6038 ieee transactions on signal processing, vol. 65,...

6038 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 65, NO. 22, NOVEMBER 15, 2017

Detection of Large-Scale Wireless Systemsvia Sparse Error Recovery

Jun Won Choi, Member, IEEE, and Byonghyo Shim, Senior Member, IEEE

Abstract—In this paper, we propose a new detection algorithmfor large-scale wireless systems, referred to as post sparse errordetection (PSED) algorithm, that employs a sparse error recoveryalgorithm to refine the estimate of a symbol vector obtained bythe conventional linear detector. The PSED algorithm operatesin two steps: First, sparse transformation converting the originalnonsparse system into the sparse system whose input is an errorvector caused by the symbol slicing; and second, the estimationof the error vector using the sparse recovery algorithm. From theasymptotic mean square error analysis and empirical simulationsperformed on large-scale wireless systems, we show that the PSEDalgorithm brings significant performance gain over classical lineardetectors while imposing relatively small computational overhead.

Index Terms—Sparse signal recovery, compressive sensing,large-scale systems, orthogonal matching pursuit, sparse trans-formation, linear minimum mean square error, error correction.

I. INTRODUCTION

A S A paradigm guaranteeing the perfect reconstruction ofa sparse signal from a small set of linear measurements,

compressive sensing (CS) has generated a great deal of interestin recent years. Basic premise of the CS is that the sparse signalscan be reconstructed from the compressed measurements as longas the signal to be recovered is sparse (i.e., number of nonzeroelements in the vector is small) and the measurement processapproximately preserves the energy of the original sparse vector[2], [3]. The CS paradigm works well in many signal processingapplications where the signal vector to be reconstructed is sparsein itself or sparse in a transformed domain. In particular, CS tech-nique has been applied to wireless communication applicationsin the context of sparse channel estimation [4]–[8] and wire-less multiuser detection [9]–[11], where the multi-dimensional

Manuscript received August 8, 2016; revised July 7, 2017; accepted August22, 2017. Date of publication September 4, 2017; date of current version Septem-ber 26, 2017. The associate editor coordinating the review of this manuscriptand approving it for publication was Prof. Ami Wiesel. This work was sup-ported in part by the research grant from Qualcomm Incorporated and in part bythe Institute for Information & Communications Technology Promotion Grantfunded by the Korea government (MSIP) (No. 2015-0-00294) and the Min-istry of Education (NRF-2017R1D1A1A09000602). This paper was presentedin part at the IEEE Global Telecommunications Conference, Austin, TX, USA,December 2014. (Corresponding author: Byonghyo Shim.)

J. W. Choi is with the Department of Electrical Engineering, Hanyang Uni-versity, Seoul 133-791, South Korea (e-mail: [email protected]).

B. Shim is with the Institute of New Media and Communications and theSchool of Electrical and Computer Engineering, Seoul National University,Seoul 151-742, South Korea (e-mail: [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/TSP.2017.2749214

quantities being estimated exhibit sparsity structure. However,not much work is available for the information vector detection(possible exception can be [12]) mainly because the informationvectors being transmitted in a typical communication system areby no means sparse so that the sparse recovery algorithm wouldnot perform better than the conventional receiver algorithm, notto mention having no performance guarantee.

It is to these types of wireless detection problem that this paperis addressed. This problem, which is seemingly unconnected tothe CS principle, is prevalent and embraces many of current andfuture detection problems in wireless communication scenariossuch as multiuser detection for Internet of Things (IoT). As anexample, consider a scenario where nt devices simultaneouslytransmit information to the access point (AP) (see Fig. 1). Oneof L quasi-orthogonal signature sequences q1 , . . . ,qL ∈ Rnr isassigned to each device for the purpose of limiting the inter-userinterference. Assuming that the channel is under flat fading, thereceived vector at the AP is

y = h1qi1 s1 + · · · + hntqin t

snt+ n

= Hs + n (1)

where H = [h1qi1 · · · hntqin t

] is the composite channel ma-trix, s = [s1 · · · snt

] is the vector of transmit symbols for nt

active devices, in is the sequence index for the nth device, hn

is the complex channel gain from the nth device to the AP, andn is the nr × 1 measurement noise vector. In this setting, themultiuser detection problem boils down to an estimation of thesymbol vector s from y with the knowledge of H.1 In the IoTscenarios, the number of devices being served nt is very largeso that it is of importance to develop low complexity multi-userdetection algorithm.

Roughly speaking, traditional way of detecting the input sig-nals falls into two categories: linear detection and nonlineardetection techniques. Linear detection techniques, such as zeroforcing (ZF) or linear minimum mean square error (LMMSE)estimation, are simple to implement and easy to use but the per-formance is in general worse than the nonlinear detectors [9].Nonlinear detection schemes usually perform better than thelinear detection but it requires significant computational over-head. Over the years, various nonlinear detection algorithms forlarge-scale systems have been proposed. These include fixed-complexity sphere decoder [13], approximate message passing(AMP)-based detector [14]–[16], and sum-product-based detec-tor [17].

1The channel gains can be estimated using the preamble or pilot signals.

1053-587X © 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See http://www.ieee.org/publications standards/publications/rights/index.html for more information.

CHOI AND SHIM: DETECTION OF LARGE-SCALE WIRELESS SYSTEMS VIA SPARSE ERROR RECOVERY 6039

Fig. 1. Multiple access system for IoT applications.

Our approach provides a comparably simple yet effectivesolution to the large-scale wireless detection problems by delib-erately combining the linear detection and a (nonlinear) sparsesignal recovery algorithm. The proposed algorithm, henceforthdubbed as post sparse error detection (PSED), is based on thesimple observation that the conventional linear detection algo-rithm performs reasonably well and thus the number of errors inthe detector output is quite small. In a nutshell, the PSED algo-rithm operates in two steps.2 In the first step, the conventionallinear detection is performed to generate a rough estimate of thetransmit symbol vector. Since the performance of conventionaldetector is acceptable in the operating regime, the error vectorobtained by the slicing of detected symbol vector can be readilymodeled as a sparse signal. Now, by a simple linear transform,a new system model whose input is the sparse error vector canbe obtained. In the second step, the sparse recovery algorithmto estimate this sparse error vector is employed. By cancellingthe recovered error vector from the conventional detector outputwhich corresponds to the sum of symbol and error vectors, weobtain more reliable estimate of the original symbol vector.

In our analysis based on the random matrix theory, we showthat the asymptotic performance of the proposed PSED algo-rithm, measured in terms of mean square error (MSE), decaysexponentially with signal-to-noise ratio (SNR), which is in con-trast to the linear or sublinear decaying behavior of the con-ventional linear detectors. In fact, we show from the empiricalsimulations that the PSED scheme outperforms the conventionallinear detectors by a large margin.

The rest of this paper is organized as follows. In Section II, wedescribe the system model and the proposed PSED algorithm.In Section III, the asymptotic performance analysis of the PSEDscheme is provided using random matrix theory. In Section V,we present the simulation results and conclude the paper inSection VI.

2In [18], two-step processing approach to correct error of real-valued signalswhen the codeword is corrupted by gross errors on a fraction of entries and asmall noise on all the entries has been proposed.

II. POST SPARSE ERROR DETECTION

A. System Model and Conventional Detectors

The relationship between the transmit symbol in the devicesand the received signal vector in the AP in IoT environmentscan be expressed as

y =√

PHs + v (2)

where y ∈ Cnr is the received signal vector, s ∈ Cnt is thevector of transmit symbols whose entries are chosen from a setΩ of finite symbol alphabet, H ∈ Cnr ×nt is the channel matrix,v ∼ CN (0, σ2

v Inr) is the noise vector, and P is the transmitted

power. As mentioned, there are two options, linear detectionand nonlinear detection schemes, in recovering the transmitinformation from the received signals. In the linear detectionscheme, an estimate s of the transmit symbol vector is obtainedby applying the weight matrix W ∈ Cnr ×nt to the receivedvector y

s = WH y. (3)

Well-known linear detection schemes include:� Matched filter (MF): W = 1√

PH

� ZF receiver: W = 1√PH(HH H

)−1

� LMMSE receiver: W = H(HH H + σ 2

v

P I)−1

.

The linear detection is simple to implement and computation-ally efficient, but the performance is typically not better than thenonlinear detection scheme. The nonlinear detectors, such asML and maximum a posteriori (MAP) detectors, exploit the ad-ditional side information that an element of the transmit vectoris chosen from the set of finite alphabets. For the lattice that thesymbol vector spans, a tree search is performed to find out thesolution minimizing the cost function. In the ML detector, forexample, a symbol vector s minimizing the ML cost functionJ(s) = mins∈Ωn t ‖y −√

PHs‖2 is chosen among all possiblecandidates. When compared to the linear detection schemes,the nonlinear detector performs better but requires higher com-putational cost. Sphere decoding (SD) algorithm, for example,performs an efficient ML detection using the closest lattice point


search (CLPS) in a hypersphere with a small radius [20]. In spiteof the substantial reduction in complexity over the brute forceenumeration, computational burden of the SD algorithm is stilla major problem, since the expected complexity is exponen-tial with the problem size [21]. Due to these reasons, in manyfuture wireless scenarios where the dimension of the system ma-trix is much larger than that of today’s wireless systems, bothlinear and nonlinear principles have their own pros and consand may not offer an elegant tradeoff between performance andcomplexity.

B. Sparse Transform via Conventional Detection

When we use conventional detectors to the system model in(2), it is clear that the detector output cannot be always identicalto the original information vector s due to the errors. In fact,in the practical SNR regime, the detector output might containan error, causing mismatches for a few entries of s.3 In orderto exploit the sparsity of the detection errors, in our approachwe convert the non-sparse system into the sparse one. Con-ventional detection together with the symbol slicing serves ourpurpose since the estimated symbol vector is roughly accurate,and hence, the resulting error vector (defined as the differencebetween the original symbol vector and the sliced estimate) canbe modeled as a sparse signal. Denoting the estimate of a symbolvector as s and its sliced version as s, we have

s = Q(s) = s − e (4)

where Q(·) is the slicing function and e is the error vector. Asmentioned, in an operational regime of communication systems,the number of nonzero entries (i.e., real errors) in e would besmall so that the error vector is readily modeled as a sparsevector. Suppose the dimension of the symbol vector s is 16 andthe symbol error rate is 10%, then the probability that 5 or lesselements being in error is 99.7%. As long as the error vectoris sparse, by transmitting this error vector, one can construct asparse system expressed in terms of the error vector e. This task,henceforth referred to as the sparse transform, can be realizedby the re-generation of the received signal from the detectedsymbol s followed by the subtraction as

y′ = y −√

PHs, (5)

where y′ is the newly obtained received vector (see Fig. 2).Then, from (2), (4), and (5), the new measurement vector y′ is

y′ =√

PH(s − s) + v

=√

PHe + v. (6)

Interestingly, by adding trivial operations (matrix-vector mul-tiplication and subtraction), one can convert the original non-sparse system into the sparse system whose input is an errorvector associated with the conventional detector. Note that thesymbol slicing is essential in sparsifying the error vector and wehave two options:

3In the operating regime of communication systems, symbol error rate (SER)is typically less than 10%.

Fig. 2. Overall structure of the proposed PSED detection algorithm.

� Hard-slicing: hard-slicing literally performs the hard de-cision of symbol estimate. The slicer function maps theinput to the closest value in the symbol set Γ (i.e.,Q(z) = arg minγ∈Γ ‖z − γ‖2). By exploiting the discreteproperty of the transmit vectors in (6), we can enforce thesparsity of the input (error vector) for the modified system.Main benefit of the hard slicing is that it entirely removesthe residual interference and noise when the estimate liesin the decision region of the original symbol.

� Soft-slicing: when the a prior information on the sourceexists, soft slicing can be considered. One possible way isto use an MMSE-based soft slicing (si = E[si |si ]), wheresi denotes the i-th element of s [22]. For example, whenthe decoder feeds back the log-likelihood ratio (LLR) in-formation Lb on the information bit, one can transformthis into the symbol prior information Pr (si). Using thenonuniform symbol probability P (si), we can estimatesi as

si = E[si |si ]

=∑

si ∈Ω

siPr (si |si) =

∑si ∈Ω siPr (si |si)Pr (si)∑si ∈Ω Pr (si |si)Pr (si)

. (7)

When the linear detectors in (3) are used to obtain s,Pr (si |si) is given by

Pr (si |si) =1

2πσ2s

exp(− 1

σ2s

(si −

√PwH

i hisi

)2)

,

(8)where σ2

s = wHi

(PHHH + σ2

v I)wi − P (wH

i hi)2 , andwi and hi are the i-th column of W and H, respectively.In the hard slicing, all entries of e are zero except forthose associated with the detection errors. Whereas, entriesunassociated with the detection errors might have smallnonzero magnitude in the soft-slicing, yielding so calledapproximately sparse vector e.4

In our analysis and derivation that follow, we will mainlyfocus on the hard-slicing based PSED technique for analyticalsimplicity.

4By approximately sparse signal, we mean a vector containing most of energyin only a few elements.


TABLE IOPERATIONS OF THE PSED DETECTOR

C. Recovery of Sparse Error Vector

Once the non-sparse system is converted into the sparse one,we can use the sparse recovery algorithm to estimate the er-ror vector. To be specific, using newly obtained measurementvector y′ and the channel matrix H, sparse recovery algorithmestimates the error vector e (see Fig. 2). There are many algo-rithms designed to recover the sparse vector in the presence ofnoise. Well-known examples include basis pursuit de-noising(BPDN) [23] and orthogonal matching pursuit (OMP) [24]. Wewill say more about this in Section II-D.

Once the output e of the sparse recovery algorithm is obtained,we add it to the sliced detector output s, generating the refinedsymbol vector ˆs

ˆs = s + e = (s − e) + e = s + (e − e). (9)

If the error estimate is accurate, i.e., e ≈ e, then the magnitudeof the error difference ε = e − e would be small so that there-estimated symbols ˆs becomes more accurate than the initialestimate s. As long as the sparsity of the error vector is ensured(i.e., number of nonzero elements in error vector e is small),an output of the sparse recovery algorithm e would be faithful(i.e., ‖e − e‖2

2 < ‖e‖22) so that the refined symbol vector will

be more reliable than the original estimate. We provide moredeliberate analysis in Section III.

It is worth mentioning that the sparse error recovery processis conceptually analogous to the decoding process of the linearblock code [26]. First, the sliced symbol vector s = s − e canbe viewed as a received vector (often denoted by r in the codingtheory) which is typically expressed as the sum of the transmitcodeword and the error vector. Also, the new observation vectory′ =

√PHe + v is similar to the syndrome, the product of

the error vector and the transpose of parity check matrix. Notethat the syndrome is a sole function of the error vector anddoes not depend on the transmit codeword. Similarly, the newobservation vector y′ is a function of e and independent of thetransmit vector s. Furthermore, in the linear block code, thedecoded error pattern is correct only when the cardinality ofthe error vector is within the error correction capability t (i.e.,‖e‖0 < t) and similar behavior occurs to the problem at handsince an output of the sparse recovery algorithm will be reliableonly when the error vector e is sparse (i.e., ‖e‖0 � nt). Finally,the error correction is performed in the decoding process byadding the reconstructed error vector e to the received vectorr and the same is true for the proposed algorithm (see (9)). InTable I and II, we summarize the proposed PSED algorithmand similarity between the PSED algorithm and the decodingprocess of the linear block code.

D. Sparse Recovery Algorithm

In this subsection, we briefly describe the sparse recoveryalgorithm used for the proposed PSED scheme. Note that anapproach often called Lasso or Basis pursuit de-noising (BPDN)formulates the problem to recover the sparse signal in the noisyscenario as [34]

mine

12‖y′ −

√PHe‖2

2 + λ‖e‖1

where λ is the penalty term to control the amount of weightgiven to the sparsity of the desired signal e. This problem isin essence a convex optimization problem and there are manyalgorithms to solve this type of problem [35]. As an alternativeto the convex optimization problem, greedy algorithms have re-ceived great deal of interest in recent years [4]. In a nutshell,greedy algorithms attempt to find the support of e (i.e., the setof columns in H constructing y′) in an iterative fashion, gener-ating a sequence of the estimate for e. In the OMP algorithm,for example, a column of H that is maximally correlated withthe modified measurement vector (residual r) is chosen as anelement of the support set E [24]. Then the estimate of the de-sired signal eE is constructed by projecting y′ onto the subspacespanned by the columns supported by E . That is,

eE =1√P

(HH

E HE)−1

HHE y′ (10)

Finally, we update the residual r so that it contains the mea-surement excluding those included by the estimated support set(r = y′ − √

PHE eE ). In case the OMP algorithm identifies thesupport E accurately (E = E), one can remove all non-supportelements in e and corresponding columns in H so that one canobtain the overdetermined system model

y′ =√

PHEeE + v. (11)

The final estimate is equivalent to the best possible estimatecalled the Oracle LS estimator,5

eE =1√P

(HH

E HE)−1

HHE y′.

While the OMP algorithm is simple to implement and alsocomputationally efficient, due to the selection of the single can-didate in each iteration, the performance depends heavily onthe selection of index. In fact, the output of OMP would bewrong if an incorrect index (index not contained in the sup-port) is chosen in the middle of the search. In order to alleviatethis drawback, various alternatives investigating multiple indiceshave been suggested. Recent developments of this approach in-clude compressive sampling matching pursuit (CoSaMP) [36],subspace pursuit (SP) [37], and generalized OMP [38].

In this work, we employ the multipath matching pursuit(MMP) algorithm, a recently proposed near-ML greedy treesearch algorithm, in recovering the sparse error vector [27].

5In case the a prior information on signal and noise is available, one canalternatively use the linear minimum mean square error (LMMSE) estimate. Forexample, if σ2

e = E [|eE |2 ] and σ2v = E [|v|2 ], we have eE = 1√

P(HH

E HE +σ 2

v

P σ 2eI)−1HH

E y′


TABLE IISIMILARITIES BETWEEN THE DECODING PROCESS OF THE LINEAR BLOCK CODE AND THE PSED ALGORITHM. IN THE DECODING PROCESS, r, c,

AND s DENOTE RECEIVED, CODEWORD, AND SYNDROME VECTORS, RESPECTIVELY, AND H IS THE PARITY CHECK MATRIX

Fig. 3. Comparison between the OMP and the MMP algorithm (L = 2 and K = 3). While OMP maintains a single candidate T k in each iteration, MMPinvestigates multiple promising candidates T k

j (1 ≤ j ≤ L) (the subscript j counts the candidate in the i-th iteration).

While aforementioned greedy recovery algorithms identify theelements of support sequentially and choose a single supportestimate, MMP performs the list tree search to find the mul-tiple promising supports (we henceforth refer to it as supportcandidates). Among multiple candidates, the best one mini-mizing the residual power is chosen as an output in the lastminute. As shown in Fig. 3, each support candidate bringsforth L child candidates in the MMP algorithm. Since multi-ple promising candidates are investigated, one can expect thatMMP outperforms the sequential greedy algorithm returning asingle support candidate. In fact, it has been shown that MMPperforms close to the best possible estimator using genie supportinformation (called Oracle estimator) for high SNR regime [27,Theorem 4.6]. The operation of MMP is summarized in Ta-ble III.

III. PERFORMANCE ANALYSIS

In this section, we analyze the performance of the proposedPSED algorithm. As a performance metric, we consider thenormalized MSE between the original symbol vector s and theoutput of PSED algorithm ˆs, which is defined as

MSE =1nt

E[‖s − ˆs‖2 ] (12)

=1nt

E[‖s − (s + e − e)‖2 ] (13)

=1nt

E[‖e − e‖2 ] (14)

One can observe that the MSE associated with the symbol vec-tor s is equivalent to the MSE associated with the error vector e.Since we use the random matrix theory for analytical tractabil-

TABLE IIITHE MMP ALGORITHM

ity, our analysis is accurate when the dimension of the channelmatrix is large. In our analysis, we assume that the sparse re-covery algorithm identifies the support e (index set of nonzeroelements) of the error vector accurately. This is true when thechannel matrix satisfies a property called the restricted isometryproperty (RIP)6 and the signal power is sufficiently higher thanthe poise power (see, e.g., [11, Theorem 4.6]).

When the support of e is identified, all non-support elementsin e and columns of H associated with these can be removed

6A matrix H ∈ Rn r ×n t is said to satisfy the RIP condition if (1 −δK )‖x‖2

2 ≤ ‖Hx‖22 ≤ (1 + δK )‖x‖2

2 .


from the system model. The resulting overdetermined systemmodel is7

y′ =√

PHe + v (15)

=√

PHEeE + v. (16)

Note that most of greedy sparse recovery algorithms use thelinear squares (LS) solution in generating the estimate of errorvector eE . In this case, the estimate eE is given by [28]

eE =1√P

(HH

E HE)−1

HHE y′. (17)

From (16) and (17), the MSE is expressed as

MSEpsed =1nt

E[‖e − e‖2 ] (18)

=1nt

E[‖eE − eE‖2 ] (19)

= EH

[trEE ,v

[1

ntP

(HH

E HE)−1HH

E vvH

· HE(HH

E HE)−1∣∣H

]], (20)

where Ex [·] is the expectation with respect to the random vari-able x. One thing to notice is that HE and v are correlated witheach other since the error pattern associated with HE dependson the realization of the noise vector v. Since this makes theevaluation of (20) very difficult, we take an alternative approachand investigate a lower bound of the MSE. First, let E be the setof indices whose elements are randomly chosen from the set ofall column indices Ω = {1, 2, . . . , n} and the cardinality of E isthe same as that of E (i.e., |E | = |E|). Then, we conjecture that

MSEpsed ≥ EH

[trEE ,v

[1

ntP

(HH

E HE)−1

HHE vvH

· HE(HH

E HE)−1

∣∣∣H]]

(21)

= EH

[1

SNRtrEE ,v

[1nt

(HH

E HE)−1

∣∣∣H]]

. (22)

where SNR = Pσ 2

v. Note that this conjecture is justified by the

fact that columns associated with E are caused by the detectionerrors while those associated with E are randomly chosen sothat the former would yield higher MSE than the latter. To judgethe effectiveness of this lower bound, we perform empiricalsimulations of two quantities (i.e., (20) and the right-hand sideof (21)) for i.i.d. complex Gaussian random matrix. We observefrom Fig. 4 that the conjecture holds true empirically and alsothe lower bound in (21) is tight across the board.

We next investigate an asymptotic behavior of the term

E[ 1SNR trE[ 1

nt(HH

E HE)−1∣∣∣H]]. We assume that the elements

of the channel matrix H are i.i.d. complex Gaussian (hij ∼

7HD is a submatrix of H that only contains columns indexed by D. Forexample, if D = {1, 4, 5}, then HD = [h1 h4 h5 ] where hj is the j-th columnof H.

Fig. 4. Comparison of the exact MSE in (20) and lower bound in the right-handside of (21) for i.i.d. complex Gaussian random matrix. We set nr = nt = 128and use BPSK modulation.

CN (0, 1/nr )) and the dimension of the channel matrix H is suf-ficiently large (i.e., nt, nr → ∞) with a constant ratio nt

nr→ β.

Let HE be the submatrix generated by randomly choosing |E|columns of H. Then we have

1|E| tr

(HH

E HE)−1 → 1

1 − β′

where β′ = |E|nr

[29, Section 2.3.3]. Notice that when the di-mension of the matrix is large, the quantity converges to thedeterministic limit depending only on the ratio β′. Therefore, itsuffices to evaluate a single matrix realization in (22) and hencethe outer expectation with respect to the channel realization isunnecessary.

We next investigate the distribution of |E| (cardinality of E)which corresponds to the number of errors in the output streamsof the linear detector. Here we consider the LMMSE detector asa conventional linear detector. Note that the event of detectionerrors is related to the post-detection signal-to-interference-and-noise-ratio (SINR) at the output of the LMMSE detector. Whilethe SINR for each output stream relies on a channel realization,when the system size is sufficiently large (i.e., nt, nr → ∞ andnt

nr→ β), the SINR for all output streams approaches to the

same quantity [29]

SINR(LMMSE)i −→ SINR(LMMSE)

∞ = SNR − F(SNR, β)4

where F(x, z) =(√

x(1 +√

z)2 + 1 −√

x(1 −√z)2 + 1

)2

and SINR(LMMSE)i is the SINR for the ith stream.

In addition, one can show that the output streams of theLMMSE receiver are asymptotically uncorrelated with eachother (see Appendix A). Thus, the detection problem for eachoutput stream can be considered separately and the number oferrors |E| is approximated as a Binomial distribution with thesuccess probability Pe , where Pe is the probability of error event


for each output stream. That is,

Pr(|E| = t) ≈(

nt

t

)

P te (1 − Pe)nt −t

When nt is large, the Binomial distribution with parameternt and Pe approaches to the Normal distribution N (ntPe,ntPe(1 − Pe)) by DeMoivre-Laplace theorem [30] and henceone can show that (see Appendix B)

Pr

(∣∣∣∣|E|nr

− Peβ

∣∣∣∣ > ε

)→ 0 (23)

Pr

(∣∣∣∣|E|nt

− Pe

∣∣∣∣ > ε

)→ 0. (24)

Our discussions so far can be summarized as follows.1) 1

|E| tr(HH

E HE)−1

converges to 11−β ′ = 1

1− |E|n r

irrespective

of the channel realization.2) |E|

nrand |E|

ntconverge in probability to Peβ and Pe , respec-

tively.Using these, one can show that

1SNR

trE[

1nt

(HH

E HE)−1

∣∣∣H]

=1

SNRE

[ |E|nt

1|E| tr

(HH

E HE)−1

∣∣∣H]

(25)

→ 1SNR

E

[|E|nt

1

1 − |E|nr

]

(26)

=1

SNRnr

ntE

[ |E|nr

1 − |E|nr

]

(27)

≥ 1SNR

nr

nt

E |E|nr

1 − E |E|nr

(28)

=1

SNR1β

Peβ

1 − Peβ(29)

=1

SNRPe

1 − Peβ. (30)

where (28) is from Jensen’s inequality. Noting that |E|nr

� 1, wesee that the obtained lower bound is tight.8

In the high SNR regime, Pe � 1 and hence

MSEpsed >1

SNRtrE

[1nt

(HH

E HE)−1

∣∣∣H]

� Pe

SNR. (31)

Since Pe is a function of SINR, the obtained lower boundof MSEpsed is a function of SNR and SINR. Indeed, whennt, nr → ∞ and nt

nr→ β, the residual interference plus noise

for the LMMSE detector approaches to Normal distribution[31], [32] so that Pe can be readily expressed as a function of the

8f (x) = x1−x is a convex function for 0 < x < 1 and hence E

[x

1−x

]≥

E [x ]1−E [x ] . In our case, x = |E|

n rand hence x � 1 so that f (x) ≈ x and the lower

bound is tight.

SINR. For example, for the binary phase shift keying (BPSK),the error probability Pe in symbol detection approaches [33]

Pe = Erf(SINR(LMMSE)∞ ) = Q

(√2SINR(LMMSE)

∞

)(32)

where Q(x) =∫∞

x1√2π

exp(− t2 )dt. Using Q(x) > x

1+x2

1√2π

e−x2 /2 , we have

MSEpsed >

Q

(√2SINR(LMMSE)

∞

)

SNR

>1

SNR

√2SINR(LMMSE)

∞1 + 2SINR(LMMSE)

∞

· 1√2π

exp(−SINR(LMMSE)

∞)

(33)

In high SNR regime, we have [19]

SINR(LMMSE)∞ ≈

{ √SNR β = 1

(1 − β)SNR + β1−β β < 1

(34)

and hence

MSEpsed

�

⎧⎪⎪⎨

⎪⎪⎩

12√

π

1SNR5/4 exp

(−√

SNR)

if β = 1,

12√

π(1 − β)1

SNR3/2 exp (−(1−β)SNR) if β < 1.

(35)

On the other hand, MSE of the conventional linear MMSEdetector is given by [29]

MSEconv =1nt

EH[tr(I + SNRHH H)−1]

−→ 1 − F(SNR, β)4βSNR

. (36)

Further, in high SNR regime, MSEconv can be expressed as[19,Chap. 6.3]

MSEconv → 1 − F(SNR, β)4βSNR

≈

⎧⎪⎪⎨

⎪⎪⎩

1√SNR

if β = 1,

1(1 − β)SNR

if β < 1.

(37)

It is interesting to compare (35) and (37), asymptotic MSEs ofthe proposed and the conventional schemes when the dimensionof the matrix goes to infinity. While the MSE of the conventionalmethod decays linearly or sublinearly with SNR, the MSE ofthe PSED decays exponentially with SNR. In Fig. 5, we plotthe MSE in (35) and (37) and the empirical MSEs as a func-tion of the SNR when the complex Gaussian system matrix isemployed. In Fig. 5(a)–(c), we use the BPSK modulation, andin Fig. 5(d), we use quadrature amplitude modulation (QAM)


Fig. 5. The MSE performance of the PSED technique and the conventional detector: (a) nr = nt = 32, (b) nr = nt = 64, and (c) nr = nt = 128 whenBPSK modulation is used. (d) The MSE performance for nr = nt = 64 system is shown for several QAM modulations.

modulations with the modulation order 4, 16 and 64. For allcases considered, we observe that the obtained bound is closeto the simulation results. We also observe that the performancedifference between the PSED algorithm and the conventionaldetector is substantial and further the difference increases withSNR. This is mainly because the accuracy of the estimated er-ror vector e for the sparse recovery algorithm improves withSNR so that the magnitude of the error difference ε = e − e isreduced sharply.

IV. COMPUTATIONAL COST OF SPARSE RECOVERY

In this subsection, we analyze the computational complexityof the PSED algorithm. We consider two versions of the PSEDalgorithm; PSED with LMMSE detection (PSED-LMMSE) andPSED with MF detection (PSED-MF). As mentioned, additionaloperations caused by the proposed method are 1) sparse trans-form and 2) sparse error recovery. While the computational

overhead of the sparse transform is fixed (matrix multiplica-tion and subtraction), that for the sparse error vector recoverydepends on the tree branching parameter L of MMP and thesparsity K. Since the number of iterations of MMP is set to thesparsity K, K matrix inverse operations (from 1 × 1 to K × Kmatrix inverse) are required. Noting that the sparsity K is muchsmaller than the dimension of symbol vector nt and the MMP al-gorithm performs well with small value of L (in our simulations,we set L = 2), additional burden of the PSED is very marginal.

Table IV summarizes the number of complex multiplicationsrequired for PSED and the conventional detectors. When theLMMSE detector is used, inversion of the covariance matrix isneeded in the weight generation process. In Table IV, we denotethe number of complex multiplications to perform the inversionof an n × n matrix by Iv (n). Since the required complexityto invert a matrix is cubic in the dimension n of the matrix(i.e., O(n3)), additional matrix inversion overhead (from 1 ×1 to K × K dimensional matrix) of PSED-LMMSE is small


TABLE IVTHE TOTAL NUMBER OF COMPLEX MULTIPLICATIONS OF SEVERAL DETECTORS

TABLE VCOMPARISON OF COMPUTATIONAL COMPLEXITY IN TERMS OF NUMBER OF COMPLEX MULTIPLICATIONS OF DETECTION SCHEMES (NUMBERS IS IN THOUSANDS)

when compared to the nt × nt dimensional covariance matrixinversion of LMMSE.9 For example, additional complexity ofPSED-LMMSE for nt = nr = 32 is only 13% and that for nt =nr = 64 is 20%. When the MF is used (PSED-MF), there is noinversion and thus the additional complexity associated withPSED might be relatively higher than that of PSED-LMMSE.In spite of the increased computational overhead, due to the low-complexity operations, overall complexity of the PSED-MF ismuch smaller than the computational burden of the LMMSEdetector.

V. SIMULATIONS AND DISCUSSIONS

A. Simulation Setup

In this section, we examine the performance of the proposedPSED algorithm (PSED-LMMSE and PSED-MF). For com-parison, we test conventional linear receivers (MF and LMMSEdetectors), K-best detection algorithm [25], and IO-LAMA [16].Also, as a performance lower bound, we consider the ML de-tector implemented via sphere decoding (SD) algorithm. Notethat the K-best detector is a sub-optimal MIMO detector and itscomplexity does not vary with channel realizations and SNR.It has been shown that IO-LAMA asymptotically achieves theperformance of individual parallel detection over random i.i.d.

9Note that gaussian elimination method for n × n inversion requires (2n3 +3n2 − 5n)/6 multiplications [40]. See [41] for more efficient implementationof matrix inversion.

Fig. 6. The SER as a function of SNR for (nr , nt ) = (32, 32) system.

Gaussian channels [16] Note also that we only present the resultof the ML detector for 32 × 32 dimensional system since it isvery hard to obtain the results when the dimension is higher thanthis.

In order to check the worst-case performance, we always turnon the sparse recovery algorithm of the PSED even in the lowSNR regime where the error signal would not be sparse. Asmentioned, we use MMP in recovering the sparse error vector.


Fig. 7. The SER as a function of SNR for (a) QPSK, (b) 16-QAM, and (c) 64-QAM modulations. The uncoded system with the size (nr , nt ) = (64, 64) isconsidered.

The branching parameter L is fixed to 2. Ideally, the sparsityparameter K should be equal to the number of errors of theconventional detector used in the first stage. Since the numberof errors is unknown, we compute analytic symbol error rate Pe

(see (32)) and set K = Pent in our simulations. In the K-bestdetector, we set the number of the symbols survived for eachlayer such that overall complexity is comparable to that of thePSED-LMMSE (see the complexity comparison in Table V).The SNR per each receive antenna is defined as

SNRr =nt

nrSNR. (38)

B. Simulation Results

1) Performance for i.i.d. Gaussian System Matrix: First, weevaluate the performance of the proposed detector for the sys-tem matrix H whose entries are chosen from i.i.d. complexGaussian (i.e., hi,j ∼ CN (0, 1

nr)). We assume that the channel

information is perfectly known in the receiver.

In order to demonstrate the efficacy of the sparse recoveryalgorithm, we compare the symbol error rate (SER) perfor-mance of several detectors for (nr , nt) = (32, 32) uncoded sys-tem. In Fig. 6, we observe that the proposed PSED-LMMSEmethod offers substantial performance gain over the conven-tional LMMSE detector. “Sparse transform + LS” refers to themethod employing sparse transform followed by LS estima-tion of error vector e. Since this scheme does not rely on thesparse recovery and thus cannot suppress the noise effectively,performance of this approach is much worse than that of thePSED-LMMSE. Note that the performance of PSED-LMMSEis within 2 dB from that of the optimal ML detector.

In Fig. 7, we plot the SER performance of the PSED algo-rithm and the conventional MIMO detectors for the uncodedsystem with the size (nr , nt) = (64, 64). In Fig. 7(a), (b), and(c), we provide the SER performance when QPSK, 16-QAM,64-QAM modulations are used. We observe that the proposedPSED-LMMSE outperforms the conventional LMMSE detec-tor by a large margin for all cases under consideration. Note


Fig. 8. The SER as a function of SNR for (a) QPSK, (b) 16-QAM, and (c) 64-QAM modulations. The coded system with the size (nr , nt ) = (64, 64) isconsidered.

that the performance gain of the PSED-LMMSE is pronouncedin high SNR regime. This is because 1) stronger sparsity, i.e.,lower K leads to better recovery performance and thus 2) thegain obtained from the post sparse error detection/correctionprocess increases sharply in this regime. Specifically, in the lowSNR regime, number of errors in a symbol vector is relativelylarge (that is, sparsity of the input error vector is high) so thatthe sparse recovery algorithm would not work properly and thegain obtained from the PSED technique would be marginal. Onthe other hand, in the high SNR regime, input error vector ofthe the sparse recovery algorithm is sparse and thus the PSEDwould return highly reliable error vector estimate, resulting inthe substantial performance gain. Note also that in contrast toPSED-LMMSE, the performance of PSED-MF is not so appeal-ing when nt is equal to nr . This is because the MF detector doesnot perform well for the SNR range of interest so that the in-put error vector after the sparse transformation is not sparse,resulting in poor reconstruction performance of the sparse

recovery algorithm. We observe that IO-LAMA outperforms theproposed PSED detector when QPSK modulation is used, but itsuffers from error floor phenomenon in high order modulationscenarios (i.e., 16QAM and 64QAM). Note that the proposedscheme outperforms the K-best detector for all cases considered.In Table V, we provide the computational complexity (i.e., thenumber of complex multiplications) of detection schemes un-der consideration. When compared to the complexity of theconventional LMMSE detector, we see that the computationalcomplexity required to perform the post processing in the PSEDdetectors is quite marginal. Note also that in spite of significantperformance gain, the complexity of PSED-LMMSE is compa-rable to that of K-best detector.

In Fig. 8, we investigate the bit error rate (BER) perfor-mance of PSED-LMMSE when the forward error correctioncode (FEC) is employed. We use the convolutional channel codewith generator polynomial (171, 131) and the Viterbi decoderto perform channel decoding [46]. The size of code block is set


TABLE VICOMPLEXITY AND PERFORMANCE OF LMMSE AND PSED-LMMSE DETECTORS FOR DIFFERENT SYSTEM SIZES

to 1536 bits and a random interleaver is used between the chan-nel encoder and the symbol modulator. For fair comparison, weused the hard-sliced symbol in the channel decoding. Overall,we observe that the performance gain of the PSED-LMMSEover the conventional LMMSE detection scheme is well main-tained for all modulation orders considered. We also see thatthe K-best detector performs worse than the LMMSE detectorfor the SNR range of interest. IO-LAMA detector works well inQPSK systems but it exhibits error floor phenomenon for highorder modulation scenarios.

Finally, we compare the performance and complexity of theLMMSE detector and PSED-LMMSE detector for various sys-tem dimensions. In Table VI, we provide the number of com-plex multiplications and the required SNR at 10−2 SER for(nr , nt) = (32, 32), (64, 64), (128, 128), and (256, 256). Notethat for all dimensions we considered, the proposed PSED-LMMSE maintains large performance gain over the LMMSE.When the system dimension increases, computational complex-ity of the PSED-LMMSE also increases due to the large numbercomputations needed for the sparse recovery algorithm.

2) Performance Evaluation for Massive MIMO Scenarios:We finally investigate the performance of the PSED in realis-tic massive MIMO scenario. We consider the uplink scenariowhere nt users with single antenna transmit the information tothe basestation with nr transmit antennas. The channel vectorbetween the kth user to the basestation is given by [47]

hk =1√P

P∑

p=1

gkpa(θp) (39)

where gkp is the channel gain for the kth user, P is the totalnumber of multi-paths, θp is the angle of arrival for the pthmulti-path signal, and a(θp) is the steering vector expressed as

a(θp) =[1 e−j2πγ sin(θp ) · · · e−j (nr −1)2πγ sin(θp )

]. (40)

Note that γ is the ratio of the antenna spacing to the wavelength.The corresponding channel matrix H is given by

H =1√P

AG. (41)

We assume that the signal powers received from each user areequal and hence the entries of the matrix G are i.i.d. Gaussianwith zero mean and unit variance. In our simulations, we setλ = 0.3 and θp = −π/2 + (p − 1)π/P where p = 1, 2, . . . , Pwhere P is the number of the multi-paths (P is set to 32) [47].

Fig. 9. The SER vs. SNR plot of the massive MIMO detectors fora) (nr , nt ) = (64, 32) and b) (nr , nt ) = (128, 32).

Fig. 9 provides the performance of the proposed detector fornt = 32 (i.e., 32 mobile users) and (a) nr = 64 and (b) nr = 128basestation antennas, respectively. We observe that the proposedPSED-LMMSE detector achieves significant performance gainover the K-best detector as well as the conventional LMMSEdetector. We also observe that as the number of basestation


Fig. 10. The SER vs. SNR plot of the IoT detectors for a) (nr , nt ) =(128, 128) and b) (nr , nt ) = (256, 256).

antennas increases, the performance gain of the proposedmethod over the conventional detectors improves significantly.

3) Performance Evaluation for IoT Multiuser Detection Sce-narios: We evaluate the performance of the proposed detectorfor the IoT scenario where nt devices access the AP and thereceiver algorithm in the AP detects the data symbols from allusers simultaneously. Each device is encoded with a pseudonoise (PN) based signature sequence with length nr and weassume that the signal transmission of all users is synchronous.

Fig. 10 shows the detection performance of the proposeddetector as a function of SNR. Two configurations (nr , nt) =(128, 128) and (256, 256) are considered in the simulations.Note that we do not include the results of the MF and PSED-MF detectors since these schemes perform poor. We show thatthe proposed PSED-LMMSE achieves significant performancegain over the existing detectors, resulting in 2.3 dB and 5.5 dBgain at 10−2 SER.

VI. CONCLUSION

In this paper, we proposed a novel detection algorithm exploit-ing sparsity of error vector in the conventional linear detection inimproving the detection quality of symbol vectors in large-scalewireless systems. Our approach operates in two steps. In the firststep, we transform the conventional communication system intothe system whose input is the sparse error vector, which is ac-complished by the conventional linear detection followed by thesymbol quantization. For the transformed measurement vector,we next apply the sparse error recovery algorithm followed bythe error cancellation to obtain the refined estimate of the trans-mit symbol vector. Our approach is simple to implement withcomparably small computational cost, yet offers substantial gainover the conventional detection schemes. We observed from theasymptotic performance analysis and empirical simulations thatthe proposed PSED algorithm achieves significant performancegain over the conventional detection schemes.

APPENDIX APROOF THAT THE OUTPUT STREAMS OF THE LINEAR MMSE

DETECTOR ARE ASYMPTOTICALLY UNCORRELATED

The ith output stream of the linear MMSE detector is givenby si = hH

i

(HHH + 1

SNR I)−1

. The correlation between theith and jth output streams is given by

E[si s∗j ] = PhH

i

(HHH +

1SNR

I)−1

hj , (42)

where i �= j. If we use a matrix inversion lemma xH (A +τxxH )−1 = xH A−1

1+τ xH A−1 x , we can show that

E[si s∗j ] =

hHi

(H[i,j ]HH

[i,j ] + 1SNR I

)−1hj

(1 + hH

i

(H[i]HH

[i] + 1SNR I

)−1hi

)

· 1(

1 + hHj

(H[j ]HH

[j ] + 1SNR I

)−1hj

) , (43)

where H[A] is the submatrix of H with the columns specifiedby the index set A are removed. According to Lemma 4 in [43],when nt, nr → ∞, the numerator in (43) converges to zero andthe denominator converges to one.

APPENDIX BPROOF OF (23) AND (24)

As mentioned, the distribution of |E| is approximatedby N (ntPe, ntPe(1 − Pe)). Thus, the quantity |E |

ntfollows

N (Pe,Pe (1−Pe )

nt) and

Pr

(∣∣∣∣|E|nt

− Pe

∣∣∣∣ > ε

)= 2Q

⎛

⎝ ε√

Pe (1−Pe )nt

⎞

⎠ (44)

< 2 exp(− ε2n2

t

2Pe(1 − Pe)

), (45)


for any positive ε. When nt → ∞, (45) goes to zero and thus|E |nr

(= β |E |nt

) also converges to Peβ.

ACKNOWLEDGMENT

The authors would like to thank the associate editor (Prof.A. Wiesel) and anonymous reviewers for their valuable com-ments and suggestions that improved the quality of the paper.

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Jun Won Choi (M’04) received the B.S. and M.S.degrees in electrical and computer engineering fromSeoul National University, Seoul, South Korea, andthe Ph.D. degree in electrical and computer engi-neering from the University of Illinois at Urbana-Champaign, Champaign, IL, USA, respectively. In2010, he joined Qualcomm, San Diego, CA, USAand participated in wireless communication sys-tems/algorithm design for commercializing LTE andLTE-A modem chipsets. Since 2013, he has been afaculty member in the Department of Electrical En-

gineering, Hanyang University, Seoul, South Korea, and leading the Signal Pro-cessing and Machine Learning Research Group. His research interests includesparsity-based signal processing, wireless communications, machine learning,and intelligent transportation systems.

Byonghyo Shim (S’95–M’97–SM’09) received theB.S. and M.S. degrees in control and instrumen-tation engineering from Seoul National University,Seoul, South Korea, in 1995 and 1997, respectively.He received the M.S. degree in mathematics and thePh.D. degree in electrical and computer engineeringfrom the University of Illinois at Urbana-Champaign,Champaign, IL, USA, in 2004 and 2005, respectively.

From 1997 and 2000, he was with the Depart-ment of Electronics Engineering, Korean Air ForceAcademy, as an Officer (First Lieutenant) and an Aca-

demic Full-time Instructor. From 2005 to 2007, he was with Qualcomm Inc.,San Diego, CA, USA, as a Staff Engineer. From 2007 to 2014, he was withthe School of Information and Communication, Korea University, Seoul, SouthKorea, as an Associate Professor. Since September 2014, he has been with theSeoul National University, where he is currently a Professor in the Departmentof Electrical and Computer Engineering. His research interests include wirelesscommunications, statistical signal processing, estimation and detection, com-pressed sensing, and information theory. He received the M. E. Van ValkenburgResearch Award from the Electrical and Computer Engineering Departmentof the University of Illinois (2005), Hadong Young Engineer Award from theIEIE (2010), and Irwin Jacobs Award from Qualcomm and KICS (2016). He isan elected member of Signal Processing for Communications and NetworkingTechnical Committee of the IEEE Signal Processing Society. He has been anAssociate Editor of the IEEE TRANSACTIONS ON SIGNAL PROCESSING, IEEEWIRELESS COMMUNICATIONS LETTERS, Journal of Communications and Net-works, and a Guest Editor of the IEEE JOURNAL ON SELECTED AREAS IN

COMMUNICATIONS.

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