PAPR-aware Massive MIMO-OFDM Downlink · In this paper, we develop a downlink transmission scheme to address the PAPR reduction problem for OFDM-based massive MIMO wireless systems,

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PAPR-aware Massive MIMO-OFDMDownlinkRAFIK ZAYANI1,2, (Member, IEEE) HMAIED SHAIEK2, (Member, IEEE), DANIEL ROVIRAS2,(Senior Member, IEEE)1University of Carthage, Sup’Com, Innov’COM, Ariana 2083, Tunisia2CNAM, CEDRIC/LAETITIA, Paris 75003, France

Corresponding author: Rafik Zayani (e-mail: [email protected]).

This work has been performed in the framework of the ADAM5 project, receiving funds from the European Union under H2020-EU.1.3.2with funding scheme MSCA-IF-EF-ST (Project ID: 796401)

ABSTRACT We investigate the peak-to-average power ratio (PAPR) reduction problem in orthogonalfrequency-division multiplexing (OFDM) based massive multi-user (MU) multiple-input multiple-output(MIMO) downlink systems. In this paper, we develop a downlink transmission scheme that performs jointlyMU precoding and PAPR reduction (PP) by exploiting the excess degrees of freedom (DoF) offered byequipping the BS by a large number of antennas. Specifically, the joint MU precoding and PAPR reductionscheme is formulated as a simple convex optimization problem solved via steepest gradient descent (GD)approach. Then, we develop a novel algorithm, referred to as MU-PP-GDm, to reduce the PAPR of thetransmitted signals by exploiting the high-dimensional null-space of the MIMO channel matrix whilemaintaining excellent transmission quality. Simulation results show that the proposed MU-PP-GDm hasa low computational complexity and can achieve substantial PAPR performance with fast convergence rate.

INDEX TERMS 5G+, massive multiple-input multiple-output (MIMO), orthogonal frequency-divisionmultiplexing (OFDM), peak-to-average power ratio (PAPR), multi-user (MU) precoding, convex optimiza-tion, gradient descent (GD).

I. INTRODUCTION

Massive multiple-input multiple-output (MIMO), also knownas large-scale multi-user (MU) MIMO, has been recognizedas a promising technology for future generations of wirelesscommunications. It has been initially introduced in [1] andpopularly studied in [2] where a large number of antennasare employed at the base station (BS) simultaneously servinga much smaller number of single-antenna users using MUprecoding. Wireless massive MIMO systems enable the useof low-cost and low-power single-antenna user devices whileexpensive equipment is only needed on the BS. In this way,potential improvements are provided in the spectral- andenergy- efficiency [3]. Furthermore, simple linear signal pro-cessing approaches, such as matched filter (MF), minimummean-squared error (MMSE) and zero-forcing (ZF), can beused as precoding techniques for massive MIMO downlinkthat have the potential to reduce the operational power con-sumption at the transmitter and enable the suppression of MUinterference (MUI) [4].

In practice, broadband wireless communications, however,encounter large delay spread, and, therefore, suffer from

frequency-selective fading. Orthogonal frequency-divisionmultiplexing (OFDM), a scheme of encoding digital sym-bols on multiple orthogonal sub-carriers, is an efficient andwell-established way to deal with frequency-selective chan-nels. Therefore, massive MIMO-OFDM is a very promisingcombination to meet the ever growing demands for higherlink readability and spectrum efficiency of next-generationwireless communication systems (5G and Beyond).

However, massive MIMO precoders exhibit transmit sig-nals with high peak-to-average power ratio (PAPR), re-gardless of whether single-carrier or OFDM transmissionand of whether a low or a high modulation order is used[4]. Consequently, the nonlinearity of the radio frequency(RF) high-power amplifier (HPA), which is necessary in atransmission chain, yields in-band distortion and out-of-bandradiation (OBR) which cause signal distortion, phase rotationand adjacent channel interference, respectively. To avoid suchheavy distortions, the transmit signals require that the poweramplifiers are backed-off and operated in their linear region(i.e., where their transfer characteristics are sufficiently lin-ear). Nevertheless, operating at lower power levels reduces

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R. Zayani et al.: A Learning Framework for PAPR-aware Massive MIMO-OFDM Downlink

the power efficiency, which would result in huge operationalexpenditure for large-scale BS having hundreds of antennas.This is not a practical solution for 5G networks since thetarget energy efficiency improvement is 100x w.r.t. 4G long-term evolution (LTE) [5]. Therefore, it is primordial to re-duce the PAPR of OFDM-based massive MIMO systems tomotivate corresponding energy- and cost- efficient massiveMIMO BS deployments. Thus, low-PAPR MU precoderswould be of paramount importance since massive MIMOsystems have the potential to reduce the PAPR of transmitsignals by exploiting the excess spatial degrees-of-freedom(DoFs) [2].

Over the recent past years, many efforts have been devotedto introduce MU-MIMO precoding based PAPR reductionschemes because joint signal processing at the receiver sideis impossible as users are spatially distributed. Low-PAPRTomlinson-Harashima precoding schemes suitable for MU-MISO and MU-MIMO downlink were initially described in[6] and [7], respectively. These schemes, however, requirespecific signal processing at the receiver side (i.e. in themobile terminal) making them less attractive. In [8], authorshave proposed a peak-signal clipping scheme for PAPR re-duction in OFDM-based massive MIMO where some of theantennas at the BS are reserved to compensate for peak-clipping signals. Another method has been introduced in [9]that aims at reserving some tones to reduce the PAPR inlarge-scale MU-MIMO-OFDM systems. Despite of their lowcomputational complexity, these methods are lose efficiencysince they sacrifice spectral efficiency by reserving some an-tennas or some tones, which are not realistic solutions for fu-ture wireless networks. In [10], Studer et al. have introduceda downlink transmission scheme for massive MU-MIMO-OFDM wireless systems. The proposed fast iterative trun-cation (FITRA) [10] algorithm aims to perform jointly MUprecoding, OFDM modulation and PAPR reduction (PMP).This method provides good PAPR performance by sacrificingthe transmission quality (MUI) and affecting the spectralpurity (i.e. OBR). Moreover, the complexity of this algorithmis high. Authors in [11] have developed an efficient approx-imate passing (AMP)-based Bayesian method (EM-TGM-GAMP) [11] for joint PAPR reduction and MUI cancellationfor massive MIMO-OFDM, which has been demonstrated toshow better performance compared to FITRA, in terms ofconvergence speed and PAPR reduction performance. Morerecently, an efficient perturbation-assisted based alterativedirection method of multipliers (ADMM) technique [12] hasbeen proposed to address PAPR reduction for large-scaleMU-MIMO-OFDM. The proposed PROXIINF-ADMM [12]algorithm has the advantage to do not cause any additionalin-band distortion (MUI) and out-of-band radiations whileits performance is very close to those provided by FITRA.However, PROXIINF-ADMM still has a quite high compu-tational complexity, even if, it has a faster convergence speedcompared to FITRA and EM-TGM-GAMP.

Motivated by these considerations, it is encouraged toconcentrate on the development of an efficient MU precod-

ing scheme based PAPR reduction that aims at performingjointly the MU precoding and the PAPR reduction withoutsacrificing neither the spectral efficiency nor the transmissionquality nor the spectral purity. It should exploit the extranull-space of the associated massive MIMO channel [13]. Abig emphasis should be put on reducing the computationalcomplexity.

In this paper, we develop a downlink transmission schemeto address the PAPR reduction problem for OFDM-basedmassive MIMO wireless systems, which leaves the pro-cessing required at each mobile terminal untouched whileaffecting only the signal processing at the BS. Our proposedmethod performs jointly MU precoding and PAPR reductionthat are formulated as a convex optimization problem andsolved online with instantaneous processed data via gradientdescent (GD) approach. On one hand, it aims at optimizingfrequency-domain precoded signals that perfectly remove allMUI, guaranteeing ideal transmission quality, and spectralpurity. On another hand, it introduces carefully, to precodedsignals, frequency-domain peak-cancelling signals that areconceived to reduce the PAPRs of the transmitted symbols.The additive peak-power signals are restricted to only thenull-space of their associated MIMO channel matrices. Thisrestriction guarantees that the additive peak-power signalsdo not cause any additional MUI. We formulate the designof these additive peak-power signals as constrained convexoptimization problems that is solved via GD approach. Theoptimization of the two sets of signals is done jointly sincethey perform in different orthogonal spaces. In this regard,we develop a novel optimization algorithm, referred to asMU precoding based PAPR reduction via gradient descentapproach (MU-PP-GDm), which is able to find efficientsolution to MU-PP for large-scale MU-MIMO-OFDM.

Compared with existing methods, e.g. [7]- [12], our pro-posed method presents the following improvements:• The substantial improvement is related to the fact that

our method has a lower computational complexity overall existing methods when considering a given PAPRperformance target. Indeed, only matrix-vector mul-tiplications are performed when using the GD-basedsolver, avoiding the computation of large-scale pseudo-inverse matrix. For example, the PROXIINF-ADMM[12] initializes the frequency-domain precoded signalsusing the zero-forcing (ZF) precoding scheme that aimsat eliminating the MUI completely, but with high com-putational complexity. A low computational complexityprecoder is an interesting merit for practical systems.

• Our proposed method guarantees perfect MUI cancella-tion and do not cause any additional out-of-band radi-ations, whereas, other proposed method (e.g., [10] and[11]) can not guarantee complete MUI cancellation andany OBR. Such precoder is of paramount importance forfuture wireless networks.

• Most PAPR reduction schemes for massive MIMO-OFDM [10] [12] use some appropriate regularizationor penalty parameters, which are manually adjusted. In

2 VOLUME 4, 2016


contrast, in our proposed algorithm all parameters areadjusted automatically leading to a faster convergenceto the optimal PAPR while maintaining excellent trans-mission quality.

The rest part of this paper is structured as follows. SectionII provides the system model, the MU precoding scheme andthe PAPR reduction problem. In section III, the proposedapproach-assisted MU precoding and PAPR reduction areformulated as convex optimization problems. The developedMU-PP-GDm algorithm is discussed in section IV. Simula-tion results are provided in section V. Finally, the conclusionis given in section IV.

Notations : Lowercase boldface letters (e.g. x) stand forcolumn vectors, bold lowercase letters with a superscript(.)t (e.g. xt) denotes row vectors, and bold uppercase letters(e.g. X) denotes matrices. We denote its transpose, conju-gate transpose, pseudo-inverse and largest singular value byXT , XH , X† and σmax (X), respectively. For a M × N -dimensional matrix X = {xmn}, we use xm to designate them-th column, and xtn to designate the n-th row. The N ×Nidentity matrix and the M × N all-zeros matrix are denotedby IN and 0M×N , respectively. We use ‖x‖2 and ‖x‖∞to denote l2-norm and l∞-norm of vector x, respectively.The cardinality and complement of set χ is |χ| and χc,respectively. E [.] stands for the expectation operator.

II. PRELIMINARIESA. SYSTEM MODELWe consider a typical OFDM-based massive MIMO down-link scenario as illustrated in Fig. 1. We assume that the BSis equipped with Mt transmit antennas and simultaneouslyserves Mr single-antenna terminals (users) over a frequency-selective channel, where Mt is significantly larger than Mr.The signal vector sn ∈ CMr×1 contains the symbols asso-ciated with the n-th tone for Mr users, where n = 1, ..., Nindexes the OFDM subcarriers, N denotes the total numberof OFDM tones. sn is chosen from a complex-valued con-stellations A. In practice, OFDM systems typically specifycertain unused subcarriers, which are used for guard-band(e.g., at both ends of the spectrum). Hence, the set of sub-carriers available are divided into two sets χ and χc, wherethe subcarriers in set χ are used for data transmission and thesubcarriers in its complementary set χc are used for guard-band. Moreover, we set sn = 0Mr×1 for n ∈ χc such that nosignal is transmitted on the guard-band.

B. MULTI-USER PRECODING SCHEMETo remove MUI at the receivers, precoding needs to beperformed at the BS since cooperative detection among usersis often impossible. The signal vectors sn, ∀n, can be linearlyprecoded as

cn =1√ςW

Wnsn (1)

where cn ∈ CMt×1 represents the precoded vector thatcontains symbols to be transmitted over the n-th subcarrierthrough the Mt antennas, Wn ∈ CMt×Mr denotes theprecoding matrix for n-th OFDM subcarrier and ςW is anormalization factor designed to obtain an average or instan-taneous transmit power equal to σ2

s .Zero-forcing (ZF) precoding scheme is considered in this

paper, which aims at cancelling the MUI completely. Notethat since Mr �Mt, there are many precoding matrices thatcan achieve perfect MUI elimination, where the most widelyused form is

Wn = HHn

(HnHH

n

)−1(2)

where Hn ∈ CMr×Mt is the MIMO channel matrixassociated with the n-th OFDM subcarrier. Here, we assumethat the channel matrices {Hn} are perfectly known at theBS, which can be determined by exploiting the channelreciprocity [14] [15] of time division multiplexing (TDD)systems.

It can be easily verified that the ZF has a high computa-tional complexity [16] [17] since it needs to compute large-scale pseudo-inverse matrix.

After precoding, the Mt-dimensional vectors cn, ∀n arereordered to Mt transmit antennas for OFDM modulation,according to the following one-to-one mapping

[xt1...x

tMt

]= [c1...cN ]

T (3)

Here, the N -dimensional vector xmt denotes thefrequency-domain signal to be transmitted from mt-th an-tenna. The time-domain signal {atmt} is obtained by applyingan inverse fast Fourier transform (IFFT) of {xtmt}. Prior thetransmission over wireless channel, a cyclic-prefix (CP) isadded to the time-domain samples of each antenna in orderto avoid inter-symbol interference (ISI).

To simplify the presentation and without loss of generality,we specify the input-output relation of wireless channel infrequency-domain only. Then, the signal received by Mr

users can be expressed as

yn = Hncn + bn, n ∈ χ (4)

where bn ∈ CMr×1 represents the receiver noise whoseentries are i.i.d circularly-symmetric complex Gaussian dis-tribution with zero-mean and variance N0/2.

Since HnH†n = IMr, transmitting cn = H†nsn perfectly

removes all MUI. Then, (4) can be transformed into Mr

independent single-stream formulation and it can be rewrittenas

yn = sn + bn, n ∈ χ (5)

Note that the precoded vectors cn needs to satisfy

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FIGURE 1. System model of the massive MIMO-OFDM downlink: Mt transmit antennas at the BS, Mr independent single-antenna terminals, N OFDM tones.The proposed MU-PP-GD, highlighted by the dashed box in the BS, combines MU precoding and PAPR reduction.

sn = Hncn, n ∈ χcn = 0Mt×1, n ∈ χc (6)

Note that this is equivalent to transmitting {xn,∀n} thatsatisfy

sn = Hnxn, n ∈ χ (7)

where xn ∈ CMt×1 denotes the frequency-domain precodedsignal associated with Mt transmit antennas at the n-thsubcarrier.

In order to ensure desirable spectral properties of thetransmitted OFDM signals, the inactive OFDM subcarrier(indexed by χc) must satisfy the following shaping con-straints

xn = 0Mt×1, n ∈ χc (8)

Consequently, the total MUI energy can be evaluated by‖Hnxn − sn‖22,∀n. Note that ZF precoding is equivalentto transmitting the solution xn to the following convex op-timization problem

minimizexn

F (xn) = ‖Hnxn − sn‖22, n ∈ χ

subject to xn = 0Mt×1, n ∈ χc (9)

This formulation inspired us to state the MU massiveMIMO precoding scheme as a convex optimization prob-lem that can be solved online with instantaneous scenarioinformation (i.e., transmit data, CSI). This would have alower computational complexity and, most importantly, itwill enable to perform the PAPR reduction in a flexible way,as discussed in Section III.

C. PEAK TO AVERAGE POWER RATIOConsider a typical OFDM system with N subcarriers. Theoversampled OFDM symbol in discrete time-domain atmt =[amt0, amt1, ..., amtLN−1] at mt-th transmit antenna , where

the oversampling factor is denoted as L, is generated by LN-point inverse fast Fourier transform (IFFT) operation as

amt(k) =1√LN

N−1∑n=0

xmt(n)ej2πnkLN , 0 ≤ k ≤ LN − 1

(10)

where xmt is the algorithm solution, k stands for a discretetime index and j =

√−1. The OFDM signal can also be

written in matrix form as

atmt︸︷︷︸[LN×1]

= FH︸︷︷︸[LN×N ]

xtmt︸︷︷︸[N×1]

, (11)

where FH denotes theLN×N IFFT matrix. A normalizationis applied to the frequency-domain signal to ensure fixedtransmit power.

The PAPR of atmt is defined as the ratio of the highestsignal peak power to its average power value. Hence, it isgiven by

PAPR(atmt) =

max0≤k≤LN−1

[|amt(k)|2

]E [|amt(k)|2]

=LN‖atmt‖

2∞

‖atmt‖22

,

(12)

Existing PAPR reduction methods, e.g. [6]- [11], for mas-sive MU-MIMO-OFDM systems performs directly throughperforming the precoded signals {cn}. The key idea behindall these methods is that, due to the redundant DoFs renderedby the large number of antennas at the BS, there exist agreat number of precoded signals that can achieve perfectMUI cancellation, from which we may find a set of precodedsignals {cn} whose time-domain counterpart signals {atmt}have a low PAPR. Recently, a solution was introduced in[12], which consists on a perturbation-assisted scheme thatdoes not rely on the precoding design to reduce the PAPR.However, either they rely on the precoding design or not,these approaches may have limited applicability because, inpractical systems, the precoding matrices can be either cho-sen from a pre-specified codebook or according a specifiedcriterion that could be performed online. In this regard, in

4 VOLUME 4, 2016


the next section, we propose a MU precoding scheme basedPAPR reduction that is transparent to the precoding design.

III. PROPOSED JOINT PRECODING AND PAPRREDUCTION ALGORITHMA. DISCUSSIONThe key idea of the proposed downlink transmission schemeis to exploit the high-dimensional null-space of the massiveMIMO channel matrix offered by equipping the BS with alarge number of antennas and to jointly perform MU precod-ing and PAPR reduction. Therefore, it aims to compute theprecoded signals {xtmt} that satisfy {Hncn = sn} (where{cn} are the reordered versions of {xtmt}) and evaluatesthe peak-canceling signals {rtmt} that reduces the PAPR ofthe resulting time-domain signal {atmt} (see Fig. 1). It isworth to mention that the peak-canceling signals {rtmt} areconstrained to lie in the null-space of their associated MIMOchannel matrices such that they do not disturb the signalsreceived by Mr users through the N subcarriers (i.e. they aretransparent to the receivers). Thus, adding the peak-cancelingsignals does not damage the transmission quality (i.e., MUIand capacity) since they vanish after propagating through thewireless channel. We next formulate the convex optimizationproblem, which jointly performs MU precoding and PAPRreduction, while enabling the use of conventional OFDMdemodulation at the receivers.

We start by specifying the necessary constraints. To com-pletely remove MUI and ensure spectral purity, the precodingconstraints in (7) and (8) must hold, which leads to the convexoptimization problem in (9). PAPR reduction is achievedby adding peak-canceling signals {rtmt} to precoded signals{xtmt}. Specifically, the N -dimensional vector rtmt is thepeak-canceling signal added to the precoded signal of them-th transmit antenna, while rn ∈ CMt×1 is the peak-canceling signal added to the precoded signal associatedto the n-th subcarrier, i.e. xn. In order to avoid any in-band distortions (i.e., MUI), the peak-canceling signals mustsatisfy the following conditions

Hnrn = 0Mr×1, n ∈ χ (13)

Note that Hn can be decomposed by using singular valuedecomposition (SVD) [18] as

Hn = UnΣnVHn , ∀n (14)

where Un is an Mr ×Mr complex unitary matrix, Σn is anMr ×Mt rectangular diagonal matrix with non-negative realnumbers on the diagonal, and VH

n =[v1n,v

2n, ....,v

Mtn

]is an

Mt ×Mt complex unitary matrix. The diagonal entries σi ofΣn are known as the singular values of Hn.

Note that MIMO transmission uses Mt×Mr-dimensionalVdn =

[v1n,v

2n, ....,v

Mrn

]as the Beamforming (BF) matrix

of the Mr data streams. On the other hand, the Mt ×(Mt −Mr)-dimensional V0

n =[vMr+1n , ....,vMt

n

]spans the

null-space of the channel since it is not used for BF of thedata streams. Therefore

HnV0n = 0, ∀n (15)

In the following, we will represent rn by using a(Mt −Mr)-dimensional vector en as

rn = V0nen (16)

Moreover, to avoid spectral regrowth, i.e. out-of-band radi-ations, the following constraint must hold on the guard bands(indexed by χc)

rn = 0Mt×1, n ∈ χc (17)

A conventional demodulation OFDM is sufficient and noadditional processing is needed at the receivers by holdingthe above constraints. That is, the peak-canceling signals aretransparent to the receivers.

Here, we wish to minimize the PAPR at each antenna.Then, we take an approach that minimizes the largest mag-nitude of the time domain-signals {atmt}. This results is asub-optimal solution that leads to a convex formulation whileit can substantially reduce the PAPR [10]. In fact, directlyoptimizing (12), i.e. minimize jointly the PAPRs associatedwith all antennas, results in a non-convex problem which isnot straightforward to solve and, to our best knowledge, thereis no efficient solution for such a non-convex problem.

In the following, we discuss how to design the peak-canceling signals {rn}. The key idea is to iteratively fit thepeak-canceling signals to their associated frequency-domainclipping-noise signals. These latter are obtained by clippingthe time-domain signals {atmt}. Given the clipping thresholdλ, the clipped signal an at the n-th subcarrier can be obtainedby

amt(k) =

{amt(k), if |amt

(k)| < λλejφ(k), if |amt (k)| > λ

, (18)

where amt(k) = |amt(k)|ejφ(k) and φ(k) is the phaseof amt(k). In order to obtain the best PAPR, the optimalclipping threshold [19] λ is closely related with the meanpower of the OFDM signal σ2

a and the ratio of the usedsubcarriers N

|χc| and is given by

λ = σa

√ln

(N

|χc|

)(19)

Evidently, the original frequency-domain clipping noiseassociated to the mt-th transmit antenna is dtmt =FFT (amt − amt). Then, the optimal peak-canceling sig-nals

V0nen = dn, ∀n ∈ χ (20)

where dn contains samples associated to the n-th subcarriercollected from the Mt vectors

(dt1,d

t2, ...,d

tMt

).

Then, the effective transmission signal xn at the n-thsubcarrier in the frequency-domain is represented as

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xn = xn + pnV0nen, n ∈ χ (21)

where pn is a regularization factor. In fact, the peak-canceling signals during the first iteration is much smallerthan that of the original clipping noises when the traditionalclipping control (CC) method is employed in OFDM sys-tems. Hence, this regularization factor aims at generatingthe optimal peak-canceling signals with fast convergence.The regularization factor is calculated using least-squaresapproximation (LSA) [20] and it is defined as

pn =

∑k |V0

nen||dn|∑k |V0

nen|2, n ∈ χ (22)

Using such regularization factor, the amplitude of peak-canceling signals V0

nen generated by LSA, almost equal tothose of the original clipping noise. Obviously, it may reducethe number of iterations to achieve a good PAPR reduction(see Section V).

However, the peak-canceling signals {V0nen} hardly equal

to {dn}, because |χ| < N . Then, we search, via the pro-posed algorithm, to optimize the solution en according to thefollowing simple convex optimization problem

minimizeen

G(en) = ‖V0nen − dn‖22, n ∈ χ

subject to en = 0(Mt−Mr)×1, n ∈ χc (23)

Note that when |χ| = N , (23) has a unique solution,i.e., en = V0

n†dn, otherwise, it leads to a poor PAPR

reduction meanwhile its computational complexity is veryhigh since we have to compute the pseudo-inverse of anMt × (Mt −Mr)-dimensional matrix V0

n.

B. PROPOSED FORMULATIONIdeally, PAPR reduction is achieved by minimizing the l∞-norm of the time-domain signals {atmt ,∀mt}. With takinginto consideration constraints in (7) and (8), The MU pre-coding scheme based PAPR reduction is cast as

minimize{atmt}

{‖at1‖∞, ..., ‖atMt‖∞}

subject to

{sn = Hnxn, n ∈ χxn = 0Mt×1, n ∈ χc (24)

Since the solution of the l∞-norm is in fact obtained bythe clipping and control, as discussed in Section III-A, weare able to state the formulation in (24) subject to constraints(17) and (20) as a simple convex optimization problem

minimize{en}

G(en) = ‖V0nen − dn‖22, n ∈ χ

subject to

sn = Hn

(xn + V0

nen), n ∈ χ

xn = 0Mt×1, n ∈ χcen = 0(Mt−Mr)×1, n ∈ χc

(25)

Note that the above formulation will yield an iterativeCF algorithm with the peak-canceling signals constrainedin the null-space of the associated MIMO channel matrices.The objective is to search for signals {en} which can helpreduce the PAPR and meanwhile satisfy the MU precoding,in-band and out-of-band constraints. It is worth noticing thatthe proposed formulation can separate the PAPR reductionproblem from the MU precoding by initializing precodedsignals {xn} according to a pre-fixed precoder or from a pre-specified codebook, and as a consequence, it is transparent tothe precoding design.

IV. MU-PP-GDM ALGORITHMIn this section, we develop a new algorithm to find aneffective solution to (25). The MU precoding scheme basedPAPR reduction is achieved by alternately repeating thePAPR reduction process using the CF method, restoring therestrictions on the PAPR reduction signal components usingnull-space MIMO channels and performing MU precoding.To make the problem tractable, the equality constraint sn =Hn

(xn + V0

nen), which is equivalent to sn = Hnxn, is

relaxed as

minimize{en,xn}

J(xn, en) = F (xn) +G(en), n ∈ χ

subject to

{xn = 0Mt×1, n ∈ χcen = 0(Mt−Mr)×1, n ∈ χc (26)

An alternating minimization strategy can be used to solve(26), in which we alternatively minimize the objective func-tion with respect to xn and en. Thus in the (l+1)-th iteration,the alternating procedure can be expressed as

x(l+1)n = argmin

{xn}J(xn, e

(l)n ), n ∈ χ (27)

e(l+1)n = argmin

{en}J(x(l+1)

n , e(l)n ), n ∈ χ (28)

By doing so, the constrained optimization problem isrelaxed, which therefore enables low-complexity first-orderalgorithm, i.e., an algorithm only requires matrix-vectormultiplications, such as the steepest gradient descent (GD)method [21] [22]. The proposed algorithm to solve (26) isreferred to as MU-PP-GD.

The search directions of the steepest descent method at theiterate x

(l)n and e

(l)n are determined by the negative gradient

of J at, respectively, x(l)n (denoted by −∇lxJ(x

(l)n , e

(l)n )) and

e(l)n (denoted by −∇leJ(x

(l+1)n , e

(l)n )), where

∇lxJ(x(l)n , e

(l)n ) =

2

LxnHHn

(Hnx(l)

n − sn

), n ∈ χ (29)

∇leJ(x(l+1)n , e(l)

n ) =2

LenV0nH(V0ne(l)

n − d(l+1)n

), n ∈ χ

(30)

6 VOLUME 4, 2016


where Lxn = 2σ2max(Hn) and Len = 2σ2

max(V0n) are the

Lipschitz constants [23] for, respectively, ‖Hnxn−sn‖22 and‖V0

nen − dn‖22.In this work, we consider the gradient descent with mo-

mentum [24] in order to enhance the convergence rate of thealgorithm. We do this by adding a fraction µ of the updatevector of the past time step to the current update vector (thealgorithm referred to as MU-PP-GDm).

A. ALGORITHM SUMMARYThe details of the proposed MU-PP-GDm is summarized inTab. Algorithm. One can note that the proposed algorithmproceeds in only one-loop where it performs the precodedsignals (Step 4) using gradients computed in Step 3, the peak-canceling signals in Step 6 and its constrained version inStep 7. Finally, in Step 9, the precoded signals are re-updatedto take into consideration the peak-canceling signals, whichare constrained to lie in the null-space of their associatedMIMO channel matrices. Also, note that the constraints in(26) are always satisfied throughout the whole iterative pro-cess. Hence, any intermediate solution can be used, withoutcausing any in-band or out-of-band radiations.

Algorithm: The MU-PP-GDm algorithmGiven a set of N modulated complex signals {sn}.1: Initialize x

(1)n = 0Mt×1,

e(1)n = 0(Mt−Mr)×1,

dx(0)n = 0Mt×1,

Lxn = 2σ2max(Hn), Len = 2σ2

max(V0n),

and set the maximal iteration number maxIterand the momentum term µ

2: for l=1,...,maxIter do

3: dx(l)n = 2

LxnHH

n

(Hnx

(l)n − sn

)+ µdx

(l−1)n , ∀n ∈ χ

4: x(l+1)n = x

(l)n − dx

(l)n , ∀n ∈ χ

5: at(l+1)mt = IFFT

(xt(l+1)mt

), ∀mt = 1...Mt

6: dt(l+1)mt = FFT

(at(l+1)mt − a

t(l+1)mt

)7: e

t(l+1)n = e

t(l)n − 2

LenV0

nH

(V0

ne(l)n − d

(l+1)n

), ∀n ∈ χ

8: pn =∑k |V

0ne

(l+1)n ||d(l+1)

n |∑k |V0

ne(l+1)n |

2 , ∀n ∈ χ

9: x(l+1)n = x

(l+1)n + pnV0

ne(l+1)n , ∀n ∈ χ

10: end for

11: return {xn = x(maxIter+1)n }

B. COMPLEXITY ANALYSISWe adopt the number of complex multiplications as a com-plexity measure to compare the complexity of the proposedalgorithm with that of PROXINF-ADMM [12] and FITRA[10]. For the proposed MU-PP-GDm, we can easily verifythat the computational cost is dominated by the gradients

∇lxJ (Line 3), ∇leJ (Line 7) and the N-points IFFT/FFT(Lines 5 and 6). The gradient in line 3 of the algorithminvolves O (2|χ|MtMr) complex multiplications. Line 7 re-quires O (2|χ|Mt(Mt −Mr)) complex multiplications forthe second gradient. The IFFT/FFT costs O (MtN log (N))complex multiplications. Therefore, the MU-PP-GDm re-quires a total of O

(2MtN log (N) + 2|χ|Mt

2)

complexmultiplications for each iteration. By contrast, the PROXINF-ADMM [12] needs O

(|χ|(MrMt

2))

for the initializationstep and O

(MtN log (N) + ImaxMr|χ|Mt

2)

for each it-eration [12], where Imax is the number iterations in theinner loop, that was flexed to 2 [12]. While, the FITRArequires about O

(MrMtN

2)

complex multiplications periteration [12]. Evidently, the number of iterations needed toachieve a desirable PAPR performance will be of paramountimportance in evaluating the complexity of these algorithms.Complexity comparison will be given, in the next section,where many PAPR performance levels are considered.

V. SIMULATION RESULTSA. PERFORMANCE EVALUATIONTo demonstrate the efficiency of the proposed algorithm forjoint MU precoding and PAPR reduction, some simulationshave been conducted in an OFDM-based MU massive MIMOsystem. This latter has Mt = 100 antennas at the BS andserving Mr = 10 single-antenna users. We consider anuncoded OFDM withN = 128 subcarriers and use a spectralmap χ, in which |χ| = 114 subcarriers are used for datatransmission. A 16 quadrature amplitude modulation (16-QAM) with Gray mapping is considered. The number ofFFT/IFFT points is set to 512, which corresponds to L = 4-oversampling in the time-domain in order to measure thePAPR levels accurately. We use the complementary cumu-lative distribution function (CCDF) to evaluate the PAPRreduction performance, which denotes the probability thatthe PAPR of the estimated signal exceeds a given thresholdPAPR0. Also, to evaluate the multi-user interference, wedefine the MUI as

MUI =

∑n∈χ ‖Hnxn − sn‖22∑

n∈χ ‖sn‖22(31)

The wireless channel is assumed to be frequency-selectiveand modeled as a tap delay line with T = 8 taps. Thetime-domain channel response matrices Ht, t = 1, ..., T ,have i.i.d. circularly symmetric Gaussian distributed entrieswith zero mean and unit variance. The equivalent frequency-domain response Hn on the n-th subcarrier can be obtainedby

Hn =

T∑t=1

Hte−j2πtnN (32)

We compare our algorithm with the conventional ZF pre-coding scheme. It is worth to mention that all parametersused in our proposed algorithm are determined automatically

VOLUME 4, 2016 7


and no parameter is defined manually, expect if we add themomentum term which is set to be µ = 0.11. In contrast toother existing algorithms, like FITRA [10] and PROXIINF-ADMM [12] where many parameters have to be fixed man-ually which is not straightforward in practical system. Inour algorithm, λ (equation 19) and {pn} (equation 22) areautomatically adjusted in each iteration that offers the bestconvergence rate to the optimal PAPR [19] [20].

To evaluate the efficiency of our algorithm in term of PAPRreduction, Fig. 2 shows the CCDF of the PAPR when differ-ent numbers of iterations are considered. Note that PAPRsassociated with all Mt antennas are taken into account incomputing the empirical CCDF. The number of channel trialsis chosen to be 1000 in our simulations. We note that our al-gorithm provides substantial PAPR reduction compared withthe conventional ZF precoding scheme. Indeed, it achieves again of 5.7dB, 6dB and 6.3dB when, respectively, 10, 20 and100 iterations are considered (at CCDF of 1%). We can alsonote that our algorithm, within few number of iterations (e.g.maxIter = 20) offers PAPR reduction performance similarthan PROXIINF-ADMM and FITRA that need to perform,respectively, 40 and 1000 iterations. However, PROXIINF-ADMM and FITRA outperform our algorithm MU-PP-GDwhen a large number of iterations is performed, e.g. 200and 2000 for, PROXIINF-ADMM and FITRA, respectively.However, this gain is at the cost of SNR loss, which causesthe symbol error rate (SER) and capacity performance losscompared to the ZF scheme. This is undesirable in practical5G and Beyond systems. This performance loss is primarilydue to an increase in the norm of the obtained solution x, i.e.increase in the transmit power ‖x‖22. It is not surprising tosee that the solution obtained by our proposed method has asmaller norm than the solution of PROXIINF-ADMM sinceour MU-PP-GDm algorithms aims at performing jointly theMU precoding (optimizing the same criterion as in ZF) andPAPR reduction. Therefore, MU-PP-GDm tends to convergeto a norm solution very close to the ZF one.

FIGURE 2. PAPR performance.

To confirm this point of view, we plot the SER perfor-mance of the proposed MU-PP-GDm algorithm in Fig. 3,where the SNR is defined as SNR = E{‖xn‖22}/N0. Incontrast to PROXIINF-ADMM and FITRA, we can note thatour algorithm incurs a negligible SNR performance loss, e.g.at SER = 10−3, 0.25dB of SNR loss is shown comparedto the ZF scheme, even when the number of iteration is large(maxIter = 200).

FIGURE 3. SER performance, 16-QAM.

We now discuss the convergence rate of the proposed MU-PP-GDm algorithm. Figs. 4 and 5 show, respectively, theMUI and the average PAPR versus the number of iterations.We note that our algorithm achieves a MUI of about−308dBwith only 30 and 80 iterations which corresponds, respec-tively, to the use of the gradient descent with or withoutmomentum. We can also note that MU-PP-GDm can achievea MUI of about −100dB when only 10 iterations are per-formed, which is a sufficient performance needed in practicalsystems. Moreover, our method has a low computationalcomplexity since this MUI is performed using simple first-degree algorithm (using matrix-vector multiplications) whichneeds O (2MtMr) per iteration. With this few number ofiterations needed, its computational complexity is consid-erably lower than initializing the precoded signal using ZFscheme (the solution adopted by PROXINF-ADMM) whichhas to compute the pseudo-inverse of a large-scale matrix.We move now to the PAPR, one can note that our proposedMU-PP-GDm performs well when the optimal solutions of{pn} is considered. We can clearly see a gain of about 100iterations for an achieved PAPR of 3dB (see Fig. 6). Theproposed MU-PP-GDm can obtain a PAPR of 4dB with only6 iterations, while PROXINF-ADMM and FITRA algorithmsrequire 20× 2 (outer loop and inner loop) and 800 iterations,respectively. Also, the proposed algorithm is able to reducethe PAPR down to 3dB with only 40 iterations, while thePROXINF-ADMM and FITRA require 100 and 1500 itera-tions, respectively, to obtain a similar result. It is worth tomention that the PROXINF-ADMM and FITRA algorithms

8 VOLUME 4, 2016


converge to a lower PAPR than the proposed MU-PP-GDmbut at the cost of transmission quality loss. This means thatthe MU-PP-GDm has the fastest convergence rate comparedto PROXINF-ADMM and FITRA but a higher PAPR thatdoes not affect a lot the transmission quality. This trade-off between PAPR reduction and transmission quality lossis assured by the joint performing of MU precoding andPAPR reduction. It is worth mentioning that the proposedalgorithm is as sensitive to the inaccuracy of the channel stateinformation (CSI) as the ZF scheme.

FIGURE 4. Convergence rate of MUI.

FIGURE 5. Convergence rate of PAPR.

Finally, we examine the efficiency of our proposed algo-rithm under different number of transmit antennas (variesfrom 20 to 120), while the number of users is fixed tobe Mr = 10. Here, we would like to show the effect ofDoF offered by equipping the BS by a large number ofantennas on the performance of our algorithm. Figs. 6 and7 illustrates, respectively, the MUI and the PAPR versusthe number of transmit antennas, where different numbersof iterations are considered. Results are averaged over 1000

independent simulations and PAPR results are averaged overPAPRs associated with all transmit antennas. We observe thatthe number of iterations required to achieve a given MUIincreases as the number of antennas decreases, e.g. for a MUIof −200dB, about 20, 40 and 80 iterations are required forMt = 100, Mt = 80 and Mt = 40. We can explain thisby the fact that the proposed algorithm achieves better MUIperformance when more DoF at the base station are available.

We move now to the PAPR (Fig. 7), one can note thatthe MU-PP-GDm can achieve lower PAPR when the null-space of the MIMO channel is larger (with higher numberof transmit antennas), i.e. more DoF at the base station areavailable.

FIGURE 6. MUI vs. number of transmit antennas.

FIGURE 7. PAPR vs. number of transmit antennas.

Simulation results showed that MU-PP-GDm is able toachieve substantial PAPR reduction performance, withoutaffecting at all the MUI and the out-of-band radiations, mean-while providing a faster convergence rate compared to theexisting algorithms. This could substantially motivate the useof low-cost and low-size radio-frequency (RF) components

VOLUME 4, 2016 9


TABLE 1. Complexity Comparison

Algo. Complexity per iteration Complexity to achieve a PAPR of 4dB Complexity to achieve a PAPR of 3dBFITRA [10] 16384000 1.3107× 1010 2.4576× 1010

PROXINF-ADMM [12] 34379200 470984000 2.3093× 109

Proposed MU-PP-GDm 2459200 14755200 98368000O(MU−PP−GDm)

O(PROXINF−ADMM)7.15% 3.13% 4.26%

O(MU−PP−GDm)O(FITRA)

15.01% 0.1% 0.4%

in future wireless massive MIMO-OFDM communicationsystems.

B. COMPLEXITY COMPARISONAccording to the aforementioned closed-form expressions(see Section IV-B) and the configuration given in SectionV-A, it is possible to numerically assess the complexity ofthe proposed MU-PP-GD algorithm. The comparison withthe existing algorithms, PROXINF-ADMM [12] and FITRA[10], is given in Tab. 1. We consider to compute the com-plexity required by each algorithm (1) for one iteration, (2) toachieve an average PAPR of 4dB and (3) to achieve an aver-age PAPR of 3dB. As discussed in Section V-A, to achieve anaverage PAPR of 4dB, the MU-PP-GDm, PROXINF-ADMMand FITRA require 6, 20 and 800 iterations, respectively.While to achieve an average PAPR of 4dB, they need 40,100 and 1500 iterations, respectively. We have consideredthese iteration numbers to numerically assess the complexityof each algorithm.

According to results illustrated in Tab. 1, we can easilynote that MU-PP-GDm algorithm complexity is substantiallyless than that of the two considered algorithms. Note that theenergy-efficiency refers to the trade-off between the PAPRreduction gain and the computational complexity of the asso-ciated method [25]. Thus, our proposed algorithm providesbetter energy-efficiency among the other algorithms.

The comparison of the proposed MU-PP-GDm algorithmto FITRA [10] and PROXINF-ADMM [12] ones is summa-rized in Tab. 2, where we illustrate the enormous advantagesof our proposed scheme compared to these methods.

VI. CONCLUSIONIn this paper, we investigated the joint MU precoding andPAPR reduction in OFDM based massive MIMO downlinksystems. We developed an algorithm to perform jointly theMUI interference cancellation and PAPR reduction. The pro-posed algorithm, referred to as MU-PP-GDm, facilitates anexplicit trade-off between PAPR and transmission quality.The motivation of MU-PP-GDm is the high-dimensionalnull-space associated to the massive MIMO downlink chan-nel matrix, which enables as to device transmit signals withlow PAPR while maintaining excellent transmission qual-ity. The joint MU precoding and PAPR reduction schemewas formulated as a simple convex optimization problemfor which an algorithm based steepest gradient descent

method was designed. The MU-PP-GDm only involves sim-ple matrix-vector multiplications at each iteration, leadingthen to the lowest computational complexity among all exist-ing algorithms. A possibility for future work is to extend thisalgorithm to take into consideration, in addition to the PAPRreduction, other RF impairments (like PA non-linearity,... ).

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of base station antennas. IEEE Transactions on Wireless Communications,9(11):3590–3600, November 2010.

[2] E. G. Larsson, O. Edfors, F. Tufvesson, and T. L. Marzetta. Massive mimofor next generation wireless systems. IEEE Communications Magazine,52(2):186–195, February 2014.

[3] H. Q. Ngo, E. G. Larsson, and T. L. Marzetta. Energy and spectralefficiency of very large multiuser mimo systems. IEEE Transactions onCommunications, 61(4):1436–1449, April 2013.

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[5] K. N. R. S. V. Prasad, E. Hossain, and V. K. Bhargava. Energy efficiencyin massive mimo-based 5g networks: Opportunities and challenges. IEEEWireless Communications, 24(3):86–94, June 2017.

[6] S. K. Mohammed, A. Chockalingam, and B. S. Rajan. A low-complexityprecoder for large multiuser miso systems. In VTC Spring 2008 - IEEEVehicular Technology Conference, pages 797–801, May 2008.

[7] Christian Siegl and Robert FH Fischer. Selected basis for par reductionin multi-user downlink scenarios using lattice-reduction-aided precoding.EURASIP Journal on Advances in Signal Processing, 2011(1):17, Jul2011.

[8] H. Prabhu, O. Edfors, J. Rodrigues, L. Liu, and F. Rusek. A low-complexpeak-to-average power reduction scheme for ofdm based massive mimosystems. In 2014 6th International Symposium on Communications,Control and Signal Processing (ISCCSP), pages 114–117, May 2014.

[9] C. Ni, Y. Ma, and T. Jiang. A novel adaptive tone reservation schemefor papr reduction in large-scale multi-user mimo-ofdm systems. IEEEWireless Communications Letters, 5(5):480–483, Oct 2016.

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TABLE 2. Downlink Transmission Schemes : Comparison Summary

Algo. Advantages DisadvantagesFITRA [10] Low PAPR Slow convergence rate

High complexityNon null MUINon null OBRAppropriate regularization parameters are neededCSI is needed

PROXINF-ADMM [12] Low PAPR Quite fast convergence rateNull MUI High complexityNull OBR Appropriate regularization parameters are needed

Requires inversion of Mr ×Mr matricesCSI is needed

Proposed MU-PP-GDm Low PAPR CSI is neededNull MUINull OBRFast convergence rateAll parameters are adjusted automatically

[17] C. Zhang, Z. Li, L. Shen, F. Yan, M. Wu, and X. Wang. A low-complexitymassive mimo precoding algorithm based on chebyshev iteration. IEEEAccess, 5:22545–22551, 2017.

[18] H. Sun, J. Guo, and L. Fang. Improved singular value decomposition(topsvd) for source number estimation of low snr in blind source sepa-ration. IEEE Access, 5:26460–26465, 2017.

[19] T. Jiang, C. Ni, C. Xu, and Q. Qi. Curve fitting based tone reservationmethod with low complexity for papr reduction in ofdm systems. IEEECommunications Letters, 18(5):805–808, May 2014.

[20] H. Li, T. Jiang, and Y. Zhou. An improved tone reservation scheme withfast convergence for papr reduction in ofdm systems. IEEE Transactionson Broadcasting, 57(4):902–906, Dec 2011.

[21] X. Qin, Z. Yan, and G. He. A near-optimal detection scheme based onjoint steepest descent and jacobi method for uplink massive mimo systems.IEEE Communications Letters, 20(2):276–279, Feb 2016.

[22] Weishui Wan. Implementing online natural gradient learning: problemsand solutions. IEEE Transactions on Neural Networks, 17(2):317–329,March 2006.

[23] G.H. Golub and C. F. van Loan. Matric Computations, 3rd edition, johnshopkins univ., sweden,.

[24] A. Botev, G. Lever, and D. Barber. Nesterov’s accelerated gradient andmomentum as approximations to regularised update descent. In 2017International Joint Conference on Neural Networks (IJCNN), pages 1899–1903, May 2017.

[25] Y. Louet, D. Roviras, A. Nafkha, H. Shaiek, and R. Zayani. Globalpower amplifier efficiency evaluation with papr reduction method for post-ofdm waveforms. In 2018 15th International Symposium on WirelessCommunication Systems (ISWCS), pages 1–5, Aug 2018.

RAFIK ZAYANI (M’09) received the Engineer,M.Sc. and Ph.D. degrees from the Ecole NationaledâAZIngenieurs de Tunis (ENIT) in 2003, 2004,and 2009, respectively. He was with the Labora-tory of Communications Systems (SysâAZCom),ENIT, from 2003 to 2005. Since 2005, he has beenwith the InnovâAZCOM laboratory, SupâAZComSchool, Tunisia. From 2004 to 2009, he was withthe Department of Telecommunication and Net-working, Institut SupÃl’rieur dâAZInformatique

(ISI), Tunis, as a contractual Assistant Professor. Since 2009, he has beenan Associate Professor (tenure position) with the ISI, Tunisia. Since 2010,he has been an Associate Researcher with the CEDRIC Laboratory, Conser-vatoire National des Arts et Metiers, France. He is an Established Researcherwith long experience in multicarrier communications, energy efficiencyenhancement by: transmitter linearization techniques (baseband DPD) andPAPR reduction; high power amplifier characterization; neural network;identification modeling and equalization; and MIMO technologies. He wasinvolved in enhanced multicarrier waveforms, such as FBMC-OQAM,UFMC, GFDM, BF-OFDM, and WOLA-OFDM. He has contributed inseveral European (EMPHATIC) and French (WONG5) projects that aim atdesigning flexible air-interfaces for future wireless communications (5G andBeyond). He has recently been awarded a H2020 Marie Sklodowska-CurieActions (MSCA) Individual Fellowships (IF) grant for his ADMA5 projectproposal.

HMAIED SHAIEK received the Engineer degreefrom the National Engineering School of Tunis in2002, and the masterâAZs degree from the Uni-versity of Bretagne Occidentale in 2003, and thePh.D. degree from the Lab-STICC CNRS Team,Telecom Bretagne, in 2007. He was with CanonInc., until 2009. He left the industry to integratewith the ÃL’cole Nationale dâAZIngenieurs deBrest as a Lecturer, from 2009 to 2010. In 2011,he joined the CNAM, as an Associate Professor

in electronics and signal processing. He has authored or co-authored threepatents, six journal papers, and over 25 conference papers. His researchactivities focus on performances analysis of multicarrier modulations withnonlinear power amplifiers, PAPR reduction, and power amplifier lineariza-tion. He contributed to the FP7 EMPHATIC European project and is in-volved in two national projects, such as Accent5 and Wong5, funded bythe French National Research Agency. He has co-supervised three Ph.D.students and four master students.

VOLUME 4, 2016 11


DANIEL ROVIRAS was born in 1958. He re-ceived the Engineer degree from SUPELEC, Paris,France, in 1981, and the Ph.D. degree fromthe National Polytechnic Institute of Toulouse,Toulouse, France, in 1989. He spent in the indus-try as a Research Engineer for seven years. Hejoined the Electronics Laboratory, Ecole NationaleSuperieure dâAZElectrotechnique, dâAZElectron-ique, dâAZInformatique, et des Telecommunica-tions (ENSEEIHT). In 1992, he joined the Engi-

neering School, ENSEEIHT, as an Assistant Professor, where he has beena Full Professor since 1999. Since 2008, he has been a Professor withthe Conservatoire National des Arts et MÃl’tiers (CNAM), Paris, France,where his teaching activities are related to radio-communication systems.He is currently a member of the CEDRIC Laboratory, CNAM. His researchactivity was first centered around transmission systems based on infraredlinks. Since 1992, his topics have widened to more general communicationsystems, such as mobile and satellite communications systems, equalization,and predistortion of nonlinear amplifiers, and multicarrier systems.

12 VOLUME 4, 2016

Documents

PAPR-aware Massive MIMO-OFDM Downlink · In this paper, we develop a downlink transmission scheme to address the PAPR reduction problem for OFDM-based massive MIMO wireless systems,