On the Performance of Mismatched Data Detection in Large MIMO Systems

Transcription

1 On the Performance of Mismatched Data Detection in Large MIMO Systems Chares Jeon, Arian Maei, and Christoph Studer arxiv: v [cs.it] Jun 016 Abstract We investigate the performance of mismatched data detection in arge mutipe-input mutipe-output MIMO) systems, where the prior distribution of the transmit signa used in the data detector differs from the true prior. To minimize the performance oss caused by this prior mismatch, we incude a tuning stage into our recenty-proposed arge MIMO approximate message passing LAMA) agorithm, which aows us to deveop mismatched LAMA agorithms with optima as we as sub-optima tuning. We show that carefuy-seected priors often enabe simper and computationay more efficient agorithms compared to LAMA with the true prior whie achieving nearoptima performance. A performance anaysis of our agorithms for a Gaussian prior and a uniform prior within a hypercube covering the QAM consteation recovers cassica and recent resuts on inear and non-inear MIMO data detection, respectivey. I. INTRODUCTION Data detection in mutipe-input mutipe-output MIMO) systems deas with the recovery of the transmit data vector s 0 O MT, where O is a finite consteation e.g., QAM or PSK), from the noisy input-output reation y = Hs 0 + n. In what foows, M T and M R denotes the number of transmit and receive antennas, respectivey, y C MR is the receive vector, H C MR MT is the MIMO system matrix, and n C MR is i.i.d. circuary symmetric compex Gaussian noise. To minimize the symbo-error rate, we are interested in soving the individuay-optima IO) data detection probem [] [4] IO) s IO = arg max s O p s y, H), = 1,..., M T. Here, s IO denotes the -th IO estimate and p s y, H) is the conditiona probabiity density function of s O given the receive vector y and the channe matrix H. The probem IO) is nown to be of combinatoria nature [] [4], and the use of an exhaustive search or spheredecoding methods resuts in prohibitive compexity for systems where M T is arge [5]. In contrast, our recenty proposed agorithm referred to as arge MIMO approximate message passing LAMA) [6], achieves IO performance using a simpe iterative procedure in the arge-system imit, i.e., where we fix the system ratio β = M T /M R and et M T. For finitedimensiona systems, LAMA was shown to deiver near-io performance at ow computationa compexity [6]. Despite a these advantages, LAMA requires repeated computations of transcendenta functions that exhibit a high dynamic range. C. Jeon and C. Studer are with the Schoo of ECE at Corne University, Ithaca, NY; e-mais: jeon@cs.corne.edu, studer@corne.edu. A. Maei is with Department of Statistics at Coumbia University, New Yor City, NY; e-mai: arian@stat.coumbia.edu. The wor of CJ and CS was supported in part by Xiinx Inc. and by the US Nationa Science Foundation under grants ECCS and CCF An extended version of this paper incuding a proofs is in preparation [1]. These computations render the design of corresponding highthroughput hardware designs that rey on finite precision e.g., fixed-point) arithmetic a chaenging tas. A. Contributions We propose a mismatched version of the LAMA agorithm short M-LAMA), which enabes the design of hardwarefriendy data detectors that achieve near-io performance. We first deveop a mismatched version of the compex Bayesian approximate message passing cb-amp) agorithm [7] that incudes a tuning stage to minimize the performance oss caused by the mismatch in the signa prior. We propose the mismatched state-evoution SE) framewor to enabe a performance anaysis in the arge-system imit. We then appy our framewor to mismatched data detection in arge MIMO systems by considering two mismatched prior distributions: i) a Gaussian prior and ii) a uniform prior within a hypercube covering the QAM consteation. We anayze the performance of the resuting mismatched agorithms in the arge-system imit and demonstrate their efficacy in finite-dimensiona systems. B. Reevant prior art The performance of zero forcing ZF) and minimum meansquare error MMSE) detection, which are both we-nown instances of mismatched detection agorithms, has been investigated in [8] [10] for the arge-system imit. The use of a uniform prior within a hypercube eads to an aternative mismatched detector for antipoda e.g., BPSK) signas [11] [13]. Corresponding theoretica resuts in [14], [15] for noiseess systems reveaed that a system ratio of β < enabes perfect signa recovery. The error-rate performance for the noisy case was derived recenty in [13]. The anaysis of the mismatched agorithms presented in our paper recovers a these resuts. The proposed M-LAMA agorithm reies upon approximate message passing AMP) [16] [18], deveoped for sparse signa recovery. The case of mismatched estimation of sparse signas via AMP was studied in [19], [0], where the performance of AMP was anayzed when the true prior is unnown. This AMP agorithm incudes a tuning stage for automated, optima parameter seection that minimizes the output mean-squared error MSE). The ey differences between M-LAMA and the resuts in [19], [0] are that i) we consider MIMO data detection and ii) we now the true signa prior and intentionay seect a mismatched prior in order to design hardware-friendy data detection agorithms that enabe near-io performance. C. Notation Lowercase and uppercase bodface etters designate vectors and matrices, respectivey. We define the adjoint of matrix H as

2 H H and use to abbreviate x = 1 N N =1 x. A mutivariate compex-vaued Gaussian probabiity density function pdf) is denoted by CN m, K), where m is the mean vector and K the covariance matrix. E X [ ] and Var X [ ] denotes the mean and variance with respect to the random variabe X, respectivey. II. MISMATCHED COMPLEX BAYESIAN AMP We start by presenting a mismatched version of the compex Bayesian approximate message passing cb-amp) agorithm [7] short mcb-amp), which enabes the use a different prior distribution p s) than the true signa prior ps 0 ). A. The mismatched cb-amp mcb-amp) agorithm Given an i.i.d. prior distribution ps 0 ) = N =1 ps 0) of the true signa s 0 and a mismatched prior p s) = N =1 p s ), the proposed mcb-amp agorithm corresponds to σ t = 1 M R r t, [ 1) τ t = arg min E S0,Z F mm S 0 + σ t Z, τ) S 0 ], ) τ 0 s t+1 = F mm s t + H H r t, τ t), 3) r t+1 = y Hs t+1 + βr t F mm s t + H H r t, τ t ), which is carried out for t max iterations t = 1,..., t max. The agorithm is initiaized by s 1 = E S 0 [S 0 ] for a = 1,..., M T, where S 0 ps 0 ), r 1 = y Hs 1, and F mm is the derivative of F mm taen by its first argument. The function F mm s, τ) = E S[ S s ] = C sp s s, τ)d s, 4) is the posterior mean with respect to the mismatched prior p s ) and the variance parameter τ, p s s, τ) = 1 C ps s, τ) p s), C is a normaization constant, and ps s, τ) CN s, τ). The functions F mm s, τ) and F mm operate eement-wise on vectors and the expectation in ) is taen with respect to the true prior distribution of S 0 ps 0 ) and Z CN 0, 1). The mcb-amp agorithm differs from the origina cb-amp agorithm derived in [7] by the additiona steps 1) and ). In every iteration, step 1) estimates the so-caed decouped noise variance σt see Section II-B) and step ) tunes the variance parameter τ t based on the estimate σt. The tuning stage ) ensures that mcb-amp converges to the soution that minimizes σt for every iteration see Section II-C). B. Decouping property of AMP-based agorithms We wi frequenty use the foowing definition. Definition 1. Assume a MIMO system with M T transmit and M R receive antennas, and et the entries of H be i.i.d. CN 0, 1/M R ). We define the arge-system imit by fixing the system ratio β = M T /M R and etting M T. As shown in [7], [17], [1], AMP-based agorithms decoupe the MIMO system into parae AWGN channes in the argesystem imit, i.e., the quantity z t = s t +H H r t can be expressed equivaenty as s 0 + w t, where w t CN 0, σt I MT ) and σt is the decouped noise variance. A ey property of AMPbased agorithms is that the decouped noise variance σt can be traced exacty by the state evoution SE) framewor. We formaize the mismatched SE framewor for mcb-amp in Theorem 1. This resut is a specific instance of the SE framewor in [17] and has been adapted to mcb-amp. Theorem 1. Assume the arge-system imit and that F mm is Lipschitz continuous. Then, the decouped noise variance σ t+1 after t mcb-amp iterations is given by the couped recursion γt = arg min Ψ mm σt, γ ), 5) γ 0 σ t+1 = N 0 + βψ mm σ t, γ t ), 6) which is initiaized by σ 1 = N 0 + β Var S0 [S 0 ]. Here, S 0 ps 0 ) and the MSE function is defined by Ψ mm σ t, γ t ) = E S0,Z[ F mm S 0 + σ t Z, γ t ) S 0 ], where the expectation is taen with respect to the S 0 and Z CN 0, 1). If the true prior is identica to the mismatched prior, i.e., ps 0 ) = p s), we can show that mcb-amp deivers the same decouped noise variance as cb-amp without prior mismatch. Lemma. If ps 0 ) = p s), then the decouped noise variance σ t+1 of mcb-amp is equivaent to σ t+1 of cb-amp [7]. The proof of Lemma foows by noting that the MSE function Ψ mm σ t, γ ) in 6) is minimized by γ = σ t in 5), and reduces to the conditiona variance []. Therefore, 6) reduces to the conventiona SE recursion of cb-amp [7]. C. Optima tuning of the variance parameter τ Before we discuss the tuning stage ) in detai, we formaize what we mean by optima tuning of the variance parameter τ t. For a iterations t = 1,..., t max, our goa is to minimize the decouped noise variance σ t max+1 given by Theorem 1, as the smaest σ t max+1 minimizes the error probabiity of our agorithm. Hence, optima tuning tries to identify a sequence of variance parameters {τ 1,..., τ tmax } so that mcb-amp utimatey eads to the smaest σ t max+1; sub-optima choices of τ t either ead to a higher σ t max+1 or cause sower convergence to the smaest σ t max+1. We wi use the foowing definition [0]. Definition. Assume the arge-system imit and denote the decouped noise variance of mcb-amp obtained from the sequence {τ 1,..., τ tmax } as σt max+1τ 1,..., τ tmax ). A sequence of parameters {τ 1,..., τ tmax } is optimay tuned at iteration t max, if and ony if for a {τ 1,..., τ tmax } with τ t [0, ) we have σ t max+1τ 1,..., τ tmax ) σ t max+1τ 1,..., τ tmax ). 7) We next show that the tuning stage in ), which is carried out at every iteration of mcb-amp, achieves the smaest σ t max+1, i.e., optimay tunes the variance parameters τ t. We omit the proof detais as it foows cosey that in [0, Sec. 4.4]. Theorem 3. Suppose {τ 1,..., τ tmax } are optimay-tuned for iteration t max. Then, for any t < t max, {τ 1,..., τ } t are aso optimay-tuned for iteration t. Thus, one can obtain t max optimay-tuned variance parameters by optimizing τ 1 at t = 1, and then, proceeding iterativey by optimizing τ t unti t = t max. In genera, the exact vaue of the decouped noise variance σ t that is needed for the tuning stage in step ) to seect τ t is unnown. We therefore use the estimate ˆσ t = 1 M R r t in

3 step 1), which was shown to converge to the true decouped noise variance σ t in the arge-system imit [18]. D. Decomposition of compex-vaued systems We now briefy discuss properties of mcb-amp in compexvaued systems that wi be necessary for our anaysis of mcb-amp in MIMO systems. In particuar, we show that for specia consteations, the compex-vaued set O can be exacty characterized by a suitaby-chosen rea-vaued set Re{O}. Definition 3. For a s O, express s as s = a + ib, where a Re{O}, b Im{O}. Then, the consteation O is separabe if ps) = pa)pb) hods for a s O and Re{O}= Im{O}. An exampe of a separabe consteation is M -QAM with equay iey transmit symbos. For such a separabe set O, the foowing emma with proof in [7]) provides the equivaence of the compex-vaued and rea-vaued SE framewor. Lemma 4. For a separabe consteation set O, the mismatched SE recursion in Theorem 1 can be expressed equivaenty by: [ F σt+1 = N 0 + β min E mm γ S0 R 0,ZR R S0 R + σ t Z R, γ ) S0 R ) ], where F mm R is posterior mean function with respect to the consteation set O R = Re{O}, S0 R O R and Z R N 0, 1/). E. Fixed-point anaysis Whie the performance of mcb-amp at every iteration can be characterized by the mismatched SE recursion equations in Theorem 1, we are interested in anayzing the performance of mcb-amp for t max. In this case, the mismatched SE in Theorem 1 converges to the foowing fixed-point equation: σ = N 0 + β min γ 0 Ψmm σ, γ ) = N 0 + βψ mm σ ). 8) Thus, as t max, σ t+1 in Theorem 1 converges to σ as given by 8). In genera, if there are mutipe fixed points, then mcb-amp converges to the argest fixed-point, which utimatey eads to a higher probabiity of error than that of the smaest fixed-point soution. To provide conditions that ensure a unique fixed-point soution to 8) see Section III), we use the foowing emma the proof is given in [7]). Lemma 5. Fix ps 0 ) and p s). The minimum recovery threshod MRT) β min for M-LAMA is defined by: β min = min σ 0 dψ mm σ ) dσ ) 1. 9) For a system ratios β < β min the fixed-point soution in 8) is unique. Furthermore, et σ = arg min dψ mm σ )/dσ ) 1. If σ 0 for any other σ σ, β min dψ mm σ )/dσ < 1, then M-LAMA aso has a unique fixed point at β = β min. III. MISMATCHED DATA DETECTION WITH TUNING We now appy the mismatched cb-amp framewor to mismatched data detection in arge MIMO systems, and refer to the agorithm as mismatched arge MIMO AMP M-LAMA). In what foows, we assume that the true prior is taen from a discrete consteation set O with equay iey symbos, i.e., ps 0 ) = 1 O a O δs 0 a), where O is the cardinaity of the set O; we aso define E S0 [ S 0 ] = E s. Note that in MIMO systems, the true signa prior is generay nown. Hence, it is natura to as why one shoud consider a mismatched prior, especiay since the LAMA agorithm [6] that uses the true prior minimizes the error probabiity. To answer this question, consider the posterior mean function 4) of LAMA [6] Fr, τ) = a O a exp 1 τ r a ) a O exp 1 τ r a ), 10) whose cacuation requires high arithmetic precision. In fact, even the use of doube-precision foating-point arithmetic becomes numericay unstabe for sma vaues of τ. Hence, the deveopment of hardware designs for LAMA that use finiteprecision arithmetic is chaenging. In contrast, suitaby-chosen mismatched priors can ead to hardware-friendy data detectors. A. Optimay-tuned data detection with a Gaussian prior We now derive an M-LAMA agorithm variant with a mismatched Gaussian prior. Without oss of generaity, we assume a standard compex Gaussian distribution for the mismatched prior, i.e., p s ) CN 0, 1) as the variance parameter τ t wi be scaed accordingy to E s in the tuning stage ). For the mismatched Gaussian prior, the posterior mean function 4) is given by F mm r, τ) = Es E s+τ r. In the arge-system imit, we can derive the foowing mismatched SE recursion given in 6) using Theorem 1: σ t+1 = N 0 + β min γ 0 E s E s+γ ) E s σ t + γ 4 ). 11) Evidenty, the mismatched SE recursion 11) ony depends on the signa energy E s and no other properties of the true prior ps 0 ). This fact aows us to optimay tune the variance parameters ony with nowedge of signa energy E s. We note that the RHS of 11) is minimized by γ = σt and thus, the mismatched SE recursion becomes: σt+1 = N 0 + β Es E s+σ σ t t. 1) By Lemma 5, M-LAMA has a unique fixed point if β 1. Interestingy, if we define signa-to-interference ratio SIR) as SIR = 1/σ and et t max, then the fixed-point soution of 1) coincides to the SIR of the inear MMSE detector in the arge-system imit [8] [10]. Hence, for a mismatched Gaussian prior, M-LAMA achieves the same performance as the inear MMSE detector. We note that the proofs given in [8] [10] use resuts from random matrix theory, whereas our anaysis uses the mismatched SE recursion in Theorem 1. Furthermore, our resut is constructive, i.e., M-LAMA is a nove, computationay efficient agorithm that impements inear MMSE detection. B. Sub-optima data detection with a Gaussian prior We can repace the optima tuning stage of τ t in ) for the M-LAMA agorithm by a fixed variance parameter choice, which eads to a sub-optima, mismatched agorithm, referred to as sub-optima M-LAMA short SM-LAMA). We now show that this approach eads to other we-nown inear data detectors. In particuar, we obtain the foowing mismatched

4 SE recursions in the arge-system imit for the foowing two choices of variance parameters γ t 0 and γ t in 5): ZF) σ t+1= N 0 + β im γ t 0 Ψ mm σ t, γ t )=N 0 + βσ t, MF) σ t+1= N 0 + β im γ t Ψ mm σ t, γ t )=N 0 + β Var S0 [S 0 ]. By Lemma 5, ZF) and MF) has a unique fixed point if β < 1 and for any finite β, respectivey. If β < 1, then the soution to the fixed-point equation of ZF) and MF) coincides exacty to the SIR given by ZF and MF detector in the argesystem imit [8] [10], respectivey. Hence, by choosing specific predefined and sub-optima variance parameters γ t, SM-LAMA can be used to perform ZF and MF data detection. C. Optimay-tuned data detection with a hypercube prior We now derive an M-LAMA agorithm variant using a mismatched uniform distribution within a hypercube around the true prior distribution of square consteations e.g., QPSK and 16-QAM). For exampe, for a QPSK system with equay iey symbos, we use a mismatched prior distributed uniformy in the interva [ 1, +1] for both the rea and imaginary part. For this mismatched prior, we use Lemma 4 to compute the posterior mean function independenty for the rea and imaginary part; the posterior mean function F mm is given by F mm s, τ) = s + τ ν s R, τ/) + iν s I, τ/)), 13) where we use s R = Re{s }, s I = Im{s }, and ν s, τ) = e 1 τ s +α) e τ 1 s α) s +α s α )), πτ Φ τ ) Φ τ with Φx)= x π e u / du. The mismatched SE recursion can be obtained from Theorem 1 and evauated numericay. There are two disadvantages of this agorithm: i) The computation of the posterior mean function 13) is not efficient from a hardware perspective as it invoves transcendenta functions. In fact, computing ν s, τ) requires simiar to that of the optima LAMA agorithm 10) excessivey high numerica precision; ii) the tuning stage ) turns out to be non-trivia and requires numerica methods to find a minimum. Hence, corresponding hardware designs are impractica. D. Sub-optima data detection with a hypercube prior Anaogousy to the ZF detector in Section III-B, we can derive a sub-optima variant of M-LAMA SM-LAMA) with the hypercube prior from Section III-C, where we repace the tuning stage in ) by the fixed choice τ t 0. This choice eads to a much simper agorithm compared to the optimay-tuned M-LAMA agorithm and enabes a detaied performance anaysis. The posterior mean function reduces to im τ 0 Fmm s, τ) = s R + sign s R ) min { α } s R, 0 + i s I + sign s I ) min { α s I, 0 }), and thus, im τ 0 F mm s, τ) and its derivative can be evauated efficienty in hardware. Furthermore, by fixing τ 0, the tuning stages in 1) and ) are no onger required. symbo error rate SER) MMSE imit) 10 3 M-LAMA-MMSE finite) M-LAMA-hypercube finite) SM-LAMA-hypercube finite) SM-LAMA-hypercube imit) LAMA finite) average SNR per receive antenna [db] Fig. 1. Symbo-error rate of M-LAMA and its variants for a arge-mimo system with 10 iterations and QPSK; finite) and imit) denote the detectors that are simuated and computed by SE in the arge-system imit), respectivey. SM-LAMA with the uniform hypercube prior performs within 1 db from LAMA [6] that achieves IO performance in the arge-system imit. We now present conditions on the system ratio β for which SM-LAMA has a unique fixed point. The foowing Lemma 6, with proof in Appendix C, shows that the MRT of SM-LAMA for M -QAM is given by β min = 1 1/M) 1. Lemma 6. Assume a M -QAM prior for S 0 with equay iey symbos. Then, the MRT for SM-LAMA is given by β min = 1 1/M) 1. Moreover, SM-LAMA has a unique fixed point at β = β min regardess of the noise variance N 0. Using Lemma 6, we obtain the same MRT for M-PAM consteations by Lemma 4. We omit the proof and refer to [7]. Coroary 7. SM-LAMA has the same MRT for M -QAM and M-PAM in a compex and rea-vaued system, respectivey. We now show that this computationay-efficient SM-LAMA variant achieves the same performance as a we-nown reaxation of the maximum ieihood data detection probem [11] [13]. The agorithm, which is nown as box reaxation BOX) detector, soves the foowing convex probem: ŝ = arg min s C M T y H s subject to s 1 14) and sices the individua entries of ŝ onto the QPSK or BSPK) consteations. The next resut shows that SM-LAMA achieves the same error rate performance as the BOX detector for β <, whie providing a simpe and computationay efficient agorithm. The proof is given in Appendix B. Lemma 8. Assume β < and the arge system imit. Then, for a compex-vaued MIMO system with QPSK or a reavaued MIMO system with BPSK), SM-LAMA achieves the same error-rate performance as the BOX agorithm in 14). IV. NUMERICAL RESULTS We now compare the error-rate performance of the M-LAMA agorithm variants proposed in this paper. Whie the mismatched

5 SE framewor in Theorem 1 enabes an exact performance anaysis in the arge-system imit, we aso carry out Monte Caro simuations in a finite-dimensiona arge MIMO system with 18 receive and 64 transmit antennas. Fig. 1 shows the symbo error-rate performance of LAMA, M-LAMA, and SM-LAMA. The optimay-tuned M-LAMA and sub-optima SM-LAMA with the uniform hypercube prior perform within 1 db of the LAMA agorithm [6], which achieves IO performance in the arge-system imit. These resuts demonstrate that carefuyseected mismatched priors enabe near-io performance in finite-dimensiona systems and in a hardware-friendy way. APPENDIX A DERIVATION OF THE MSE FUNCTION Ψ QAM σ ) We wi compute the MSE function for Ψ QAM σ ) by first computing the MSE function Ψ PAM σ ) for a rea-vaued M-PAM system with equay iey symbos and then, use Lemma 4 to express Ψ QAM σ ) for M -QAM. We start by defining F α s ) = s + signs ) min{α s, 0}. Note that for equay iey symbos, the M-PAM consteation can be expressed by ps ) = 1 M/ M = M/+1 δs 1)). Thus, by using symmetry for equay iey M-PAM symbos: Ψ PAM σ ) = M/ M =1 ΨPAM σ ), 15) where we introduced Ψ PAM σ ) that is defined by: Ψ PAM σ ) = E Z [F α 1) + σz) 1)) ] = σ + ᾱ σ )Q ᾱ ) σ + α σ )Q α ) σ σ π ᾱ e ᾱ σ σ π α e α σ, with ᾱ = α 1), α = α + 1), Z N 0, 1), and Qx) = x π e u / du. By Lemma 4, Ψ QAM can be computed by Ψ QAM σ ) = Ψ PAM σ /) with α = M 1. APPENDIX B PROOF OF LEMMA 8 We start by stating the foowing resut from [13, Thm..1] that estabishes the error-rate performance of BOX detector. Theorem 9 Theorem.1 [13]). Assume a rea-vaued BPSK system with β <. The bit-error rate in the arge-system imit converges to Q1/τ ), where τ is the unique soution to τ = arg min τ>0 gτ), where gτ) is defined by ) gτ) = τ 1 β 1 + N0 βτ + ) τ x τ π e x dx. We note that as τ is the unique, minima soution to gτ), we have g τ ) = 0. Rearranging terms in g τ ) = 0 resuts in the fixed-point equation τ = N 0 + βψτ ) with Ψτ ) = 1 τ τ π e τ τ ) + Q τ 4 τ ). 16) The function Ψτ ) in 16) is identica to 15) for a BPSK system, i.e., M = with σ = τ. This impies that the BOX-reaxed method in [13] and SM-LAMA with the uniform hypercube prior achieves the same fixed-point 8). Moreover, due to the decouping property of SM-LAMA detaied in Section II-B, the error-rate of a rea-vaued BPSK system in the arge-system imit is given by Q1/τ ). The resut of Theorem 9 can be generaized to compex-vaued QPSK systems and is equa to fixed-point soution of SM-LAMA by Lemma 4. APPENDIX C PROOF OF LEMMA 6 As shown in Appendix A, we compute dσ Ψ QAM σ ) and d observe that dσ Ψ QAM σ ) 1 1/M, where equaity is achieved ony when σ 0. Thus, β min = 1 1/M) 1. To show that M-LAMA aso has a unique fixed point when β = β min, we use Lemma 5 and observe that no other σ > 0 satisfies β min = dψσ )/dσ ) 1 at σ = σ. REFERENCES [1] C. Jeon, A. Maei, and C. Studer, Mismatched data detection in arge MIMO systems, in preparation. [] S. Verdú, Mutiuser Detection. Cambridge University Press, [3] D. Guo and S. Verdú, Mutiuser detection and statistica mechanics, in Commun., Inf. and Netw. Security. Springer, 003, pp [4], Randomy spread CDMA: Asymptotics via statistica physics, IEEE Trans. Inf. Theory, vo. 51, no. 6, pp , Jun [5] D. Seethaer, J. Jadén, C. Studer, and H. Böcsei, On the compexity distribution of sphere decoding, IEEE Trans. Inf. Theory, vo. 57, no. 9, pp , Sept [6] C. Jeon, R. Ghods, A. Maei, and C. Studer, Optimaity of arge MIMO detection via approximate message passing, in Proc. IEEE Int. Symp. Inf. Theory ISIT), Jun. 015, pp [7], Optima data detection in arge MIMO, in preparation. [8] S. Verdú and S. Shamai, Spectra efficiency of CDMA with random spreading, IEEE Trans. Inf. Theory, vo. 45, no., pp , Mar [9] D. Tse and S. Hany, Linear mutiuser receivers: effective interference, effective bandwidth and user capacity, IEEE Trans. Inf. Theory, vo. 45, no., pp , Mar [10] S. Shamai and S. Verdu, The impact of frequency-fat fading on the spectra efficiency of CDMA, IEEE Trans. Inf. Theory, vo. 47, no. 4, pp , May 001. [11] P. H. Tan, L. K. Rasmussen, and T. J. Lim, Constrained maximumieihood detection in CDMA, IEEE Trans. Commun., vo. 49, no. 1, pp , Jan [1] A. Yener, R. D. Yates, and S. Uuus, CDMA mutiuser detection: A noninear programming approach, IEEE Trans. Commun., vo. 50, no. 6, pp , Jun. 00. [13] C. Thrampouidis, E. Abbasi, W. Xu, and B. Hassibi, BER anaysis of the box reaxation for BPSK signa recovery, IEEE Int. Conf. Acoust., Speech, Signa Process. ICASSP), 016. [14] D. L. Donoho and J. Tanner, Counting the faces of randomy-projected hypercubes and orthants, with appications, Discrete Comput. Geometry, vo. 43, no. 3, pp , Apr [15] O. Mangasarian and B. Recht, Probabiity of unique integer soution to a system of inear equations, Eur. J. Oper. Res., vo. 14, no. 1, pp. 7 30, Oct [16] D. L. Donoho, A. Maei, and A. Montanari, Message-passing agorithms for compressed sensing, Proc. Nat. Acad. Sci. USA, vo. 106, no. 45, pp , Nov [17] M. Bayati and A. Montanari, The dynamics of message passing on dense graphs, with appications to compressed sensing, IEEE Trans. Inf. Theory, vo. 57, no., pp , Feb [18] A. Montanari, Graphica modes concepts in compressed sensing, Compressed Sensing Y.C. Edar and G. Kutynio, eds.). Cambridge University Press, 01. [19] A. Mousavi, A. Maei, and R. G. Baraniu, Parameteress Optima Approximate Message Passing, arxiv: [cs.it], Oct [0], Consistent Parameter Estimation for LASSO and Approximate Message Passing, arxiv: [math.st], Nov [1] A. Maei, Approximate message passing agorithms for compressed sensing, Ph.D. dissertation, Stanford University, Jan [] D. Guo, Y. Wu, S. Shamai, and S. Verdú, Estimation in Gaussian noise: Properties of the minimum mean-square error, IEEE Trans. Inf. Theory, vo. 57, no. 4, pp , Apr d