x (1) l,n } = {x l,n } 5: for p=1,...,N iter do 6: ȃt(p) m = IF F T x t(p) m , ∀ m = 1...M 7: ϵ t(p) m = F F T āt(p) m - ȃt(p) m 8: ϵ (p) l,n = 0 M ×1 , for n ∈ χ c 9: ν l = En m |V l,n ϵ (p) l,n ||ϵ (p) n | m |V l,n ϵ (p) l,n | 2 10: r (p) l,n = ν l V l,n ϵ (p) l,n , for n ∈ χ 11: x (p+1) n = x (p) l,n + r (1)

F. Complexity Analysis

As a complexity measure, we adopt the number of complex multiplications. For the developed localPAPRfree algorithm, one can easily note that the added computational cost, compared with the original CF-mMIMO precoding scheme, is dominated by the N -points IFFT/FFT (Steps 6 and 7) and the projection operation in Step 10. The IFFT/FFT costs O(M N log (N )) and the projection costs O(M 2 |Ξ|). Therefore, the localPAPRfree requires a total of O(2M N log (N ) + M 2 |Ξ|) complex multiplications for each iteration. Evidently, the number of iterations should be chosen judiciously to offer good balance between PAPR reduction and computational complexity. Note that since the computational complexity of the local FZF precoding scheme is about M N log(N )+N

3τ 2 p M 2 + τpM 2 + τ 3 p -τp 3
[11], the added complexity related to our PAPR reduction algorithm is quite negligible.

V. SIMULATION RESULTS

This section provides numerical results to evaluate the performance of OFDM based cell-free massive MIMO in terms of DL SE and DL EE, in presence of nonlinear power amplifiers. In addition, the benefit of our proposed PAPR reduction algorithm is evaluated.

A. Simulation Scenario

Let us introduce the considered CF-mMIMO-OFDM system setup. We assume that the L APs and the K UEs The large-scale fading coefficients {β l,k } are modeled as [START_REF] Interdonato | Local partial zero-forcing precoding for cell-free massive mimo[END_REF] β l,k = PL l,k .10

σ sh z l,k 10 , (48)
where PL l,k denotes the path-loss and 10 σ sh z l,k 10 models log-normal shadow fading with standard deviation σ sh and z l,k ∼ N (0, 1). Note that, in this investigation, we consider the 3GPP Urban Microcell path-loss model, which is given by equation (49) when assuming a 2 GHz carrier frequency [11][39].

PL l,k [dB] = -30.5 -36.7log 10 d l,k 1 m , (49)
where d l,k denotes the distance between the l-th AP and the k-th UE including AP and UE's heights.

The simulation settings are reported in Table I. Note that the PA characteristics are modeled by the memoryless modified Rapp model, defined by the 3GPP in [START_REF] Nokia | Realistic power amplifier model for the new radio evaluation[END_REF], with parameters G = 16, V sat = 1.9, p = 1.1, A = -345, B = 0.17 and q = 4. Its corresponding complex-valued polynomial model parameters, which capture the AM/AM and AM/PM conversions, can be computed as in [START_REF] Dakhli | Theoretical analysis and compensation for the joint effects of hpa nonlinearity and rf crosstalk in vblast mimo-ofdm systems over rayleigh fading channel[END_REF].

June 20, 2022 uses all the degrees of freedom to suppress interference towards all the available orthogonal directions, resulting in a small array gain M -τ p . By the way, it would be better to cancel only interference towards UEs (< τ p ) with good channel gains and increase the array gain M -τ p .

Note that RZF can be improved by taking into account the NLD power in computing the regularization matrices {P l }, which is out of scope of this paper. The reason is because we aim to show the relevance of our proposed PAPR reduction algorithm in improving the performance of this traditional RZF.

2) Effect of the number of APs (L): for the different values of L, validating our derived analytical expression in [START_REF] Shaiek | Analytical analysis of ser for beyond 5g post-ofdm waveforms in presence of high power amplifiers[END_REF]. From these results, it is clear that the SE of the system impacted by PA is degraded w.r.t. the system with ideal PA. Again, we can note the higher sensitivity of the RZF against the nonlinear distortions compared to FZF, which is as more pronounced as the number of APs increases. It is worth mentioning that, with FZF, the DL SE performance does not tend to an upper bound at large L, while, with RZF, the DL SE performance is upper bounded.

Lastly, from results depicted in Fig. 7, one can see that the DL SE loss, caused by power amplifiers, gets larger when the number of APs gets higher. For an IBO = 3, the FZF-related SE loss gets higher from 17.32% to 25.98%

when the number of APs grows from 60 to 180. While the SE loss achieved by the RZF increases from 45.55%

to 56.51%.

June 20, 2022 3) Effect of the number of pilots (τ p ): The implementation versatility of RZF versus the limitation of FZF is shown in Fig. 8. When M is fixed, RZF precoding can be performed by any number of pilots τ p , while FZF constraints the number of orthogonal pilots, such that τ p < M has to be verified (else the FZF inverse peudomatrix is not defined). For instance, Figs. 8(a) and 8(b) show the ability of RZF to achieve much higher SEs in the operation regime in which FZF cannot be implemented. Moreover, we can observe from these results a loss in the performance of the FZF when τ p goes from 12 to 15. In linear case, it gets lower from 4.68 to 3.99 kbps/user.

Such degradation is caused by decreasing the array gain M -τ p when τ p tends to M . This confirm the intuition saying to only cancel interference towards UEs (< τ p ) with good channel gains, leading to increase the array gain M -τ p . In nonlinear case, FZF performs better than RZF in its implementable operation regime, else it does not.

For τ p = 12, FZF has a gain of 60% and 57% for, respectively, an IBO of 6dB and 3dB. performance. The CCDF denotes the probability that the PAPR of the transmitted signal is higher than a given threshold PAPR0.

In Figs. 9(a) and 9(b), we show the CCDF of the PAPR for, respectively, FZF and RZF, when different numbers of iterations are considered. Note that PAPRs associated with all the antennas of all the APs are taken into account in computing the empirical CCDF. The number of trials is chosen to be 5000 in our simulations. Looking at these results, one can clearly note that the localPAPRfree algorithm offers substantial PAPR reduction compared with the original local FZF and RZF precoding schemes. Indeed, for the FZF, it achieves a gain of 3.09dB, 4.94dB and 5.71dB when, respectively, 1, 3 and 5 iterations are considered (at a CCDF of 1%). That is really obvious that it converges so fast, thanks to the regularization factor ν l computed in equation (47).

In order to see the impact of τ p and τ s on the PAPR performance, we plot in 9(a) results achieved when considering FZF with τ p = 15 and τ s = 8 and in 9(b) results achieved when considering RZF with τ p = 20 and τ s = 10. One can see a slight better performance when τ s is larger, i.e. the projection space is larger. It is worth to mention that these values of τ s have chosen empirically, such that we have the best PAPR reduction without damaging the transmission quality. We recall that the localPAPRfree algorithm can be applied in a distributed and scalable fashion, i.e., implemented at each AP and executed locally, making benefits of scalable and high energy-efficient CF-mMIMO-OFDM.

In addition to the performance in terms of PAPR reduction, it is important to study the impact of the developed algorithm on the DL SE. To clearly see the impact of the PAPR reduction, we assume that the power amplifiers are perfectly linearized, i.e, a perfect digital predistortion (DPD) is added before each PA [41][42]. Then, this latter acts as a limiter.

To this end, Figs. equal to 2dB, it is about 41% and 86% for, respectively, FZF and RZF.

To further evaluate the effectiveness of the proposed localPAPRfree algorithm, we compare the average spectral efficiency and energy efficiency of our PAPR reduction method with the classical local FZF and RZF precoding schemes without any PAPR reuction (see Figs. 13 and 14). Moreover, the gain in terms of energy efficiency is also given versus the number of APs and the IBO is equal to 3dB. Looking at these results, with our proposed PAPR reuction method, the DL SE and EE improve significantly. In particular, the provided gain increases while the number of APs (L) is increasing. When L varies from 100 to 200, it goes from 23% to 38% and from 40% to , for FZF where:

• in (a) the diagonal matrix K 0 whose elements are equal to the complex coefficient K 0 as we assume the adoption of the same power amplifier at each antenna branch of each AP.

• in (b) α l,k,k is equal to (M -τ p )γ l,k for FZF precoding scheme as explained in [START_REF] Interdonato | Local partial zero-forcing precoding for cell-free massive mimo[END_REF].

Next, we compute

K t=1 E L l=1
√ η l,t h H l,k,n K 0 w l,n,it η l,t (β l,k -γ l,k ), for FZF for more details concerning (c), the reader can refer to [11][36].

June 20, 2022

The NLD variance Ψ k,n can be approximated as In (e), the average distortion power can be approximated only by intra-user interference since the local FZF precoding scheme is able to considerably reduce the inter-user interference, when τ p → M . Therefore, (e) can be resumed in (f). Normalizing (e) w.r.t. to the noise power σ 2 b , we find equation [START_REF] Dakhli | Theoretical analysis and compensation for the joint effects of hpa nonlinearity and rf crosstalk in vblast mimo-ofdm systems over rayleigh fading channel[END_REF]. Plugging 51, 52 and 53 into 29 gives 34.

Ψ k,n = E | L l=1 h H l,

Fig. 1 .

 1 Fig. 1. System model of the OFDM based CF-mMIMO downlink: M transmit antennas at the AP, K independent single-antenna terminals, N OFDM subcarriers. The proposed PAPR reduction algorithm, highlighted by the dashed box in blue, applied independently at each AP.

Fig. 2 .

 2 Fig. 2. A radio frame with Nc time-domain OFDM symbols and N subcarriers in frequency-domain that are grouped into N rb RBs. The time-frequency resource grid of a single RB with N sb subcarriers is shown. The UL training, UL payload and DL payload are also shown.

Algorithm:

  The proposed localPAPRfree algorithm Given: -a set of N modulated complex signals {s k,n } -the channel estimates { Hl,n }, for n = 1, .., N -the UEs path-losses {β l,k } 1: Define:

  , ∀n ∈ χ 12: end for 13: return {x (N iter+1) l,n } June 20, 2022

  are independently and uniformly distributed within an area of size D × D squared meters, which is wrapped around at the edges to avoid boundary effects. We consider an uncoded OFDM with IFFT/FFT of N = 256 and use a spectral map Ξ, in which N a = |Ξ| = 248 are used for data transmission and N gb = |Ξ c | = 4 are used as guard-band at each side. Thus N = N a +2N gb = 256.The IFFT/FFT size is set to 1024, which corresponds to 4-oversampling in the time-domain to evaluate the PAPR accurately. Note that We consider an OFDM radio frame of duration 1 ms (corresponding to the channel coherence time), which consists of 14 OFDM symbols. From the frequency-domain perspective, each RB contains N rb = 12 subcarriers with 15 kHz of bandwidth each.

Fig. 3 . 20 Fig. 4 .

 3204 Fig. 3. CDFs of DL per-RB SE achieved by local FZF precoder: M = 16, K = 20, τp = 15, power control is based on (13). Solid curves denote analytical results obtained in closed-form (equation (34)) while markers indicate the results obtained by MCS.

Fig. 6 Fig. 5 .

 65 Fig. 5. CDFs of DL per-RB SE achieved by FZF and RZF precoders: L = 200, M = 16, K = 20, τp = 15, power control is based on (13).

Fig. 6 .

 6 Fig. 6. Median DL per-RB SE, achieved by FZF and RZF precoders, averaged over several large-scale fading realizations, M = 16, K = 20, τp = 15. Solid curves denote results obtained by MCS while circle markers indicate the results obtained in closed-form (equation (34) for FZF.

Fig. 7 .

 7 Fig. 7. DL SE loss caused by nonlinear distortion on FZF and RZF precoders, averaged over several large-scale fading realizations, M = 16, K = 20, τp = 15.

Fig. 8 .

 8 Fig. 8. Median DL per-RB SE, achieved by FZF and RZF precoders, averaged over several large-scale fading realizations, M = 16, K = 20, τp = 15. Only results obtained by MCS are provided.

10 Fig. 9 .

 109 Fig. 9. CCDFs of PAPR: L = 200, M = 16, K = 20, power control is based on 13.

Fig. 11 .

 11 Fig. 11. CDFs of DL per-RB SE achieved RZF precoders: L = 200, M = 16, K = 20, τp = 20, τs = 10, power control is based on (13).

10 Fig. 12 .Fig. 13 .Fig. 14 .

 10121314 Fig. 12. Median DL per-RB SE achieved by FZF and RZF precoding schemes, averaged over several large-scale fading realizations, L = 200, M = 16, K = 20.

60%

  for, respectively, FZF and RZF (as shown by Figs. 13(b) and ??(b)). June 20, 2022

2 L l=1 √

 2l=1 -τ p )|K 0 | η l,k γ l,k2

√|K 0 | 2 √ 2 +

 022 η l,t h H l,k,n K 0 w l,n,it √ η l,t ĥH l,k,n w l,n,it + hH l,k,n w l,n,it η l,t η l′,t α l,k,t α l′,k,t + E hH l,k,n w l,n,it w H l′,n,it hl′,k,n(c) = (M -τ p )|K 0 | 2 t∈P k L l=1 √ η l,t γ l,k |K

σ 2 d

 2 k,n d l,n | 2 ,n d l,n d H l′,n h l′,k,n l,k,n,m d l,n,m d * l′,n,m′ h l′,k,n,m′ l,k,n,m d l,n,m d * l,n,m h l,k,n,m + l,k,n,m d l,n,m d * l′,n,m′ h l′,k,n,m′ l,k,n,m σ 2 d h l,k,n,m +The two terms in (d) represents, respectively, the intra-AP and inter-AP distortions. The cross term can be neglected since channels are uncorrelated to the distortion noise. The first term in (d) contains inter-user and intrauser interference. (e) is obtained by assuming that the channel is constant during a time-interval (channel coherence time).

  to derive the downlink SE and EE.

			K
	s k,n +	PGU k,n	s k,n +	MUI kt,n
	Coherent precoding	Precoding gain uncertainly	t̸ =k	Multi-user interference

A. Downlink Spectral Efficiency

Inspired by works

[START_REF] Björnson | Massive MIMO Networks: Spectral, Energy, and Hardware Efficiency[END_REF]

[START_REF] Interdonato | Local partial zero-forcing precoding for cell-free massive mimo[END_REF]

[25]

[START_REF] Larsson | Fundamentals of Massive MIMO[END_REF]

that apply a bounding technique, referred to as hardening bound, we derive the achievable downlink SE, i.e., a lower bound on the ergodic capacity, for OFDM based CF-mMIMO under PA non-linearity and the assumption of independent Rayleigh fading channel. Towards this end, we rewrite

[START_REF] Zayani | PAPR-Aware Massive MIMO-OFDM Downlink[END_REF]

, in frequency-domain, as y k,n = CP k,n s t,n + PAI k,n PA impairments

  perform the best PAPR reduction performance while keeping good transmission quality. It consists in fitting, iteratively, these PCSs to time-domain clipping-noise signals, denoted by {e n , ∀n}, that give the desired PAPR. These latter are computed by clipping the time-domain signals {a t m , ∀m}, at each antenna. The clipped signals can be obtained as follows

Table Algorithm .

 Algorithm One can clearly see that the proposed algorithm proceeds in only one-loop where it performs the OFDM modulated signals in Step 6, the clipping-noise in Step 7, the OOB constraint in Step 8 and the optimal value of ν l in Step 9. The peak-cancelling signals {r l,n } are performed in Step 10. Finally, the precoded signals {x l,n } are updated in Step 11. Note that any intermediate solution can be used, ensuring good in-band transmission quality and without causing any OOB radiation.

TABLE II OOB

 II [DB] RADIATION PERFORMANCE.In order to measure the amount of signal power that is transmitted outside the active subcarriers Ξ, we define the OOB radiation as followsOOB [dB] = 10log10 n∈Ξ c ∥x n ∥ 2 intended future work. In addition, improving the EE analysis through the use of realistic power consumption models for power amplifiers and back-hauling is of paramount importance. |PGU k,n | 2 and K t̸ =k E |MUI kt,n | 2 .

	IBO [dB]		1	3	6	9	12
	FZF/RZF w/o PAPR red.	-10.92 -14.86 APPENDIX A	-25.46 -46.49 -301.5
	FZF/RZF w localPAPRfree (τp = 15, τs = 8) From equation (29), we need first to compute L l=1	-17.77 -41.45 √ η l,k E h H l,k,n K 0 w l,n,i k -304	-304	-304
	RZF w localPAPRfree (τp = 20, τs = 10)		-20.14 -297.21 -304	-304	-304
	L	√ η l,k E h H l,k,n K 0 w l,n,i k
	C. Out-of-band (OOB) radiation performance l=1

2

, E

Finally, random pilot assignment is adopted for simplicity. So, each UE randomly select a pilot sequence from a predefined set of τ p < K orthogonal pilot sequences of length τ p samples.

B. Performance Evaluation

In order to evaluate the performance of the considered CF-mMIMO-OFDM system, we adopt the cumulative density function (CDF) of the SE, which corresponds to the SE values collected over different random realizations of the AP/UE locations for 500 network snapshots. ). Second, we can see the significant impact of the PA non-linearity on the SE performance, especially for high percentiles. The reason is that the PA distortion is beam-formed towards UEs with good channel conditions, since the control power scheme boosts the energy towards these UEs.

In Fig. 4, the CDFs of SEs achieved by the local RZF are given. It is worth noticing the higher sensitivity of the local RZF precoding scheme to the PA non-linearity compared to the FZF one. It is more clear now that only UEs with good channels (i.e., upper SE percentiles) are impacted.

In order to compare the impact of the non-linear PA on the performance of the two precoding schemes, local FZF and local RZF, Fig. 5 shows the CDFs of SEs when considering two values of IBO (6 and 9dB). Again, we can note that these latter suffer from the distortion caused by power amplifiers. Moreover, in linear case (Ideal PA), we observe that the RZF performs better than FZF, for low and high percentiles. The reason is that RZF can manage better the interference by taking into account channel estimation error and pilot contamination, thanks to the regularization matrix. Lastly, it is clear that RZF is more sensitive than FZF against the nonlinear distortions, especially at high SE percentiles. The reason is that the RZF boosts power towards UEs with good channel conditions, leading to high NLD power towards this UEs. Moreover, FZF still has worse performance at low SE percentiles because it [START_REF] Price | A useful theorem for nonlinear devices having gaussian inputs[END_REF] We recall that the user grouping, related to the PAPR reduction, is based on the user path-loss: the τ s users having the highest path-losses belong to Ω l (set of strong users), while τ p -τ s users belong to Ω c l (set of weak users). It is worth mentioning that if UE k ∈ Ω l , any UE t ∈ P k belongs to Ω l , such that the users using the same pilot undergoes that same effect. In this comparison, the simulation setup consists in L = 200, M = 16, K = 20. For FZF, τ p = 15, τ s = 8, while τ p = 20, τ s = 10 for RZF. Here, the results achieved by the use of the localPAPRfree algorithm (dashed blue curves) corresponds to PAPR performance, given in Figs. 10 and11, with N iter = 5. We can clearly observe the robustness of the proposed PAPR-aware precoding scheme against the saturation effects compared to the traditional local FZF and RZF precoding schemes (red curves). With the proposed scheme, we can achieve the same DL SE as with ideal PAs (benchmark, black curves) from an IBO of 3dB. Moreover, it performs better than the case without PAPR reduction when the PAs are operated with lower IBO values (1 and 2dB). Most importantly, although the sensitivity of RZF against PA-related NLD, it caches up and gains in performance (see Fig. 11).

When coming to Figs. 12(a) and 12(b) other comments can be highlighted. Indeed, we can clearly see the ability of the localPAPRfree algorithm to enhance the median DL SE. The maximum gain is achieved when the IBO is June 20, 2022

Note that for FZF and RZF precoding schemes, we have OOB = -∞ (dB), as no data is transmitted over the guard-band. Nevertheless, we have OOB > 0 in presence of non-linear power amplifiers. Interestingly, the proposed localPAPRfree algorithm does not ignore the OOB radiation and it ensures that the signal power on subcarriers Ξ c is zero, such that no interference is caused by the PAPR reduction, in the guard-band (the constraint on the OOB is given in the algorithm's Step 8, as shown in Table Algorithm). Table II gives the OOB radiation performance of the proposed algorithm compared to the classical FZF/RZF without any PAPR reduction, with PAs acting as limiters operating at IBO of 1, 3, 6, 9 and 12. One can clearly note that the localPAPRfree algorithm can perform satisfactory OOB radiation performance from an IBO of 3dB (OOB radiation < -41dB), making our solution useful in practice.

VI. CONCLUSION

The proposed localPAPRfree algorithm is a versatile distributed PAPR reduction method that can significantly reduce the PAPR of transmitted signals and then improve spectral efficiency and the energy efficiency of an OFDM based cell-free massive MIMO system, compared to the traditional precoding schemes. Specifically, the localPAPRfree algorithm can anticipate the PA related distortions and adds frequency domain peak-cancelling signals to the transmitted ones (generated by the traditional precoding scheme), reducing the PAPR and avoiding the PA distortions. Interestingly, the localPAPRfree can, via the added PCSs, control the radiated distortion. This latter is optimized such that it is negligible towards the UEs with good channel gains while it can be tolerable towards UEs with poor channel gains. Indeed, it is not necessary to cancel distortion towards all the available directions, a finding that have have shown through the theoretical study conducted within this investigation. Moreover, we have shown that the DL SE increases unlimitedly as number of APs grows while a SE loss is increasing compared to the linear case.

It is worth mentioning that the application of the proposed algorithm with the other existing local precoding schemes (like local partial ZF and local protective partial ZF [START_REF] Interdonato | Local partial zero-forcing precoding for cell-free massive mimo[END_REF]) is straightforward, certainly constitutes an June 20, 2022