low-complexity compressive sensing detection for spatial...

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 63, NO. 7, JULY 2015 2565

Low-Complexity Compressive Sensing Detectionfor Spatial Modulation in Large-Scale

Multiple Access ChannelsAdrian Garcia-Rodriguez, Student Member, IEEE, and Christos Masouros, Senior Member, IEEE

Abstract—In this paper, we propose a detector, based on thecompressive sensing (CS) principles, for multiple-access spatialmodulation (SM) channels with a large-scale antenna base sta-tion (BS). Particularly, we exploit the use of a large number ofantennas at the BSs and the structure and sparsity of the SMtransmitted signals to improve the performance of conventionaldetection algorithms. Based on the above, we design a CS-baseddetector that allows the reduction of the signal processing load atthe BSs particularly pronounced for SM in large-scale multiple-input–multiple-output (MIMO) systems. We further carry outanalytical performance and complexity studies of the proposedscheme to evaluate its usefulness. The theoretical and simulationresults presented in this paper show that the proposed strategyconstitutes a low-complexity alternative to significantly improvethe system’s energy efficiency against conventional MIMO detec-tion in the multiple-access channel.

Index Terms—Spatial modulation, large-scale MIMO, multipleaccess, compressive sensing, energy efficiency.

I. INTRODUCTION

THE exponential growth of the data rates in wireless com-munications has caused a significant increase in the total

energy consumption required to establish the communicationlinks [1]. This is because novel transceiver structures with ahigher number of antennas, transmission power or complexityin their signal processing algorithms have been designed toaccommodate this growth [1], [2]. For this reason, the energyefficiency (EE) of the multi-user wireless transmission con-stitutes one of the main areas of research interest at present[1], [2]. Technologies such as large-scale MIMO and SM havebeen developed with the main objective of satisfying the EE re-quirements of future wireless communication systems [3], [4].

Massive MIMO technologies increase the EE by incorporat-ing a large number of antennas at the BSs [4]–[7]. This leadsto communication systems in which the use of conventional

Manuscript received July 14, 2014; revised March 4, 2015; accepted May 8,2015. Date of publication May 18, 2015; date of current version July 13,2015. This work was supported in part by the Royal Academy of Engineering,U.K., and in part by EPSRC under Grant EP/M014150/1. The material in thispaper was presented in part at the IEEE International Conference on Acoustics,Speech and Signal Processing, Brisbane, Australia, April 2015, and the IEEEInternational Conference on Communications, London, U.K., June 2015. Theassociate editor coordinating the review of this paper and approving it forpublication was A. Ghrayeb.

The authors are with the Department of Electronic and Electrical Engineer-ing, University College London, London WC1E 6BT, U.K. (e-mail: [email protected]; [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TCOMM.2015.2434817

linear detection and precoding techniques becomes optimal inthe large-scale limit [4], [5]. However, the increased numberof radio frequency (RF) chains have a considerable influenceon the EE, hence severely affecting the large-scale benefitsfrom this perspective [8]. To alleviate this impact, SM posesas a reduced RF-complexity scheme by exploiting the transmitantenna indices as an additional source of information [3], [9].Intuitively, instead of activating all the antennas simultaneouslyas in conventional MIMO transmission, SM proposes to switchon a subset of them and modify the receiver’s operation todetect both the active antenna indices and the amplitude-phasesymbols. This reduces the number of RF chains when comparedto conventional MIMO systems at the cost of decreasing themaximum achievable rates [3], [9], [10].

So far, the literature of SM has mostly focused on de-veloping strategies for point-to-point links [3], [9]–[15]. Forinstance, several low-complexity detectors that approach theperformance of the optimal maximum likelihood (ML) esti-mation have been proposed [15]–[18]. In this context, the useof a normalized CS detection algorithm as a low-complexitysolution for space shift keying (SSK) and generalized spaceshift keying (GSSK) peer-to-peer (P2P) systems was introducedin [18]. In this work, the authors apply a normalization to thechannel matrix before the application of the greedy compressivedetector to improve performance. However, the authors restrictits application to single-user SSK and GSSK systems, whichconstrains its use to low data rate transmission. The perfor-mance of the normalized CS detection for SSK and GSSKhas been recently enhanced in [19], where the improvement isobtained by pre-equalizing the received signal.

More recently, the use of SM has been extended to themultiple access channel (MAC) as a way of enhancing theachievable rates of the conventional single-antenna devicesconsidered in this setting [20]–[22]. The literature related tothe study of multiple access SM builds upon [20], [21], wherea characterization of the bit error rate (BER) probabilities ofthe optimal ML detector for both spatial and SSK modulationsis performed. However, the signal processing load of the MLdetector makes it impractical in the MAC as it grows exponen-tially with this parameter [20]. In this setting, several detectionschemes to reduce the complexity of the ML detector in theMAC have been proposed and studied in [22], where the focusis on Q-ary SSK modulation.

A number of related works have concentrated on the designof detection schemes to account for the particularities of thelarge-scale MAC. The development of detection schemes in

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2566 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 63, NO. 7, JULY 2015

this setting is motivated by the intractable complexity of non-linear detectors such as the sphere decoder when a high numberof antennas and users are considered [4], [15]. The use of amessage passing detection (MPD) algorithm is proposed in[23] for the MAC with a high number of antennas at theBS. This algorithm offers a performance improvement withrespect to conventional MIMO systems with the same spectralefficiency. However, both the storage requirements and the totalnumber of operations are conditioned by the high number ofmessages transmitted between all the nodes, which must beupdated in every iteration [24]. A more complex local searchdetection algorithm based on finding the local optimum in termsof the maximum likelihood cost is also introduced in [23].An iterative detector for large-scale MACs is developed in[25]. Here, the authors decouple the antenna and symbol es-timation processes to reduce the global detection complexity.The algorithm introduced in [26] accounts for the sparsity andsignal prior probability of SM transmission in the MAC. In thiswork, the use of stage-wised linear detection is discarded dueto its high complexity and the authors propose a generalizedapproximate message passing detector. A related approach hasbeen developed in [27] to deal with quantized measurementsand spatial correlation. Still, the above algorithms do not fullyaccount for the particularities of iterative detection processesand the complexity benefits that can be obtained by leveragingthe principles behind CS algorithms.

In this paper we propose a low-complexity detector basedon CS for the MAC of SM systems with large-scale BSs. Inparticular, we show that the signal structure of SM in the MACcan be exploited to provide additional information and improvethe performance of the CS algorithms [28], [29]. Moreover, theuse of a high number of antennas in the MAC allows us to elim-inate the error floor that greedy CS techniques show in noisyscenarios for practical uncoded BERs [18]. Indeed, contrary tothe common CS knowledge, in this paper we show by meansof a thorough complexity analysis that the trade-off betweencomplexity and performance is especially favorable for CS-based detection schemes in scenarios with a high number ofreceive antennas.

Furthermore, in this paper we compare the EE and signalprocessing (SP) complexity of the proposed strategy with theconventional zero forcing (ZF) and minimum mean squareerror (MMSE) linear detectors. In particular, we show thatthe improvements offered by the proposed technique allowenhancing the EE achieved with these detectors while re-ducing their computational complexity. In fact, the detailedcomplexity analysis leads us to derive interesting conclusionsregarding the algorithms that must be used to solve the ZF andMMSE detection problems in large-scale SM-MIMO systems.To summarize, the contributions of this paper can be stated asfollows:

• We propose and validate the use of CS-based algorithmsas an efficient alternative to recover SM signals underlarge-scale MIMO conditions. Moreover, we design aCS-based detector specifically tailored for the MAC ofSM systems to improve the performance and the EE ofthe detectors conventionally used for large-scale MIMO.

• As opposed to the previous literature, we perform ananalytical characterization of the computational com-plexity of the iterative CS algorithm to determine theconditions in which the use of CS-based detection in SMis especially beneficial.

• We carry out a mathematical analysis of the performanceand convergence of the CS greedy algorithms whenapplied to the proposed large-scale MAC scenario.

• We use the above complexity and performance analysesto characterize the EE of the proposed, which is shownto be superior to that of the existing linear detectors.

II. PRELIMINARIES

A. System Model of the Multiple Access Channel (MAC)

The model considered throughout this paper characterizesthe MAC of a multi-user MIMO system comprised of K mobilestations (MSs) with nt antennas each, and a single BS with Nreceive antennas. The total number of antennas allocated at theMSs is denoted as M = K × nt. The behavior of the multipleaccess system can be described by

y = Hx + w, (1)

where y ∈ CN×1 is the signal received at the BS, and x ∈ C

M×1

denotes the signal transmitted by the MSs. Moreover, w ∈C

N×1 ∼ CN (0, σ 2n IN) denotes the standard additive white-

Gaussian noise vector with variance σ 2n , and H ∈ C

N×M ∼CN (0, IN ⊗ IM) is a matrix whose m, n-th complex coefficient,hm,n, represents the frequency flat fading channel gain betweenthe n-th transmit antenna and the m-th receive antenna. In theprevious expressions, ∼ indicates “distributed as”, IA is anA × A identity matrix, and ⊗ denotes the Kronecker product[30]. As typically assumed in the SM detection literature, theBS is expected to have a perfect knowledge of the communica-tion channel H [3], [4].

Throughout this paper we assume that the data symbolstransmitted by the active antennas belong to a normalizedQ-QAM constellation satisfying E[Es] = 1, where Es refers tothe symbol energy and the operator E denotes the expectedvalue. Based on this, the total average signal-to-noise ratio(SNR) of the MAC ρ can be expressed as

ρ = E[xHx]σ 2

n= S · E[Es]

σ 2n

, (2)

where (·)H denotes the Hermitian transpose and S ≤ M is thetotal number of antennas simultaneously active amongst the MSs.

B. Multiple Access Spatial Modulation

SM and its generalized version reduce the hardware com-plexity of multiple-antenna devices by limiting the number ofactive antennas per user and conveying additional informationonto their spatial position [3], [9]. In this section we focus ondescribing the operation of generalized SM transmission witha single RF chain since particularization to conventional SMis straightforward by letting S = K and forcing the number of

GARCIA-RODRIGUEZ AND MASOUROS: CS DETECTION FOR SM IN LARGE-SCALE MULTIPLE ACCESS CHANNELS 2567

antennas per user to be a power of two [9], [31]. Throughout thispaper we use the term SM when referring to both conventionaland generalized SM for ease of description. In SM, each trans-mitter conveys the same constellation symbol by activating agiven number of antennas na according to the input bit sequence[3], [9], [31]. Without loss of generality, in the following weassume that the users activate the same number of antennas,i.e. we have S = na × K. Mathematically, the transmit sig-nal xu ∈ C

nt×1 of the u-th SM transmitter can be expressedas [9]

xu =[0 · · · sq

l1· · · sq

lk· · · 0

]T, (3)

where lk ∈ [1, nt] denotes the active antenna index and sq

represents the q-th symbol of the transmit constellation B. Thenumber of bits that can be encoded on the antenna indices is�log2

(ntna

)� [8], [31]. In the previous expression,(··)

denotes thebinomial operation and �·� is the floor function. Therefore, thenumber of possible antenna combinations at the transmitter isgiven by r = 2b, where b � �log2

(ntna

)�. This determines thecardinality of A, the set comprised of the possible antennagroups with Al being the l-th element. Note that there may beinvalid antenna groups to preserve an integer length of the bitstream. The composite transmit vector x ∈ C

M×1 is obtained byconcatenating the transmit signals as x = [xT

1 xT2 · · · xT

K]T.

When compared with conventional MIMO technologies,SM reduces the inter-user interference and the circuit powerconsumption of the MSs for the same number of antennas.However, this causes a reduction of the achievable rates withrespect to conventional MIMO for the same number of antennas[3]. Particularly, while a conventional MIMO transmitter is ableto convey BMIMO = nt · log2(Q) bits, a single SM transmitterencodes BSM = b + log2(Q) bits in every channel use. At thereceiver, detection schemes exploit the channel knowledge todetermine the active antennas and the conveyed constellationsymbols [3]. Among these, the optimum detector follows theML criterion and its output reads as

x = arg minsp

∥∥y − Hsp∥∥2

2 . (4)

Here, the signal sp ∈ CM×1 belongs to the set that includes

all the possible transmit signals � and ‖ · ‖p denotes the �p

norm. The cardinality of �, |�| = (Q × r)K , exponential withthe number of users K, establishes an upper bound on thecomplexity of SM detection.

C. Large-Scale MIMO and Low-Complexity Detection

The large-scale MIMO theory focuses on analyzing the ben-efits of communication systems with a high number of antennasat the BSs [4], [5]. One of the fundamental results in this fieldstates that, provided that N M, the received signal after lineardetection g ∈ C

M×1 satisfies

g = D(Hx + w)a.s.−−−−→

N→∞, M=const.x, (5)

where D ∈ CM×N is a linear detection matrix that for the

matched filter (MF), ZF and MMSE detectors read as [32]

DMF = HH, (6)

DZF = H† = (HHH)−1

HH, (7)

DMMSE = (HHH + ςI)−1HH . (8)

In the above, ς = M/ρ [32], (·)† denotes the pseudoinverseoperation and (·)−1 refers to the inverse matrix. Let g{u} be thedecision vector corresponding to the u-th user. In the followingwe adopt the sub-optimal but low-complexity approach ofdecoupling the estimation of the spatial and amplitude-phasemodulated symbols [22], [33]. Specifically, the estimated activeantenna indices Al and the transmitted constellation symbol qfor the u-th user are obtained from (5) as

Al = arg maxl

∥∥∥g{u}Al

∥∥∥2, (9)

q =D(

g{u}Al

), (10)

where g{u}{Al,Al} represent the entries of the decision vector of the

u-th user g{u} determined by the sets Al and Al respectively,and D denotes the demodulation function.

Note that the necessary increase in the number of transmitantennas when SM is used degrades the performance of theabove-mentioned detectors due to the worse conditioning of thechannel matrix. For this reason, in this paper we propose a low-complexity solution inspired by CS to take advantage of thelarge-scale MIMO benefits while, at the same time, reducingthe detection complexity. In other words, in the following welook at scenarios where N M does not necessarily hold butN K brings the massive MIMO effect.

III. THE TRIVIAL APPROACH: DIRECT APPLICATION

OF THE CS ALGORITHMS FOR SM DETECTION

The main issue with the conventional ZF and MMSE lineardetectors when applied to SM and generalized SM detectionis that the entire channel matrix H ∈ C

N×M must be used fordetection even though only S columns contribute to the acquisi-tion of the amplitude-phase signal information. We circumventthis by exploiting the sparsity of SM signals to reduce thecomplexity of linear detectors. The signals conveyed by SMare defined as S-sparse because they only contain S � M non-zero entries equal to the number of antennas simultaneouslyactive S [18]. This property has been exploited by CS toimprove signal estimation from compressive measurements.Specifically, CS capitalizes on signal sparsity to guarantee areliable signal recovery with efficient algorithms [34], [35].The CS measurements y ∈ R

N×1 of a sparse signal x can beexpressed as [35]–[37]

y = �x + e, (11)

where x ∈ RM×1 represents the original sparse signal, � ∈

RN×M is the measurement matrix, and e ∈ R

N×1 is a


measurement error term. Note that the complex-valued systemin (1) can be straightforwardly re-expressed to resemble thereal-valued one in (11) [18]. In this paper, we exploit thesimilarity between (1) and (11) to improve the detection per-formance of the conventional linear MIMO detectors.

In CS, the restricted isometry property (RIP) determineswhether signal recovery guarantees are fulfilled or not for anycommunication channel � = H [35], [36]. For the case of theMIMO channel, the RIP of order S is satisfied for a channelmatrix H if, for any S-sparse signal x, the relationships

(1 − δS)‖x‖22 ≤ ‖Hx‖2

2 ≤ (1 + δS)‖x‖22 (12)

hold for a constant δS ∈ (0, 1). For instance, a matrix comprisedof independent and identically distributed Gaussian randomvariables is known to satisfy δS ≤ 0.1 provided that N ≥cS log(M/S), with c being a fixed constant [36]. Note that thiskind of channel communication matrix conventionally arises inrich scattering environments with Rayleigh fading [4].

Once the signal measurements are acquired and contrary tothe ML detector given in (4), the detection of the SM signalsin CS relies on the sparsity of SM transmission to generate anestimate. For the case with δ2S <

√2 − 1 [37] we solve (4) in a

low-complexity CS-inspired fashion as

minimize ‖x‖1

subject to ‖Hx − y‖2 ≤ μ, (13)

In the above expression the constant μ limits the noise power‖w‖2 ≤ μ. Although the above optimization problem can besolved with well-known convex approaches, these alternativesare often computationally intensive, so faster techniques thatoffer a trade-off between performance and complexity such asgreedy algorithms are commonly used instead [36].

From the vast variety of CS greedy algorithms, in this sectionwe choose one of the most efficient schemes to approximate thesolution of (13): the Compressive Sampling Matching Pursuit(CoSaMP) [38]. The CoSaMP is a low-complexity algorithmthat goes through an iterative reconstruction process to recoverboth the active antenna indices and the amplitude-phase in-formation of the transmitted signals. Moreover, this algorithmprovides optimal error guarantees for the detection of sparsesignals since, similarly to the more complex convex algorithms,a stable signal recovery is guaranteed under noisy conditionswith a comparable number of receive antennas [38]. In ourresults we show that the large number of antennas at the BS ben-efits the use of this algorithm for SM detection by performinga thorough complexity analysis as opposed to [18], [38]. More-over, as the structure of the transmitted signals in the MAC isnot accounted for in the generic CS detection, in the followingwe consider an approach specifically tailored for the consideredscenarios to improve the detection performance.

IV. PROPOSED ENHANCED CS TECHNIQUE: SPATIAL

MODULATION MATCHING PURSUIT (SMMP)

One of the key characteristics of the conventional greedy CSalgorithms is that no prior knowledge of the sparse signal other

Algorithm 1 Spatial Modulation Matching Pursuit

Inputs: H, y, S, na, imax.1: Output: xiend � S-sparse approximation2: x0 ← 0, i ← 0 {Initialization}3: while halting criterion false do4: r ← y − Hxi {Update residual}5: i ← i + 16: c ← HHr {MF to estimate active antenna indices}

{7–11: Detect na indices with highest energy per user}7: � ← ∅

8: for j = 1 → K do9: M ← {(j − 1) · nt, . . . , j · nt − 1}

10: � ← M(arg max{|c|M|}na) ∪ �

11: end for{12–13: Detect remaining k− S highest-energy indices}

12: c(�) ← 013: � ← arg max{|c|}(k−S) ∪ �

14: T ← � ∪ supp(xi−1) {Merge supports}15: b|T ← H†

T y {Least squares problem}16: b|T C ← 0

{17–21: Obtain next signal approximation}17: xi ← 018: for j = 1 → K do19: M ← {(j − 1) · nt, . . . , j · nt − 1}20: xi|M(arg max {|b|M|}na ) ← max {|b|M|}na

21: end for22: end while

than the number of non-zero entries is assumed. However, whenapplied to the proposed scenario, this condition can generatesituations in which the output of the detector does not havephysical sense. For instance, the detected signal could havemore than one active antenna per user, which is not possiblewhen conventional SM modulation is used [3]. This undesiredoperating condition is caused by the noise and inter-user in-terference effects that arise in the MAC. To mitigate these, inthis sub-scheme we incorporate the additional prior knowledgeabout the distribution of the non-zero entries in the transmittedsignal to further enhance performance [28].

The detection algorithm considered in this paper is referredto as spatial modulation matching pursuit (SMMP) to explicitlyindicate that it corresponds to a particularization to SM op-eration of the structured CoSaMP iteration developed in [29].In particular, SMMP reduces the errors in the identification ofthe active antennas by exploiting the known distribution of thenon-zero entries [28], [29]. This also improves the convergencespeed of the algorithm as less iterations are required to deter-mine the active antenna indices. While it is intuitive that theconcept behind this strategy could be incorporated to other CSdetection algorithms, in the following we focus on the CoSaMPalgorithm for clarity.

The pseudocode of the proposed strategy is shown inAlgorithm 1 for convenience and its operation can be describedas follows [29], [38]. The algorithm starts by producing anestimate of the largest components of the transmitted signal to


identify the active antennas [38]. For this, the algorithm em-ploys the residual signal r ∈ C

N×1 given by

r � y − Hxi = H(x − xi) + w, (14)

where xi ∈ CM×1 is the approximation of the transmit signal at

the i-th iteration. The residual signal concentrates the energy onthe components with a largest error in the estimated receivedsignal y = Hxi [25], [39]. The decision metric used to deter-mine the plausible active antennas c ∈ C

M×1 is obtained as theoutput of a MF and can be expressed as

c = HHr. (15)

From this decision metric, the active antenna estimation processforms a set � of decision variables with cardinality |�| =k ≥ S. The set � provides an estimate of the plausible activeantennas. Note that these entries do not have to correspond tothe S coefficients with highest estimated energy in the transmit-ted signal as in the CoSaMP algorithm [38]. For instance, letx = [0, 0,−1, 0|0, 1, 0, 0] be the signal transmitted in a MACwith K = S = 2 users with nt = 4 antennas each and BPSKmodulation. The total number of transmit antennas is M = nt ×K = 8. Then, the set � in the conventional CoSaMP algorithmcould be formed by any S = 2 entries of the set of integersϒCoSaMP = {1, 2, . . . , 8} without any restriction. This contrastswith the considered SMMP algorithm, in which the set � isformed by at least na = 1 entry of the set ϒ1

SMMP = {1, 2, 3, 4}and another one from ϒ2

SMMP = {5, 6, 7, 8}. In simple terms,the proposed algorithm forces that at least na entries per userare selected as we exploit the knowledge that

‖xk‖0 = na, k ∈ [1, K], (16)

where ‖ · ‖0 determines the number of non-zero entries [34],[36]. This is represented in lines 7–11 of Algorithm 1, wherethe arg max{·}p and max{·}p functions return the indices andthe entries of the p components with largest absolute valuein the argument vector, and ∅ denotes the empty set. Theremaining k − S entries are instead selected as the entries withhighest energy independently of the user distribution. Theseadditional entries aim at improving the support detection pro-cess by involving the LS problem, which offers an enhancedperformance in the antenna identification with respect to theestimate provided by the MF [4].

Once the entries with highest error energy in the currentresidual have been estimated, the set T is obtained as

T � � ∪ supp(xi−1), (17)

where supp(·) identifies the indices of the non-zero entries. Thisset provides a final estimate of the plausible active antennasused for transmission by incorporating the ones considered inthe previous iteration [39]. Therefore, the set T determines thecolumns of the matrix H used to solve the unconstrained leastsquares (LS) problem given by

minimizeb|T

‖HT b|T − y‖22 → b|T = H†

T (Hx + w), (18)

where b|T denotes the entries of b ∈ CM×1 supported in T .

This notation differs from HT , which refers to the submatrixobtained by selecting the columns of H determined by T .

The LS approximation is a crucial step as the complexityreduction and the performance improvement w.r.t. the conven-tional linear alternatives depend on the efficiency of this process[38]. This procedure can also be seen as implementing a ZFdetector in which, instead of inverting all the columns of thechannel matrix, only the columns that have been previouslyincluded in the support are inverted. This allows exploitingthe large-scale MIMO detection benefits as the equivalent ZFdetector generally satisfies N 2k [5]. Due to the possibilityof using different algorithms to solve this problem, a detailedanalysis of their complexity is developed in Section V-A.

After solving the LS problem, the signal approximation ofthe i-th iteration xi is built by selecting the entries with highestenergy at the output of the LS problem based on a user-by-user criterion following (16). Finally, the sparse output of thealgorithm xiend is obtained after the algorithm reaches the maxi-mum pre-defined number of iterations imax or a halting criterionis satisfied [38]. Overall, although sub-optimal for a large butfinite number of antennas, the proposed scheme exploits thehigh performance offered by linear detection schemes togetherwith the structured sparsity inherent to SM transmission toreduce complexity. We also note that, although shown viaiterative structures in Algorithm 1, the additional operationsrequired in the considered algorithm can be implemented viavector operations with reduced computational time.

Regarding the compromise between complexity and perfor-mance of the considered technique, it should be noted thatseveral parameters can be modified to adjust this trade-off [38]:

• Maximum number of iterations of SMMP (imax): Thetotal number of iterations determines the complexity anddetection accuracy of the algorithm. This parameter canbe used to adjust the performance depending on thecomputational capability of the BS as shown hereafter.

• Number of entries detected at the output of the MF (k):The parameter k determines the dimensions of the LSproblem, hence severely affecting the SP complexity.It also influences the detection performance since thesolution is conditioned by the LS matrix HT .

• Maximum number of iterations of the iterative LS (ilsmax):The accuracy of the LS solution is improved in everyiteration when iterative algorithms are used [41]. Hence,there exists a trade-off between complexity and perfor-mance that can be optimized at the BS depending on thecommunication requirements. As this parameter greatlyaffects the global complexity, a detailed study of the re-quired number of iterations is developed in Section VI-A.

At this point we also point out that the main difference ofthe proposed approach with respect to the typical CS is thatwe also consider over-determined scenarios, i.e. communica-tion systems where N ≥ M. As shown hereafter, this entailsthat SMMP is not only able to reduce the complexity of theconventional linear detectors in scenarios where N ≥ M, but italso provides performance guarantees when this condition doesnot hold.


TABLE ICOMPLEXITY IN NUMBER OF REAL FLOATING-POINT OPERATIONS (FLOPS) TO SOLVE A m × n LEAST SQUARES PROBLEM

V. COMPLEXITY ANALYSIS

The analysis of the computational complexity is commonlyperformed by determining the complexity order. This is the ap-proach adopted in [18], [23], [38] to determine the complexityof the proposed algorithms. Instead, in this paper we adopt amore precise approach since, as shown in the following, thecomplexity order does not provide an accurate characterizationof the total number of operations due to the iterative natureof the greedy CS-based algorithms. In fact, as opposed to theresults obtained in [18] and [38], here we conclude that the LSproblem can dominate the global complexity.

A. Complexity of the Least Squares Problem

An efficient implementation of the LS algorithm is requiredto reduce the complexity of the proposed approach sinceotherwise it can dominate the total number of floating pointoperations (flops) [38]. In general, the methods to solve the LSproblem can be classified according to their approach to obtainthe solution into direct and iterative procedures [41]:

• Direct methods include the QR and the Cholesky decom-positions and they are based on producing a system ofequations that can be easily solved via backward andforward substitutions. The total number of flops is con-ditioned by the costly decompositions that must be per-formed at the beginning of each coherence period [43].

• Iterative methods solve the LS problem by refining aninitial solution based on the instantaneous residual [41].These approaches prevent the storage intensive decom-positions required by direct methods, an aspect especiallybeneficial in the proposed scenario due to the largedimensions of the channel matrix H.

The number of flops of the QR decomposition, Choleskydecomposition, Richardson iteration and conjugate gradient(CG) LS algorithms are detailed in Table I. Note that thefact that each complex calculus involves the computation ofseveral real operations have been taken into account in Table I.Moreover, for simplicity, it has been considered that a realproduct (division) has the same complexity of a real addition(subtraction), a usual assumption in the related literature [40].

Regarding the complexity of the Richardson iteration, notethat it depends on a parameter 0 < α < 2/λ2

max(A) that de-termines the convergence rate of the algorithm [41]. In thelast expression, λmax(A) denotes the maximum singular valueof an arbitrary matrix A. The determination of this parameterbecomes necessary if a high convergence speed is required andthese additional operations are included in Table I. Concerningthe CG algorithm, Table I shows that the complexity differsdepending on the availability of an initial approximation to theLS solution [41]. This difference must be considered because,as opposed to the other detectors, the complexity reductionof the greedy CS-based approach is based on the increasingaccuracy of the approximations as the algorithm evolves, henceimproving the convergence speed as explained in Section IV.

The comparison between the direct LS methods shows that,even though the QR decomposition is more numerically accu-rate, it also is more complex than the Cholesky decomposition[43]. Therefore, the Cholesky decomposition is preferred forthe considered scenario due to the high dimensions of the matri-ces involved in the LS problem [43]. Regarding the complexityof the iterative methods, the results in Table I describe thereduction of the complexity order when compared to the directmethods pointed out in [38]. However, this improvement doesnot guarantee a reduction in the number of operations as thetotal complexity is highly dependent on the number of iterationsrequired until convergence as shown in Section VIII.

B. Overall Complexity of the Proposed Algorithm

Based on the above analyses, a tight upper bound on the totalnumber of real flops of the ZF and SMMP detectors is shown inTable II. The complexity of the ZF detector is solely determinedby the operations required to solve the LS problem. Therefore,the increase in the number of transmit antennas of SM may havea significant impact on the total complexity. This is because,while M = K for the conventional large-scale MIMO scenarioswith single-antenna users, the relationship M = K × nt is satis-fied for SM transmitters in the MAC. However, the performanceimprovements offered by SM justify the complexity increaseat the BSs, where computational resources are expected to beavailable [20].


TABLE IICOMPLEXITY IN NUMBER OF REAL FLOATING-POINT OPERATIONS (FLOPS) OF DIFFERENT LARGE-SCALE MIMO DETECTORS

TABLE IIICOMPLEXITY AND BER OF A BASE STATION WITH N = 128,

K = S = 12, nt = 8, imax = 2 AND SNR = 5 dB

Additionally, we point out that the flop calculation for theproposed algorithm has been performed for the worst-casescenario in which none of the S entries from the previousSMMP iteration coincide with the k coefficients selected atthe output of the MF. This means that the average number ofoperations is generally smaller than the one shown in Table II.In spite of this, this tight upper bound allows us to determinethe conditions under which CS-based detection is convenient.

The total number of real flops and the BERs for a specificlarge-scale MIMO scenario are represented in Table III asan illustrative example. The relevant system parameters areN = 128, K = S = 12, nt = 8, imax = 2 and SNR = 5 dB. Thenumber of CG iterations ilsmax is set to ensure the maximumattainable performance. Overall, it can be seen that the SMMPalgorithm offers a considerable performance improvement withreduced complexity. The results of Table III, which are furtherdeveloped in Section VIII, also lead us to conclude that the useof iterative LS algorithms for conventional linear detection maybe able to reduce the detection complexity over a channel coher-ence time. Note that the solution of the LS problem is not exactin these cases, which influences the resulting BERs as shownin Table III. This results in a performance improvement for theconventional ZF detector due to the energy concentration onthe components that correspond to the active antennas. Indeed,the significant complexity benefits attained by the use of iter-ative LS algorithms motivate the study of their convergencespeed in the following section.

VI. CONVERGENCE AND PERFORMANCE ANALYSES

A. Convergence Rate of the Iterative LS in Large-Scale MIMO

The complexity of the proposed algorithm highly dependson the number of iterations required to obtain a solution tothe LS problem. Although the required number of iterations to

achieve convergence cannot be known in advance due to therandom nature of the communication channel, in this sectionwe derive two expressions to determine the number of iterationsdepending on the required output accuracy: a straightforwardbut less accurate one based on an asymptotic analysis, and amore complex and precise one that resorts to the cumulative dis-tribution function (CDF) of the condition number. This offersa more refined and intuitive approach than the one adopted in[38] as the resultant expressions directly depend on the numberof antennas of the communication system. Furthermore, the fol-lowing study can be used to determine whether it is convenientfrom a numerical perspective to use iterative LS algorithmsto solve the ZF problem when applied to large-scale MIMOscenarios without SM.

The convergence rate of the iterative LS methods is deter-mined by the condition number of the LS matrix HL ∈ C

N×|L|[38], [41]. Throughout this section, L is a set defined as L � Twhen the SMMP algorithm is used whereas it is given byL � {1, 2, . . . , M} for ZF. The standard condition number � ∈[1,∞] of the matrix HL is defined as [41], [44]

�(HL) = λmax(HL)

λmin(HL)=√

σmax(HH

LHL)

σmin(HH

LHL) = √ (W), (19)

where λmax and λmin denote the maximum and minimum sin-gular values of the argument matrix respectively, σmax and σmindenote the maximum and minimum eigenvalues, and (W)

refers to the modified condition number defined in [44]. In thelast equality, W ∈ C

s×s is a Wishart matrix with t degrees offreedom W ∼ CWs(t, Is) defined as [44], [45]{

W � HLHHL , if N ≤ |L|,

W � HHLHL, otherwise.

(20)

In the above, s = min(N, |L|) and t = max(N, |L|).The conditions of large-scale MIMO greatly favor the use

of the iterative algorithms thanks to the reduced differencebetween the maximum and minimum singular values of thecommunication channel, which in turn ensures a fast con-vergence [4], [41]. This effect is enhanced by the proposedalgorithm due to the operation with a better conditioned matrixthan conventional ZF or MMSE detectors. In other words, eventhough several LS problems are solved when the SMMP algo-rithm is used, the overall complexity can be reduced w.r.t. the


conventional ZF or MMSE detectors because the dimensionsof the LS matrices are smaller and the convergence speed of thealgorithm is faster. In fact, we argue that the effectiveness ofthe CS greedy algorithms in terms of complexity is maximizedwhen a high number of antennas at the BS is considered due tothe faster convergence when compared to conventional lineardetectors.

To characterize the above-mentioned convergence rate, inthis paper we use the results of conventional and large-scaleMIMO theory related to the distribution of the condition num-ber of complex Gaussian and Wishart matrices [44], [45].Throughout, we focus on the CG algorithm for brevity, but sim-ilar conclusions can be derived for other iterative LS methods[38], [41]. In particular, the residual generated by each iterationof the CG algorithm satisfies [41]∥∥∥bi − H†

Ly∥∥∥

2≤ 2 · �i ·

∥∥∥b0 − H†Ly∥∥∥

2, (21)

where the index i denotes the iteration number, bi is the LSapproximation at the i-th iteration and � is defined as

� = �(HL) − 1

�(HL) + 1. (22)

Based on the above result, an upper bound in the number of CGiterations can be expressed as [41]

ils <1

2�(HL) · log

(2

ε

), (23)

where ε is the relative error defined as

ε =∥∥∥bi − H†

Ly∥∥∥

2∥∥∥b0 − H†Ly∥∥∥

2

. (24)

From the above expressions it can be concluded that the con-vergence speed of the iterative LS methods increases when asmaller number of entries are selected at the output of the MFin the proposed algorithm. This is because the high number ofantennas available at the BS greatly favors the convergence ofthe CG due to the reduced condition number. This is character-ized by the following proposition on the convergence of the CGalgorithm in the large antenna number limit.

Proposition 1: An upper bound in the number of iterationsrequired by the LS CG algorithm to achieve a relative errorreduction of ε under large-scale MIMO conditions is given by

ils <1

2·∣∣∣∣1 + √

β (|L|)1 − √

β (|L|)∣∣∣∣ · log

(2

ε

), (25)

where the function β(V), V ∈ Z+ is defined as{

β(V) = N/V if N ≥ V,

β(V) = V/N otherwise.(26)

Proof: The large-scale limit theory establishes that thecondition number of a channel matrix H ∈ C

N×k with entrieshm,n ∼ CN (0, 1) independent and identically distributed (i.i.d.)

Fig. 1. Empirical CDF and limit condition number for (a) N = 128, |L| = 8and (b) N = 128, |L| = 32.

converges almost surely in the asymptotic limit of transmit andreceive antennas to [45, Theorem 7.3]

�(HT ) −−−−→N,|L|→∞

1 + √1/β (|L|)

1 − √1/β (|L|) =

∣∣∣∣1 + √β (|L|)

1 − √β (|L|)

∣∣∣∣ . (27)

Equation (27) provides a useful approximation to determine themaximum condition number of a Rayleigh channel with a highnumber of receive antennas [4]. This can be seen in Fig. 1,where the CDF of the condition number of a Rayleigh fadingchannel matrix with N = 128 receive antennas is depicted.The number of columns is |L| = 8 and |L| = 32 for Fig. 1(a)and (b) respectively. From the results of this figure it can beconcluded that the condition number of the channel matrix isbelow the bound shown in (27) with a high probability. Toconclude the argument, (27) is substituted into (23). �

In spite of being valid in a high number of cases, theprobability of requiring a higher number of iterations for certainbadly conditioned channels cannot be quantified with the aboveapproximation. For this reason, in the following we resort tothe analysis of the CDF of the modified condition numberF (ξ) = P( ≤ ξ) with ξ ≥ 1 developed in [44]. To do so, wefirst define σ � [σ1, σ2, . . . , σk] as the ordered eigenvalues ofthe Wishart matrix defined in (20) so that 0 < σ1 ≤ . . . ≤ σk.Moreover, let V(σ ) be the Vandermonde matrix with m, n-thentry vm,n = σm−1

n [30, Section 4.6]. The CDF of the modifiedcondition number of an uncorrelated central Wishart matrixis given by [44, Equation (9)]

F (ξ) =[

k∏m=1

(s − m)!k∏

n=1

(t − n)!]−1 k∑

l=1

∫ ∞

0|ϒ |dσk. (28)

Here, |ϒ| denotes the determinant of the matrix ϒ, whose m,

n-th entry υm,n is defined as

υm,n =⎧⎨⎩

γ (t − k + m + n − 1, ξσk)

−γ (t − k + m + n − 1, σk) , m �= lv2

m,nσt−kl e−σl , m = l

(29)

where γ (a, b) is the lower incomplete gamma function givenby γ (a, b) = ∫ b

0 e−tta−1dt. Equations (28) and (29) allows us to


estimate the CDF of the condition number of a communicationchannel, which in turn is necessary to determine the number ofLS iterations as shown in the following. Using the above results,the following theorem can be stated:

Theorem 1: The probability that a given number of LSiterations ilsmax suffices to achieve a relative error reduction ofε in the LS CG algorithm is given by

P(

ils ≤ ilsmax

)= F

⎛⎜⎝⎡⎣ 2 · ilsmax

log(

2ε

)⎤⎦2⎞⎟⎠ . (30)

Proof: The first step to derive (30) is to note that thedefinition of the condition number used in [44] varies withrespect to the one employed in [41], [45]. Attending to theirrelationship, which is shown in (19), the CDF of the standardcondition number � can be expressed as

F�(θ) = P(� ≤ θ) = F (θ2). (31)

In plain words, F (θ2) gives the probability that for a givenconstant θ , the condition number of the LS matrix is below thatvalue. The solution of the above expression can be immediatelyobtained via numerical integration [44]. Once the CDF of thestandard condition number has been characterized, the numberof iterations of the CG algorithm in the considered scheme canbe determined. Particularly, by using (23) the probability thata given number of LS iterations ilsmax achieves a relative errorreduction of ε can be expressed as

P(

ils ≤ ilsmax

)= P

⎛⎝� ≤ 2 · ilsmax

log(

2ε

)⎞⎠ . (32)

The proof is completed by substituting (32) into (31). �The result of Theorem 1 can be used to show the trade-off

between accuracy and complexity of the CG when the numberof LS iterations is varied. In other words, (30) characterizes theimpact of varying the number of iterations of the CG algorithmin the performance of the ZF and greedy CS detectors.

B. Convergence Rate and Error Analysis of CS-BasedDetection in Large-Scale MIMO

The evolution of the error at the i-th iteration between theoriginal signal and the sparse approximation is studied in[38] provided that the assumption δ4S < 0.1 on the restrictedisometry constant is satisfied. In this section we derive a moreintuitive metric to characterize the error reduction based on thenumber of antennas used for signal transmission and recep-tion. In particular, we apply the results on the maximum andminimum singular values of a random Gaussian matrix to theanalysis developed in [38] to derive the following bound.

Theorem 2: The Euclidean norm of the error between thesparse signal at the i-th iteration of the generic and SMMP CSalgorithms and the transmitted signal is upper bounded by

‖x − xi‖2 ≤ c1(N, S)‖x − xi−1‖2 + c2(N, S)‖w‖2. (33)

Fig. 2. Theoretical and empirical evolution of the maximum Euclidean normof the error vs. number of iterations for N = 128, K = 16, nt = 4, na = 1,imax = 4, k = 2K, SNR = 4 dB, 4-QAM and 105 channel realizations.

Here, c1(N, S) and c2(N, S) depend on the number of activeantennas for transmission and reception and are given by

c1 =⎛⎜⎝2 +

T4S

(2+4

√β(4S)

β(4S)

)T3S

(1 − 1√

β(3S)

)2

⎞⎟⎠

×T4ST2S

(1+2

√β(4S)

β(4S)

)+(

1+2√

β(2S)β(2S)

)(

1 − 1√β(2S)

)2, (34)

c2 =⎛⎜⎝2 +

T4S

(2+4

√β(4S)

β(4S)

)T3S

(1 − 1√

β(3S)

)2

⎞⎟⎠⎛⎜⎝

(2 + 2√

β(2S)

)√

T2S

(1 − 1√

β(2S)

)2

⎞⎟⎠+ 2

√T3S −

√T3S√

β(3S)

, (35)

where TC � max(N, C), C ∈ R+.

Proof: The proof is shown in Appendix A. �Note that, as opposed to [38], (40) and (41) show the direct

relationship between the norm bounds and the dimensions ofthe measurement matrix. In other words, the RIP hypothesis isnot longer necessary thanks to the use of the random matrixtheory results. The bound derived in (33) allows characterizingthe error reduction per iteration as a function of the numberof receive antennas. In other words, the convergence speedand performance of the algorithm is upper bounded by (33)as a smaller value of the c1(N, S) will allow obtaining a fasterconvergence whereas c2(N, S) characterizes the error floor. Thisis shown in Fig. 2, where the maximum empirical Euclideannorm of the error for the proposed algorithms is obtainedover 105 channel realizations and compared with the analyt-ical bound. The theoretical result is obtained by using (33)with a number of receive antennas that guarantees algorithmicconvergence [38]. In this figure, it can be seen that the useof a theoretical large-scale approximation allows us to upper


bound the evolution of the empirical maximum �2 norm of theerror for a practical number of iterations. Moreover, Fig. 2 alsodepicts the faster convergence and reduction in the maximumEuclidean norm of the error offered by the SMMP algorithmw.r.t. the straightforward CS approach.

VII. ENERGY EFFICIENCY

The study of the EE becomes especially important in theuplink of multi-user scenarios due to the necessity of findingenergy-efficient schemes that allow increasing the battery life-time [1]. In this section, we define the EE model that will beused hereafter to characterize the EE improvements offered bythe proposed technique in the MAC when compared to otherdetection schemes. Towards this end, we express the EE as therate per milliwatt of total consumed power by using the metric[26], [46]–[52]

ε = Se∑Ku=1 Pu

subject to BER ≤ BERobj. (36)

Here, BERobj is the objective average BER, Se refers to thespectral efficiency in bits per channel use (bpcu), and Pu is thetotal power consumption of the u-th MS in milliwatts requiredto achieve a given BERobj. The total power consumption perMS can be expressed as [47]

Pu = PCu + PTu = [Pψ + P� ] +⎡⎣⎛⎝ nt∑

j=1

pu,j

⎞⎠ · ζ

⎤⎦ . (37)

In the previous expression, PCu = Pψ + P� denotes the totalcircuit power consumption excluding the power amplifier (PA)and it is divided into two components: Pψ that representsthe circuit power consumption that depends on the number ofactive antennas, and P� that corresponds to the static powerconsumption and it is fixed to a reference value of 5 mWper MS [47]. In particular, Pψ comprises the additional powerconsumption required to activate the circuitry of the RF chainsand the digital signal processors for transmission. Moreover,PTu =∑nt

j=1 pu,j · ζ refers to the power consumption of the PAsand it depends on the factor ζ = ν

ηand the power of the signal

that is required to be transmitted by each of the antennas pu,j.In the last expression, the factor ν is the modulation-dependentpeak to average power ratio (PAPR) and η corresponds to thePA efficiency [46]. Based on the above, the global EE can beexpressed as

ε = Se∑Ku=1

{[Pψ + P� ] +

[(∑ntj=1

νη

· pu,j

)]} ,

subject to BER ≤ BERobj. (38)

For the simulations of this work, the efficiency factor corre-sponds to the one of a class-A amplifier, η = 0.35, which iscommonly used in this setting due to the linearity required totransmit QAM signals [46].

Fig. 3. BER vs. SNR for N = 128, K = S = 16, nt = 4, na = 1, imax = 2,k = K and Se = 64 bpcu.

VIII. SIMULATION RESULTS

Monte Carlo simulations have been performed to charac-terize the performance and complexity improvements of theproposed technique. To maintain the coherence with the SMliterature, we compare conventional massive MIMO systemsand SM systems with the same spectral efficiency. The per-formance and energy efficiency results of spatial multiplexingsystems with the same number of transmit antennas have notbeen represented in these figures to avoid congestion andbecause they exhibit a worse behavior when compared to thespatial multiplexing systems considered here. In the following,SM-ZF and SM-MMSE refer to the linear ZF and MMSEdetectors introduced in Section II-C, SM Generic CS denotesthe CoSaMP algorithm without the improvement described inSection III, and the techniques with MIMO on their descriptioncorrespond to those of a conventional MIMO scenario withoutSM. For reasons of brevity, in this section we select k = S forthe generic and SMMP CS algorithms. Note that this decisionentails a minimization of the computational complexity forthe CS-based algorithms following the results of Table II.Additionally, we also consider the MPD algorithm in spite ofits increased computational complexity [23]. The performanceresults, unless stated otherwise, have been obtained with anexact solution of the LS problem via Cholesky decompositionfor clarity.

Fig. 3 characterizes the performance of the above-mentioneddetectors in a scenario with N = 128, K = 16, nt = 4, na = 1corresponding to the single active antenna SM, and a resultingspectral efficiency of Se = 64 bpcu. The results of this fig-ure have been obtained with imax = 2 iterations of Algorithm1, which ensures a reduced complexity when compared tothe algorithms developed in [23]. Note that an error floor isexpected due to the non-feasible solutions produced by thegeneric CS algorithm. This, however, cannot be appreciatedin these results since the high number of antennas consideredat the BS moves the error floor to very low BER values [18].Fig. 3 shows that the proposed algorithm is able to reducethe required transmission power by more than 4 dB w.r.t.


Fig. 4. BER vs. SNR for N = 128, K = 16, nt = 7, na = 2, imax = 3, k =S = 2K and Se = 96 bpcu.

conventional MIMO with nt = 1 and 16-QAM to achieve atarget BER of 10−4. Moreover, it is portrayed that SMMPapproaches the performance of the more complex MPD andoutperforms the SM-ZF detector. This is because the proposedstrategy is able to iteratively identify the active antenna indicesand then perform a selective channel inversion with a matrix ofreduced dimensions. Instead, the performance of conventionalMIMO with two active antennas per user approaches the per-formance of SMMP. However, we remark that in this case thepower consumption and complexity of the MSs is increased dueto the additional RF chains implemented.

Fig. 4 shows the performance of the considered detectorsin a scenario with N = 128, K = 16, nt = 7, generalized SMwith na = 2 active antennas per user, and a resulting spectralefficiency of Se = 96 bpcu. The results of this figure showthat generalized SM systems with SMMP detection are capableof improving the performance of conventional MIMO systemsemploying the same number of RF chains (‘MIMO-ZF. nt =1’). The use of SMMP also allows outperforming conventionalMIMO systems with a pair of RF chains (‘MIMO-ZF. nt = 2’)for the range of practical BERs. Moreover, it can be seenthat the SMMP algorithm clearly improves the performanceof other linear detectors such as the SM-MMSE detector. Inthe following we focus our attention on single active antennaSM for reasons of brevity, although it is clear that the resultantconclusions also extend to generalized SM transmission.

The number of users is increased from K = 16 of Fig. 3 toK = 32 in Fig. 5, which shows a scenario especially favorablefor the proposed technique. Fig. 5 shows that the proposedstrategy is able to provide an enhancement of up to three ordersof magnitude in the BER at high SNR values w.r.t. conventionallarge-scale MIMO, and both ZF and MMSE detectors in SM.The performance improvement offered by SM when comparedto conventional MIMO transmission is coherent with the behav-ior described in [20], [53]. The effect of acquiring inaccuratechannel state information (CSI) on the performance is alsoshown in Fig. 5(b). The imperfect CSI is modeled followingH = √

1 − τ 2H + τB, where τ ∈ [0, 1] regulates CSI quality

Fig. 5. BER vs. SNR for N = 128, K = S = 32, nt = 4, na = 1, imax = 3,k = K and Se = 128 bpcu with (a) perfect and (b) imperfect CSI (τ = 0.25).

Fig. 6. BER vs. complexity for N = 128, K = S = 16, nt = 4, na = 1,SNR = 6 dB, Se = 64 bpcu, imax = 2, k = K with different LS methods.

and B ∈ CN×M ∼ CN (0, IN ⊗ IM) characterizes the channel

estimation error [54]. The results of this figure for τ = 0.25highlight the robustness of SMMP when compared to tradi-tional CS-based detection, which makes it the best alternativein terms of performance under imperfect CSI.

Fig. 6 shows the trade-off between performance and com-plexity in real flops for a MAC with N = 128, K = 16, na = 1,Se = 64 bpcu, imax = 2, k = K and SNR = 6 dB. The numberof antennas per user is nt = 4 when SM is used, whereas it isnt = 1 and nt = 4 for conventional MIMO. Note that the com-plexity and performance of the CG algorithm varies depend-ing on the number of iterations as explained in Section V-A.From the results of this figure it can be concluded that theSMMP approach offers the best performance with a restrainedcomplexity. When compared to large-scale MIMO systems withthe same spectral efficiency and alike number of RF chains(‘MIMO-ZF, nt = 1’), the use of SM increases the complexitydue to the higher number of antennas at the MSs, but in turnoffers a performance improvement of more than two orders of


Fig. 7. BER vs. complexity for N = 128, K = S = 8, nt = 4, na = 1, SNR =2 dB, Se = 32 bpcu, imax = 2, k = K with different LS methods.

magnitude. This improvement is not as significant with respectto the spatial multiplexing system with the same number ofantennas (‘MIMO-ZF, nt = 4’), where instead the complexityis dramatically reduced. Fig. 6 also shows the complexityimprovements that can be obtained when the iterative CG algo-rithm termed ‘ZF-SM. Conjugate Gradient’ considered in thispaper is used to solve the LS problems against traditional directmethods such as ‘ZF-SM. QR decomposition’ and ‘ZF-SM.Cholesky decomposition’ in Fig. 6. The number of iterationsand, inherently, the complexity can be adjusted depending onthe BER required at the BS. This leads us to conclude that theuse of the CS-based detection is especially convenient in fastfading scenarios with a reduced channel coherence period. Thisis because whereas CS-based detection methods perform thesame computations in every channel use, the linear detectorssolved with direct methods focus the intensive computations atthe beginning of the coherence period and reduce the complex-ity afterwards [2].

A similar conclusion can be obtained from the results ofFig. 7, where we focus on controlling the attainable perfor-mance by varying the number of LS iterations in the CGalgorithm. We also note that, as opposed to the conclusionsachieved in [18], [38], the iterative LS algorithm accounts for60% of the global detection complexity, hence justifying theneed of an accurate complexity characterization. Overall, itcan be concluded that the proposed strategy offers significantperformance and complexity improvements with respect to con-ventional detection schemes in systems with the same numberof antennas.

Regarding the EE of the CS-based detection, Fig. 8 showsthis metric for a MAC with N = 128 and a varying numberof users. The transmission power is varied depending on thenumber of users so that BERobj = 10−3 in (36). The circuitpower consumption depending on the number of active anten-nas is set to a realistic value of Pψ = 20 mW [46], and the noisevariance is fixed to σ 2 = 0.01. We note that the PAPR factor ofthe single-antenna users is increased w.r.t. SM due to the useof a higher modulation order Q. From the results of this figureit can be concluded that the use of SM allows to significantly

Fig. 8. EE vs. number of users K to achieve BER = 10−3. N = 128, nt = 4,na = 1. Pψ = 20 mW.

Fig. 9. EE vs. number of users K to achieve BER = 10−3. N = 128, nt = 4,na = 1. Pψ = 10 mW.

increase the EE of the conventional large-scale MIMO for lowand intermediate system loading factors. This improvementdue to the reduced circuit power consumption and PAPR wasalready noticed in [55] for the downlink of small-scale P2Psystems. Moreover, the proposed technique outperforms therest of conventional detectors, hence constituting an energy-efficient alternative in the MAC.

Fig. 8 also shows that, in spite of the transmission energysavings that can be obtained when nt = 2 and 4-QAM are usedin MIMO systems, the increased circuit power consumptioncaused by the higher number of RF chains penalizes the EE.To show this effect, in Fig. 9 we reduce the power consumedby the RF chains Pψ to half, i.e., Pψ = 10 mW. By doing so,now it can be seen that the use of a MIMO system with nt = 2outperforms the option of having single-antenna devices. Still,the EE of the SM alternatives is significantly higher than the oneof MIMO strategies for a low and intermediate number of usersdue to the reduced transmission power required to compensatethe inter-user interference.


IX. CONCLUSION

In this paper, a low-complexity detection algorithm for SMhas been presented. The proposed strategy is based upon CS byincorporating the additional structure and sparsity of the trans-mitted signals in the MAC. Our complexity and performanceanalyses show that the benefits of the proposed are maximizedwhen a high number of receive antennas at the BS are used dueto its faster convergence and improved performance. Overall,the results derived in this paper confirm that the CS-baseddetection for SM constitutes a low-complexity alternative toincrease the EE in the MAC. Possible future work can becarried out in the analytic characterization of the bit error rateperformance of the proposed scheme in the large-scale regime.

APPENDIX APROOF OF THEOREM 2

The proof of Theorem 2 commences by recalling the re-sults regarding the maximum and minimum singular valuesof Wishart matrices with large dimensions [4]. Let W be theWishart matrix defined in (20) with the entries of the matrixHL ∈ CN×|L| satisfying hm,n ∼ CN (0, 1). For these expres-sions, no assumptions on the definition and cardinality of Land R are adopted. For large N and |L|, the maximum andminimum eigenvalues of W converge to

σmin(W) −→N,|L|→∞ T|L| ×

(1 − 1√

β (|L|))2

σmax(W) −→N,|L|→∞ T|L| ×

(1 + 1√

β (|L|))2

(39)

where β(V) was defined in (26) and TC � max(N, C), C ∈R

+. Based on the above expressions, we can redefine somerelationships that will be useful in the sequel∥∥HH

Ly∥∥

2 ≤√T|L| ×(

1 + 1√β (|L|)

)‖y‖2,

∥∥∥H†Ly∥∥∥

2≤⎛⎝ 1√

T|L| ×(

1 − 1√β(|L|)

)⎞⎠ ‖y‖2,

∥∥HHLHLx

∥∥2 � T|L| ×

(1 ± 1√

β (|L|))2

‖x‖2,

∥∥∥(HHLHL

)−1x∥∥∥

2�

⎛⎝ 1√T|L| ×

(1 ± 1√

β(|L|))⎞⎠2

‖x‖2. (40)

Note that the last two inequalities can determine upper andlower bounds [38]. Moreover, the following relationship is alsosatisfied ∥∥HH

RHL∥∥ ≤ TB

β(B)+ 2TB√

β(B), (41)

where the set G with cardinality B is given by G � R ∪ L.As the derivation of the previous expression is based upon theideas developed in [38], but considering instead that the spectral

norm of HHRHL is always smaller than the one of HH

G HG − Tand that the function β : Z+ → R

+, here we avoid a detaileddescription for brevity.

The objective of the proof is to establish an upper bound onthe �2 norm of the error between the transmitted signal and theapproximation obtained by the generic CS algorithm at the i-thiteration [38], i.e.,

‖x − xi‖2 ≤ Rmax, (42)

where Rmax is the upper bound to be derived and xi is the esti-mated signal at the i-th iteration. The upper bound is obtainedby following the arguments developed in [38] and applying theresults obtained in (40) and (41) where convenient. To preservethe coherence with [38], in the following it has been assumedthat |T | ≤ 3S or, in other words, |�| = 2S entries are selectedat the output of the MF per iteration.

To derive the desired result, first note that the differencebetween the S-sparse transmitted signal x and the output ofthe LS problem b is bounded when selecting the best S-sparseapproximation xi [38]. Formally,

‖x − xi‖2 ≤ ‖x − b‖2 + ‖b − xi‖2 ≤ 2‖x − b‖2, (43)

where the basic norm property ‖a + z‖ ≤ ‖a‖ + ‖z‖ has beenapplied. With the purpose of deriving an upper bound to theright-hand side of (43), we first express ‖x − b‖2 as

‖x − b‖2 ≤ ‖x|T C‖2 + ‖x|T − b|T ‖2, (44)

where the inequality holds because b is supported on T fol-lowing (18). We now focus on deriving an upper bound for‖x|T − b|T ‖2, which is given by

‖x|T − b|T ‖2

(a)=∥∥∥x|T − H†

T (Hx|T + Hx|T C + w)

∥∥∥2

(b)≤∥∥∥(HH

T HT)−1

HHT Hx|T C

∥∥∥2+∥∥∥H†

T w∥∥∥

2

(c)≤⎛⎜⎝(

T4Sβ(4S)

+ 2T4S√β(4S)

)T3S

(1 − 1√

β(3S)

)2

⎞⎟⎠ ‖x|T C‖2 + ‖w‖2√T3S −

√T3S√

β(3S)

.

(45)

In the above expressions,(a)= holds by definition (18), and

(b)≤follows from the definition of the pseudoinverse matrix and by

noting that H†T Hx|T = x|T . Moreover,

(c)≤ is obtained by usingthe relationships derived in (40) and (41) along with the factthat, by definition, |T | ≤ 3S and x is S-sparse. Substituting thelast inequality of (45) into (44) we obtain the desired upperbound

‖x−b‖2 ≤⎛⎜⎝1+

(T4S

β(4S)+ 2T4S√

β(4S)

)T3S

(1− 1√

β(3S)

)2

⎞⎟⎠‖x|T C‖2+ ‖w‖2√T3S−

√T3S√

β(3S)

.

(46)


To derive an upper bound on ‖x|T C‖2, first define the sig-nal m = x − xi−1. Considering this, the following relationshipholds [38]

‖x|T C‖2(a)=∥∥∥(x − xi−1)|T C

∥∥∥2

= ‖m|T C‖2

(b)≤ ‖m|�C‖2, (47)

where(a)= is satisfied because the signal xi−1 is supported in T

whereas(b)≤ holds because � ⊂ T .

Based on (47), the next step consists on upper bounding‖m|�C‖2. Let the set P be the support of the signal m withcardinality |P| ≤ 2S. Since � is formed by the 2S entries withhighest energy from the output of the MF c, the relationship

‖c|P−�‖2 ≤ ‖c|�−P‖2 (48)

is satisfied. Recall from (14) that the residual signal at the startof the i-th iteration satisfies r = Hm + w. By making use of(40) and (41), it can be seen that

‖c|P−�‖2

= ∥∥HHP−�r

∥∥2

= ∥∥HHP−�(Hm + w)

∥∥2

≥ ∥∥HHP−�Hm|P−�

∥∥2 − ∥∥HH

P−�Hm|�∥∥

2 − ∥∥HHP−�w

∥∥2

(a)≥ T2S

(1 − 1√

β(2S)

)2

‖m|P−�‖2 −(√

T2S +√

T2S√β(2S)

)× ‖w‖2 −

(T2S

β(2S)+ 2T2S√

β(2S)

)‖m‖2, (49)

where(a)≥ follows from |P − �| ≤ 2S. The same expressions

can be used to derive an upper bound to ‖c|�−P‖2, yielding

‖c|�−P‖2

= ∥∥HH�−Pr

∥∥2 = ∥∥HH

�−P (Hm + w)∥∥

2

≤ ∥∥HH�−PHm

∥∥2 + ∥∥HH

�−Pw∥∥

2

(a)≤(

T4S

β(4S)+ 2T4S√

β(4S)

)‖m‖2+

(√T2S +

√T2S√

β(2S)

)‖w‖2.

(50)

Here,(a)≤ is obtained by noting that |(� − P) ∪ P| ≤ 4S. By

using (48), (49) and (50) the following threshold is derived

‖m|P−�‖2 = ‖m|�C‖2 ≤(

2 + 2√β(2S)

)√

T2S

(1 − 1√

β(2S)

)2‖w‖2

+⎛⎜⎝(

T4Sβ(4S)

+ 2T4S√β(4S)

+ T2Sβ(2S)

+ 2T2S√β(2S)

)T2S

(1 − 1√

β(2S)

)2

⎞⎟⎠ ‖m‖2. (51)

The proof of (33) is completed by combining the results from(43)–(51). Formally,

‖x − xi‖2(43)≤ 2‖x − b‖2

(46)−(47)≤⎛⎝ 2‖w‖2√

T3S−√

T3S√β(3S)

⎞⎠+⎛⎜⎝2+

(2T4Sβ(4S)

+ 4T4S√β(4S)

)T3S

(1− 1√

β(3S)

)2

⎞⎟⎠‖m|�C‖2

(47)−(51)≤⎛⎜⎝2 +

(2T4Sβ(4S)

+ 4T4S√β(4S)

)T3S

(1 − 1√

β(3S)

)2

⎞⎟⎠×⎡⎢⎣⎛⎜⎝(

T4Sβ(4S)

+ 2T4S√β(4S)

+ T2Sβ(2S)

+ 2T2S√β(2S)

)T2S

(1 − 1√

β(2S)

)2

⎞⎟⎠ ‖x − xi−1‖2

+(

2 + 2√β(2S)

)√

T2S

(1 − 1√

β(2S)

)2‖w‖2

⎤⎥⎦+ 2‖w‖2√T3S −

√T3S√

β(3S)

= c1(N, S)‖x − xi−1‖2 + c2(N, S)‖w‖2. (52)

ACKNOWLEDGMENT

The authors would like to thank Dr. Francesco Renna for thehelpful discussions throughout the development of this work.

REFERENCES

[1] G. Li et al., “Energy-efficient wireless communications: Tutorial, survey,and open issues,” IEEE Wireless Commun., vol. 18, no. 6, pp. 28–35,Dec. 2011.

[2] E. Bjornson, L. Sanguinetti, J. Hoydis, and M. Debbah, “Optimal designof energy-efficient multi-user MIMO systems: Is massive MIMO theanswer?” IEEE Trans. Wireless Commun., vol. 14, no. 6, pp. 3059–3075,Jun. 2015.

[3] M. Di Renzo, H. Haas, A. Ghrayeb, S. Sugiura, and L. Hanzo, “Spatialmodulation for generalized MIMO: Challenges, opportunities, and imple-mentation,” Proc. IEEE, vol. 102, no. 1, pp. 56–103, Jan. 2014.

[4] F. Rusek et al., “Scaling up MIMO: Opportunities and challenges withvery large arrays,” IEEE Signal Process. Mag., vol. 30, no. 1, pp. 40–60,Jan. 2013.

[5] J. Hoydis, S. ten Brink, and M. Debbah, “Massive MIMO in the UL/DL ofcellular networks: How many antennas do we need?” IEEE J. Sel. AreasCommun., vol. 31, no. 2, pp. 160–171, Feb. 2013.

[6] C. Masouros, M. Sellathurai, and T. Ratnarajah, “Large-scale MIMOtransmitters in fixed physical spaces: The effect of transmit correlation andmutual coupling,” IEEE Trans. Commun., vol. 61, no. 7, pp. 2794–2804,Jul. 2013.

[7] C. Masouros and M. Matthaiou, “Space-constrained massive MIMO: Hit-ting the wall of favorable propagation,” IEEE Commun. Letters, vol. 19,no. 5, pp. 771–774, May 2015.

[8] D. Ha, K. Lee, and J. Kang, “Energy efficiency analysis with circuit powerconsumption in massive MIMO systems,” in Proc. IEEE 24th Int. Symp.PIMRC, Sep. 2013, pp. 938–942.

[9] P. Yang, M. Di Renzo, Y. Xiao, S. Li, and L. Hanzo, “Design guidelinesfor spatial modulation,” IEEE Commun. Surveys Tuts., vol. 17, no. 1,pp. 6–26, 1st Quart. 2015.

[10] M. Di Renzo, H. Haas, and P. M. Grant, “Spatial modulation for multiple-antenna wireless systems: A survey,” IEEE Commun. Mag., vol. 49,no. 12, pp. 182–191, Dec. 2011.

[11] C. Masouros, “Improving the diversity of spatial modulation in MISOchannels by phase alignment,” IEEE Commun. Lett., vol. 18, no. 5,pp. 729–732, May 2014.

[12] C. Masouros and L. Hanzo, “Constellation-randomization achievestransmit diversity for single-RF spatial modulation,” IEEE Trans. Veh.Technol., to be published.


[13] C. Masouros and L. Hanzo, “Dual layered downlink MIMO transmis-sion for increased bandwidth efficiency,” IEEE Trans. Veh. Technol.,to be published.

[14] A. Garcia-Rodriguez, C. Masouros, and L. Hanzo, “Pre-scaling optimiza-tion for space shift keying based on semidefinite relaxation,” IEEE Trans.Commun., to be published.

[15] A. Younis, S. Sinanovic, M. Di Renzo, R. Mesleh, and H. Haas, “Gen-eralised sphere decoding for spatial modulation,” IEEE Trans. Commun.,vol. 61, no. 7, pp. 2805–2815, Jul. 2013.

[16] S. Sugiura, C. Xu, S. X. Ng, and L. Hanzo, “Reduced-complexity coherentversus non-coherent QAM-aided space-time shift keying,” IEEE Trans.Commun., vol. 59, no. 11, pp. 3090–3101, Nov. 2011.

[17] C. Xu, S. Sugiura, S. X. Ng, and L. Hanzo, “Spatial modulation and space-time shift keying: Optimal performance at a reduced detection complex-ity,” IEEE Trans. Commun., vol. 61, no. 1, pp. 206–216, Jan. 2013.

[18] C.-M. Yu et al., “Compressed sensing detector design for space shiftkeying in MIMO systems,” IEEE Commun. Lett., vol. 16, no. 10,pp. 1556–1559, Oct. 2012.

[19] C.-H. Wu, W.-H. Chung, and H.-W. Liang, “OMP-based detector de-sign for space shift keying in large MIMO system,” in Proc. IEEEGLOBECOM, 2014, pp. 4072–4076.

[20] N. Serafimovski, S. Sinanovic, M. Di Renzo, and H. Haas, “Multi-ple access spatial modulation,” EURASIP J. Wireless Commun. Netw.,vol. 2012, pp. 1–20, Sep. 2012.

[21] M. Di Renzo and H. Haas, “Bit error probability of space-shift keyingMIMO over multiple-access independent fading channels,” IEEE Trans.Veh. Technol., vol. 60, no. 8, pp. 3694–3711, Oct. 2011.

[22] L.-L. Yang, “Signal detection in antenna-hopping space-division multiple-access systems with space-shift keying modulation,” IEEE Trans. SignalProcess., vol. 60, no. 1, pp. 351–366, Jan. 2012.

[23] T. Narasimhan, P. Raviteja, and A. Chockalingam, “Large-scale multiuserSM-MIMO versus massive MIMO,” in Proc. ITA, Feb. 2014, pp. 1–9.

[24] D. L. Donoho, A. Maleki, and A. Montanari, “Message-passing algo-rithms for compressed sensing,” Proc. Nat. Academy Sci., vol. 106,no. 45, pp. 18914–18919, Nov. 2009.

[25] J. Zheng, “Low-complexity detector for spatial modulation multiple ac-cess channels with a large number of receive antennas,” IEEE Commun.Lett., vol. 18, no. 11, pp. 2055–2058, Nov. 2014.

[26] S. Wang, Y. Li, M. Zhao, and J. Wang, “Energy efficient and low-complexity uplink transceiver for massive spatial modulation MIMO,”IEEE Trans. Veh. Technol., to be published.

[27] S. Wang, Y. Li, and J. Wang, “Multiuser detection in massive spatialmodulation MIMO with low-resolution ADCs,” IEEE Trans. WirelessCommun., vol. 14, no. 4, pp. 2156–2168, Apr. 2015.

[28] Y. C. Eldar and M. Mishali, “Robust recovery of signals from a struc-tured union of subspaces,” IEEE Trans. Inf. Theory, vol. 55, no. 11,pp. 5302–5316, Nov. 2009.

[29] R. G. Baraniuk, V. Cevher, M. F. Duarte, and C. Hegde, “Model-based compressive sensing,” IEEE Trans. Inf. Theory, vol. 56, no. 4,pp. 1982–2001, Apr. 2010.

[30] G. H. Golub and C. F. Van Loan, Matrix Computations. Baltimore,MD, USA: The Johns Hopkins Univ. Press, 2012, vol. 3.

[31] A. Younis, N. Serafimovski, R. Mesleh, and H. Haas, “Generalisedspatial modulation,” in Proc. Conf. Rec. 44th ASILOMAR, Nov. 2010,pp. 1498–1502.

[32] C. Peel, B. Hochwald, and A. Swindlehurst, “A vector-perturbation tech-nique for near-capacity multiantenna multiuser communication—Part I:Channel inversion and regularization,” IEEE Trans. Commun., vol. 53,no. 1, pp. 195–202, Jan. 2005.

[33] R. Zhang, L.-L. Yang, and L. Hanzo, “Generalised pre-coding aidedspatial modulation,” IEEE Trans. Wireless Commun., vol. 12, no. 11,pp. 5434–5443, Nov. 2013.

[34] R. Baraniuk, “Compressive sensing [lecture notes],” IEEE Signal Process.Mag., vol. 24, no. 4, pp. 118–121, Jul. 2007.

[35] E. J. Candès and M. B. Wakin, “An introduction to compressive sam-pling,” IEEE Signal Process. Mag., vol. 25, no. 2, pp. 21–30, Mar. 2008.

[36] E. Candès, “Compressive sampling,” in Proc. Int Congr. Math., Madrid,Spain, Aug. 22–30, 2006, pp. 1433–1452.

[37] E. Candes, J. Romberg, and T. Tao, “Stable signal recovery from incom-plete and inaccurate measurements,” Commun. Pure Appl. Math., vol. 59,no. 8, pp. 1207–1223, Aug. 2006.

[38] D. Needell and J. A. Tropp, “CoSaMP: Iterative signal recovery fromincomplete and inaccurate samples,” Appl. Comput. Harmonic Anal.,vol. 26, no. 3, pp. 301–321, May 2009.

[39] X. Rao and V. K. Lau, “Distributed compressive CSIT estimation andfeedback for FDD multi-user massive MIMO systems,” IEEE Trans.Signal Process., vol. 62, no. 12, pp. 3261–3271, Jun. 2014.

[40] M. Arakawa, “Computational workloads for commonly used signal pro-cessing kernels,” Def. Tech. Inf. Center Document, Fort Belvoir, VA,USA, Tech. Rep., 2006.

[41] A. Björck, Numerical Methods for Least Squares Problems. Philadelphia,PA, USA: SIAM, 1996.

[42] R. Hunger, “Floating point operations in matrix-vector calculus,” Tech-nische Universitat Munchen, Munchen, Germany, Tech. Rep. [Online].Available: https://mediatum.ub.tum.de/doc/625604/625604.pdf

[43] L. Vandenberghe, “Applied numerical computing,” Univ. Calif.,Los Angeles, CA, USA, Univ. Lecture, 2011.

[44] M. Matthaiou, M. McKay, P. Smith, and J. Nossek, “On the condi-tion number distribution of complex Wishart matrices,” IEEE Trans.Commun., vol. 58, no. 6, pp. 1705–1717, Jun. 2010.

[45] A. Edelman, “Eigenvalues and condition numbers of random matrices,”Ph.D. dissertation, Yale Univ., New Haven, CT, USA, 1988.

[46] S. Cui, A. J. Goldsmith, and A. Bahai, “Energy-constrained modu-lation optimization,” IEEE Trans. Wireless Commun., vol. 4, no. 5,pp. 2349–2360, Sep. 2005.

[47] G. Miao, “Energy-efficient uplink multi-user MIMO,” IEEE Trans.Wireless Commun., vol. 12, no. 5, pp. 2302–2313, May 2013.

[48] K. Ntontin, M. Di Renzo, A. Perez-Neira, and C. Verikoukis, “Towards theperformance and energy efficiency comparison of spatial modulation withconventional single-antenna transmission over generalized fading chan-nels,” in Proc. IEEE Int. Workshop CAMAD, Sep. 2012, pp. 120–124.

[49] C. Masouros, M. Sellathurai, and T. Ratnarajah, “Maximizing energyefficiency in the vector precoded MU-MISO downlink by selective pertur-bation,” IEEE Trans. Wireless Commun., vol. 13, no. 9, pp. 4974–4984,Sep. 2014.

[50] C. Masouros, M. Sellathurai, and T. Ratnarajah, “Vector perturbationbased on symbol scaling for limited feedback MISO downlinks,” IEEETrans. Signal Process., vol. 62, no. 3, pp. 562–571, Feb. 2014.

[51] A. Garcia-Rodriguez and C. Masouros, “Power-efficient Tomlinson-Harashima precoding for the downlink of multi-user MISO systems,”IEEE Trans. Commun., vol. 62, no. 6, pp. 1884–1896, Jun. 2014.

[52] A. Garcia-Rodriguez and C. Masouros, “Power loss reduction for MMSE-THP with multidimensional symbol scaling,” IEEE Commun. Lett.,vol. 18, no. 7, pp. 1147–1150, Jul. 2014.

[53] M. Di Renzo and H. Haas, “Bit error probability of SM-MIMO overgeneralized fading channels,” IEEE Trans. Veh. Technol., vol. 61, no. 3,pp. 1124–1144, Mar. 2012.

[54] S. Wagner, R. Couillet, M. Debbah, and D. T. M. Slock, “Large systemanalysis of linear precoding in correlated MISO broadcast channels underlimited feedback,” IEEE Trans. Inf. Theory, vol. 58, no. 7, pp. 4509–4537,Jul. 2012.

[55] A. Stavridis, S. Sinanovic, M. Di Renzo, and H. Haas, “Energy evaluationof spatial modulation at a multi-antenna base station,” in Proc. IEEE 78thVTC—Fall, Sep. 2013, pp. 1–5.

Adrian Garcia-Rodriguez received the M.S. degreein telecommunications engineering from Universi-dad de Las Palmas de Gran Canaria, Las Palmas,Spain, in 2012. He is currently working toward thePh.D. degree in the Department of Electronic andElectrical Engineering, University College London,London, U.K. His research interests lie the fieldof signal processing and wireless communications,with emphasis on energy-efficient and multiantennacommunications.

Christos Masouros (M’06–SM’14) received theDiploma degree in electrical and computer engineer-ing from the University of Patras, Patras, Greece, in2004 and the M.Sc. degree by research and the Ph.D.degree in electrical and electronic engineering fromThe University of Manchester, Manchester, U.K., in2006 and 2009, respectively.

He is currently a Lecturer with the Departmentof Electronic and Electrical Engineering, UniversityCollege London, London, U.K. He has been a Re-search Associate with the University of Manchester

and a Research Fellow with Queen’s University Belfast, Belfast, U.K. He holdsa Royal Academy of Engineering Research Fellowship 2011–2016 and is thePrincipal Investigator of the EPSRC project EP/M014150/1 on large-scale an-tenna systems. His research interests lie in the field of wireless communicationsand signal processing, with particular focus on green communications, large-scale antenna systems, cognitive radio, interference mitigation techniques forMIMO, and multicarrier communications.

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