4196 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 64, NO. 10, OCTOBER 2016

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1 4196 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 64, NO. 10, OCTOBER 2016 Linear Precoding ain for Large MIMO Configurations With QAM and Reduced Coplexity Thoas Ketseoglou, Senior Meber, IEEE, and Ender Ayanoglu, Fellow, IEEE Abstract In this paper, the proble of designing a linear precoder for ultiple-input ultiple-output (MIMO) systes in conjunction with quadrature aplitude odulation (QAM) is addressed. First, a novel and efficient ethodology to evaluate the input output utual inforation for a general MIMO syste as well as its corresponding gradients is presented, based on the auss Herite quadrature rule. Then, the ethod is exploited in a bloc coordinate gradient ascent optiization process to deterine the globally optial linear precoder with respect to the MIMO input output utual inforation for QAM systes with relatively oderate MIMO channel sizes. The proposed ethodology is next applied in conjunction with the coplexityreducing per-group processing technique to both perfect channel state inforation (CSI) at the transitter as well as statistical CSI (Statistical CSI) scenarios, with large transitting and receiving antenna sizes, and for constellation size up to M = 64. We show by the nuerical results that the precoders developed offer significantly better perforance than the configuration with no precoder as well as the axiu diversity precoder for QAM with constellation sizes M = 16, 32, and 64 and for MIMO channel size up to Index Ters Linear precoding, MIMO, utual inforation, quadrature aplitude odulation. I. INTRODUCTION THE concept of Multiple-Input Multiple-Output (MIMO) systes still represents a prevailing research direction in wireless counications due to its ever-increasing capability to offer higher rate, ore efficient counications, as easured by spectral utilization, and under low transitting or receiving power. Within MIMO research, the proble of linear precoding design has been widely studied. Early wor on MIMO precoding can be found in [1] [3]. In addition, a variety of precoding concepts and under ultiple scenarios can be found in [3] [10]. Within ore recent MIMO precoding research with realistic, finite alphabet inputs, the proble of designing an optial linear precoder toward axiizing Manuscript received March 6, 2016; revised July 6, 2016; accepted August 1, Date of publication August 9, 2016; date of current version October 14, The associate editor coordinating the review of this paper and approving it for publication was V. Raghavan. T. Ketseoglou is with the Electrical and Coputer Engineering Departent, California State Polytechnic University, Poona, CA USA (e-ail: tetseoglou@csupoona.edu). E. Ayanoglu is with the Center for Pervasive Counications and Coputing, Departent of Electrical Engineering and Coputer Science, University of California at Irvine, Irvine, CA USA (e-ail: ayanoglu@uci.edu). Color versions of one or ore of the figures in this paper are available online at Digital Object Identifier /TCOMM the utual inforation between the input and output was considered in [11] and [12] where the optial power allocation strategies were presented (e.g., Mercury Waterfilling (MWF)), together with general equations for the optial precoder design. Furtherore, in [13] a channel pairing approach to MIMO precoding was presented. In addition, [14] also considered precoders for utual inforation axiization and showed that the left singular vectors of the optial precoder can be set equal to the right singular vectors of the channel. Finally, in [15], a utual inforation axiizing precoder for a parallel layer MIMO detection syste was set forth reducing the perforance gap between axiu lielihood and parallel layer detection. Recently, globally optial linear precoding techniques were presented [16], [17] for scenarios eploying perfect channel state inforation available at the transitter (CSIT) 1 with finite alphabet inputs, capable of achieving utual inforation rates uch higher than the previously presented MWF [11] techniques by introducing input sybol correlation through a unitary input transforation atrix in conjunction with channel weight adjustent (power allocation). In addition, [18] presented an iterative algorith for precoder optiization for su rate axiization of Multiple Access Channels (MAC) with Kronecer MIMO channels. Furtherore, ore recent wor showed that when only Statistical CSI 2 is available at the transitter, in asyptotic conditions when the nuber of transitting and receiving antennas grows large, but with a constant transitting to receiving antenna nuber ratio, one can design the optial precoder by looing at an equivalent constant channel and its corresponding adjustents as per the pertinent theory [21], and applying a odified expression for the corresponding ergodic utual inforation evaluation over all channel realizations. This developent allows for a precoder optiization under Statistical CSI in a uch easier way [21]. However, existing research in the area has not provided any results of optial linear precoders in the case of QAM with constellation size M 16, with the exception of [22]. In past research wor, a ajor ipedient toward developing optial precoders for QAM has been a lac 1 Under CSIT the transitter has perfect nowledge of the MIMO channel realization at each transission. 2 Statistical CSI pertains to the case in which the transitter has nowledge of only the MIMO channel correlation atrices [19], [20] and the theral noise variance IEEE. Personal use is peritted, but republication/redistribution requires IEEE perission. See for ore inforation.

2 KETSEOLOU AND AYANOLU: LINEAR PRECODIN AIN FOR LARE MIMO CONFIURATIONS 4197 of an accurate and efficient technique toward input-output utual inforation evaluation, its gradients, and evaluation of the input-output iniu ean squared error (MMSE) covariance atrix, as required by the precoder optiization algorith and other algoriths involved, e.g., the equivalent channel deterination in the Statistical CSI case [21]. In this paper, we propose optial linear precoding techniques for MIMO, suitable for QAM with constellation size M 16. An additional advantage of these techniques is their ability to accoodate MIMO configurations with very large antenna sizes, e.g., The only related wor in this area is [22] which has antenna sizes up to We show in the sequel that the proposed ethod is faster than the one in [22]. Carrying out this calculation has been very difficult to do until now due to the coplexity involved in tacling this proble. Our approach entails a novel application of the auss-herite quadrature rule [23] which offers a very accurate and efficient way to evaluate the capacity of a MIMO syste with QAM. We then apply this technique within the context of a bloc gradient ascent ethod [24] in order to deterine the globally optial linear precoder for MIMO systes, in a siilar fashion to [16], for systes with CSIT and sall antenna size. We show that for M = 16, 32, and 64 QAM, the optial linear precoder offers 50% better utual inforation than the axial diversity precoder (MDP) of [25] and the no-precoder case, at low signal-to-noise ratio (SNR) for a standard 2 2 MIMO channel, although the absolute utilization gain achieved is lower than 1 bps/hz. We then proceed to show that significantly higher gains are available for different channels, e.g., a utilizationgainof1.30 bps/hz at SNR = 10 db, when M = 16. We then eploy larger antenna configurations, e.g., up to with CSIT and M = 16, 32, and 64 together with the coplexity reducing technique of per-group processing (PP) which was originally presented in [26], and show very high gains available with reduced syste coplexity. Finally, we also eploy Statistical CSI scenarios in conjunction with PP and show very significant gains for large antenna sizes, e.g., and M = 16, 32, 64. The ain advantages of our wor copared with other interesting proposals for large MIMO sizes, e.g., [22], lie over four ain directions: a) It offers a precoder solution very close to the globally optial one for each subgroup, instead of a locally optial one, b) It is faster, c) It allows for larger constellation size, e.g., M = 32, 64, and d) It allows larger MIMO configurations, e.g., The paper is organized as follows: Section II presents the syste odel and proble stateent. Then, in Section III, we present a novel auss-herite approxiation to the evaluation of the input-output utual inforation of a MIMO syste that allows for fast, but otherwise very accurate evaluation of the input-output utual inforation of a MIMO syste, and thus represents a ajor facilitator toward deterining an alost globally optial linear precoder for MIMO. In Section IV, we present nuerical results for the globally optial precoder that ipleents the auss-herite approxiation in the bloc coordinate gradient ascent ethod. Finally, our conclusions are presented in Section V. II. SYSTEM MODEL AND PROBLEM STATEMENT The instantaneous N t transit antenna, N r receive antenna MIMO odel is described by the following equation y = Hx + n, (1) where y is the N r 1 received vector, H is the N r N t MIMO channel atrix, is the precoder atrix of size N t N t, x is the N t 1 data vector with independent coponents each of which is in the QAM constellation of size M, n represents the circularly syetric coplex Additive White aussian Noise (AWN) of size N r 1, with ean zero and covariance atrix K n = σn 2I N r,wherei Nr is the N r N r identity atrix, and σ 2 = 1 SNR, SNR being the (coded) sybol signal-tonoise ratio. In this paper, a nuber of different channels are considered, e.g., channels coprising independent coplex aussian coponents or spatially correlated Kronecer-type channels [19] (including those siilar to the 3PP spatial correlation odel (SCM) [27]), or ore generally canonical [28] or Weichselberger channels [20]. The precoding atrix needs to satisfy the following power constraint tr( h ) = N t, (2) where tr(a), A h denote the trace and the Heritian transpose of atrix A, respectively. An equivalent odel called herein the virtual channel is given by [16] y = H V h x + n, (3) where H and are diagonal atrices containing the singular values of H,, respectively and V is the atrix of the right singular vectors of. When a capacity approaching code, e.g., the LDPC code is eployed in this MIMO syste, the overall utilization in bps/hz is deterined by the utual inforation between the transitting sybols x and the receiving ones, y [29], [30]. It is shown [16] that the utual inforation between x and y, for channel realization H, I (x; y), is only a function of W = V 2 H 2 Vh.The optial CSIT precoder is found by solving: called the original proble, and axiize I (x; y) subject to tr( h ) = N t, (4) axiize I (x; y) V, subject to tr( 2 ) = N t, (5) called the equivalent proble, where the reception odel of (3) is eployed. The solution to (4) or (5) results in exponential coplexity at both transitter and receiver, and it becoes especially difficult for QAM with constellation size M 16 or large MIMO configurations. A ajor difficulty in the QAM case stes fro the fact that there are ultiple evaluations of I (x; y) in the bloc coordinate ascent ethod eployed for deterining the globally optial precoder. More specifically, for each bloc coordinate gradient ascent iteration, there are two line bactracing searches required [16], which deand one I (x; y) plus its gradient evaluations per search

3 4198 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 64, NO. 10, OCTOBER 2016 trial, and one additional evaluation at the end of a successful search per bactracing line search. Thus, the need of a fast, but otherwise very accurate ethod of calculating I (x; y) and its gradients prevails as instruental toward deterining the globally optial linear precoder for CSIT. In the Statistical CSI case, the corresponding optiization proble becoes axiize E H {I (x; y)} subject to tr( h ) = N t, (6) where the expectation is perfored over all the channels H. The ground-breaing wor of [21] has shown that the proble in (6) for large antenna sizes can be solved by an approxiate way of calculating the ergodic utual inforation E H {I (x; y)} for a fixed precoding atrix through well-deterined paraeters of a deterinistic channel. More specifically, when N t, N r while the ratio N t /N r is ept constant, [21] has showed that an asyptotic ethodology to approxiate the ergodic utual inforation exists, it also presented a ethodology to deterine the asyptotically optial precoder for the Statistical CSI case for Weichselberger channels. The evaluation of the utual inforation and the optial asyptotic precoder, under these conditions, entail evaluations of equivalent deterinistic channel utual inforation ultiple ties. In [22] the sae approach was used for Kronecer channels requiring the evaluation of an equivalent deterinistic channel. These paraeters include the utual inforation of a corresponding deterinistic channel defined in [22, eq. (19)], i.e., siilar to CSIT scenario utual inforation evaluation. Thus, ethods that offer siplification of CSIT utual inforation evaluation, I (x; y), are also iportant in the Statistical CSI case toward deterining the globally optial linear precoder for MIMO. III. ACCURATE APPROXIMATION TO I (x; y) FOR MIMO SYSTEMS BASED ON AUSS-HERMITE QUADRATURE In Appendix A we prove that by applying the auss-herite quadrature theory for approxiating the integral of a aussian function ultiplied with an arbitrary real function f (x), i.e., F. = + exp( x 2 ) f (x)dx, (7) which is approxiated in the auss-herite approxiation with L weights and nodes as F c(l) f (v l ) = c t f, (8) l=1 with c =[c(1) c(l)] t, {v l } L l=1,andf =[f (v 1) f (v L )] t, being the vector of the weights, the nodes, and function node values, respectively (see Appendix A), a very accurate approxiation is derived for I (x; y) in a MIMO syste, as presented in the following lea. Let us first introduce soe notations that ae the overall understanding easier. Let n e denote the equivalent to n, real vector of length 2N r derived fro n by separating its real and iaginary parts as follows n e =[n r1 n i1 n rnr n inr ] t, (9) with n rv, n iv being the values of the real, iaginary part of the vth (1 v N r ) eleent of n, respectively. Let us also define the real vector v({ rv, iv } N r v=1 ) of length 2N r defined as follows v({ rv, iv } N r v=1 ) =[v r1 v i1,,v rnr v inr ] t, (10) with rv, iv (1 v N r ) being perutations of indexes in the set {1, 2,, L}. Then the following lea is true concerning the auss-herite approxiation for I (x; y). Lea 1: For the MIMO channel odel presented in (1), the auss-herite approxiation for I (x; y) with L nodes per receiving antenna is given as where ˆ f = with I (x; y) N t log 2 (M) ( ) 1 Nr r1 =1 i1 =1 N r log(2) 1 rnr =1 inr =1 =1 c( r1 )c( i1 ) ˆ f, (11) c( rnr )c( inr )g (σ n r1,σn i1,,σn rnr,σn inr ), (12) g (σ v r1,σv i1,,σv rnr,σv inr ) (13) being the value of the function ( ) log 2 exp( 1 σ 2 n H(x x ) 2 ) (14) evaluated at n e = σ v({ rv, iv } N r v=1 ). The proof of this lea is presented in Appendix A, together with a derivation ethodology available for this expression in the N t = N t case. In Appendix A, we also show that fˆ is an approxiation to the physical quantity Input-Dependent Output Entropy (IDOE) which relates to the negative MIMO output entropy when the MIMO input is x = x. Let us stress that, the presented novel application of the auss-herite quadrature in the MIMO odel allows for efficient evaluation of I (x; y) for any channel atrix H, and precoder, as required in the precoder optiization process, as explained below. IV. LOBAL OPTIMIZATION OVER TOWARD MAXIMUM I (x; y) FOR QAM A. Description of the lobally Optial Precoder Method Siilarly to [16], we follow a bloc coordinate gradient ascent axiization ethod to find the solution to the optiization proble described in (4), eploying the virtual odel of (3). It is proven in [16] that I (x; y) is a concave function over W and 2. It thus becoes efficient to eploy two different gradient ascent ethods, one for W, and another one for 2.Weeploy and to denote Vh and 2, respectively, evaluated during the execution of the optiization algorith. The value of the step of each iteration over W and 2 is deterined through bactracing line searches, one for each of the variables W and 2. We describe the

4 KETSEOLOU AND AYANOLU: LINEAR PRECODIN AIN FOR LARE MIMO CONFIURATIONS 4199 bactracing line search for W first, then the one over 2. At each iteration of the globally optial algorith over W, the gradient of I (x; y) over W, W I, is required. We eploy a novel ethod in deterining W I at each iteration, as described in detail in the next subsection. We then select two paraeters, α 1 and β 1, both saller than one and positive, and perfor the bactracing line search as follows: At each new trial, a paraeter t 1 > 0 that represents the step size is updated by ultiplying it with β 1. The initial value for t 1 is equal to 1. Then the algorith checs if I (W + t W I)>I (W) + α 1 t W I 2 F, (15) where W I F is the Frobenius nor of W I [31] and we used the notation I (W) to ean the value of I (x; y) at a fixed W. If the condition is satisfied, the algorith proceeds with calculating a new NEW fro W NEW = W + t 1 W I, as follows: W NEW = h NEW 2 H 2 NEW (16) by eploying the eigenvalue decoposition (EVD) of W NEW, it updates V h = NEW, and then it proceeds to the bactracing line search over 2. If the condition is not satisfied, the search updates t 1 to its new and saller value, β 1 t 1, and repeats the chec on the condition, until the condition is satisfied or a axiu nuber of attepts in the first loop, n 1, has been reached. Then, the bactracing line search on 2 taes place in a fashion siilar to the search described for W, but with soe raifications. First, based on the second bactracing line search loop paraeters α 2 and β 2, the bactracing line search is as follows: At each new trial, a paraeter t 2 > 0 that represents the step size is updated by ultiplying it with β 2. The initial value for t 2 is equal to 1. Then the algorith checs if I (W + t 2 I)>I (W) + α 2 t 2 I 2 F, (17) where 2 I F is the Frobenius nor of 2 I [31]. If the condition is satisfied, the algorith proceeds with updating to the new NEW, but after setting any negative ters in the ain diagonal of NEW to zero and renoralizing the reaining ain diagonal entries. If the condition is not satisfied, the search updates t 2 to its new and saller value, β 2 t 2 and repeats the chec on the condition, until the condition is satisfied or a axiu nuber of attepts in the second loop, n 2 has been reached. We thus see that there will in general be ultiple evaluations of I (x; y), until the searches satisfy the conditions set or the axiu nuber of attepts allowed in a search has been reached. This explains the iportance behind the requireent for an algorith capable of efficient calculation of I (x; y). In addition, as the paraeters α 1, α 2, β 1, β 2 need to be optiized for faster and ore efficient execution of the globally optial precoder optiization, this requireent becoes even ore essential. Finally, the role of n 1, n 2 is also very iportant as when the nuber of attepts within each loop grows, the corresponding differential value of the paraeter decreases and after a few attepts, the corresponding value of the step size is alost zero. By eploying the proposed approach the possibility of finding Algorith 1 lobal Precoder Optiization Algorith With t Iterations 1: procedure PRECODER( H ) 2: while i t do 3: Deterine W NEW = h 2 through bactracing line search (16) 4: Set V h = 5: Deterine W NEW = V 2 2 H Vh 6: Deterine 2 NEW, through bactracing line search (17) 7: Set negative entries on the diagonal of 2 NEW, to zero 8: Noralize 2 NEW, to a trace equal to N t 9: Set = NEW, 10: Deterine W NEW = V 2 2 H Vh 11: Set W = W NEW 12: Evaluate I (W) 13: end while 14: return I (W) 15: end procedure the globally optial precoder for large QAM constellations with M 16 or large MIMO configurations becoes reality, as our results deonstrate. The algorith s pseudocode for a nuber of iterations t is presented under the heading Algorith 1. B. Deterination of W I, 2 I We first set M = W 1 2. Then, it is easy to see that I is a function of M (see, e.g., [21] where the notion of sufficient statistic is eployed to show that I (x; y) depends on W). The derivation of W I is presented in Appendix B. The proof is based on the following theore. 3 Theore 1: Substituting M = V H V h = W 1 2 for H in (11) results in the sae value of I (x; y).inotherwords, since M is a function of H,, My is a sufficient statistic for y. Proof: The proof of the theore is siple. First, recall that the virtual channel odel in (3) is equivalent to the following odel, which results by ultiplying (3) by the unitary atrix V on the left, resulting in ỹ = V y = V H V h x + V n, (18) where the odified noise ter V n has the sae statistics with n, because V is unitary. By applying the auss-herite approxiation to (18), we see that we get the desired result, i.e., the value of I (x; y) reains the sae, since both channel anifestations represent equivalent channels, i.e., the original one and its equivalent, thus their utual inforation is the sae. This copletes the proof of the theore. Note that using this theore, an alternative proof of part of [16, Th. 1] can be developed, naely the fact that I (x; y) is only a function of W, asmy is a sufficient statistic for 3 The theore applies without loss of generality to the N t = N r case. If N t = N r,then H needs to be either shrun, or extended in size, by eliination or addition of zeros, respectively.

5 4200 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 64, NO. 10, OCTOBER 2016 y and M is a function of W. Assue without loss of generality that N t = N r. The gradient of I (x; y) with respect to M can be found (see Appendix B for the derivation) fro the auss- Herite expression presented in (11) as follows M I = 1 log(2) 1 ( 1 ) Nr r1 =1 i1 =1 rnr =1 inr =1 c( r1 ) c( inr )R (σ v r1,σv i1,,σv rnr,σv inr ), (19) where R(σ v r1,σv i1,,σv rnr,σv inr ) is the value of the N t N t atrix 1 exp( 1 n M(x σ 2 x ) 2 ) exp( 1 σ 2 n M(x x ) 2 ) ((n M(x x ))(x x ) h + ((n M(x x ))(x x ) h ) h ) (20) evaluated at n e = σ v({ rv, iv } N r v=1 ). The required W I for the execution of the optiization process can be found fro Appendix B as per the next lea, using an easily proven equation. Using the fact that for a Heritian atrix such as M, we need to add the Heritian of the differential above in order to evaluate the actual gradient (see [31]), we get the desired result as follows (see Appendix B). Lea 2: For the MIMO channel odel presented in (1), the auss-herite approxiation allows to approxiate W I as follows. W I reshape((vec( M I) T ( (M ) I + I M ) 1 ), N t, N t ), (21) where reshape(a,, n) is the standard reshape of a atrix A (with total nuber of eleents n) to a atrix with rows, n coluns, and where denotes Kronecer product of atrices. Standard reshape eanates fro the vector vec(a) of atrix A which encopasses all coluns of A starting fro the leftost one to the rightost. Then, as I (x; y) is a concave function of W [21], we can axiize over W in a straightforward way using closed for expressions. This is based on the fact that the approxiated I (x; y) through the auss-herite approxiation is very accurate, as shown in the next section. The utilization of this lea goes beyond the exploitation of the gradient of the utual inforation in the precoder optiization algorith. As it is well nown fro [16], W I =, where = E { E{(x E(x y))(x E(x y)) h y} }, is the iniu ean squared error (MMSE) covariance atrix of the channel. Thus, by using the current lea, an accurate estiate of the MMSE covariance atrix of the MIMO channel can be achieved. This is very useful especially when dealing with, e.g., Statistical CSI cases where, the MMSE covariance atrix becoes instruental in deriving the asyptotically optial precoder [21]. Thus, based on the proposed auss-herite approxiation, accurate, but otherwise siplified derivation TABLE I COMPUTATIONAL COMPLEXITY PARAMETERS OF AUSS-HERMIT QUADRATURE I (x, y) EVALUATION of the MMSE covariance atrix of the channel becoes possible. Finally, since fro [32] we have that 2 I = diag(v h W IV 2 H ), we can easily evaluate it through the procedure presented above. C. Coplexity of the lobally Optial Precoder Deterination Method The coplexity of the globally optial precoder deterination ethod depends ainly on the evaluations of I (x; y) as required by the optiization algorith described above. By eploying the auss-herite approxiation described herein it becoes possible to significantly accelerate the evaluations of I (x; y) at each iteration of the optiization algorith. However, the coplexity is still high: Each evaluation of I (x; y) through the auss- Herite approxiation requires, as per the developent in Appendix C, eploying L weights and L nodes per each real and iaginary coponent of the noise vector n, resulting in 2N t L total weight and node diensions. Then, evaluation of f in (30) requires 2N r nested DO loops, each of length L, resulting in L 2N r eory paraeter values overall. For a good approxiation in auss-herite quadrature, a value of L 3 is required (please see the next section for relevant results) and thus the overall eory requireent becoes 3 2N r. As far as the coputational coplexity in the nuber of operations (including both (coplex) suations and ultiplications) involved in calculating I (x; y) through the auss-herite approxiation, the corresponding coplexity is shown in Appendix B to be M 2N t (2L 1) (2N r ) L 4 N r (2N t + N r 1). Thus, for exaple, going fro M = 16 to M = 32 QAM will result in about 4 ties higher coplexity with N t and N r held constant. On the other hand, increasing N r has an even ore profound effect on the coplexity, due to its ore coplicated presence in this coplexity equation. For exaple, increasing N r fro 2 to 3 while eeping all other paraeters constant, will increase the coplexity by a factor of (2L 1) 2 or for L = 3 by 25 ties. Our systeatic nuerical evaluations corroborate these nubers very closely. As we observe by coparing the figures in Table 1 to the nubers presented in [22], we see that the auss-herite approach is about 7 ties faster per iteration of the precoder for M = 16 and PP with groups of size 2 2, i.e., the presented approach is faster. This is one of the ajor iproveents due to the proposed ethodology. V. NUMERICAL RESULTS The results presented in this subsection eploy QAM with 16, 32, or 64 constellation sizes. We eploy MIMO systes with N t = N r = 2 when global precoding optiization is

6 KETSEOLOU AND AYANOLU: LINEAR PRECODIN AIN FOR LARE MIMO CONFIURATIONS 4201 Fig. 1. Results for I (x; y) without precoding for the H 1 channel and QAM M = 16 odulation. perfored. We have used an L = 3 auss-herite approxiation which results in 3 2N r total nodes due to MIMO. The ipleentation of the globally optiizing ethodology is perfored by eploying two bactracing line searches, one for W and another one for 2 at each iteration, in a fashion siilar to [21]. For the results presented, it is worth entioning that only a few iterations (e.g., typically < 8) are required to converge to the optial solution results as presented in this paper. We apply the coplexity reducing ethod of PP [26] which offers sei-optial results under exponentially lower transitter and receiver coplexity [26]. PP divides the transitting and receiving antennas into independent groups, thus achieving a uch sipler detector structure while the precoder search is also draatically reduced as well. Finally, we address both the CSIT and Statistical CSI cases. We divide this section into five subsections. In the first subsection, we exaine the accuracy of the proposed auss-herite approxiation and provide a coparison with the lower bound technique presented in [32]. In the second subsection we show results for the globally optial precoder for M = 16, 32 based on the approxiation in conjunction with independent aussian channels and CSIT. In the third subsection we present results for Statistical CSI channels siilar to the ones in 3PP SCM [27], with antenna size up to with PP and odulation size M = 16, 32. In the fourth subsection, we present results for CSIT jointly with PP and higher size of antennas and odulation. Finally, in the last subsection, we present results for a MIMO syste with 100 base station antennas and 4 user antennas with M = 16, 64. A. Accuracy of the auss-herite Approxiation Technique In Fig. 1 we present results for I through siulation, the auss-herite approxiation (H) with L = 3,and the lower bound developed in [32] versus the signal-to-noise ratio per bit (SNR b ) in db, for QAM with M = 16, and for the coonly used channel [16], [25], [ ] 2 1 H 1 =. 1 1 In the sae figure, we also show a lower bound for I (x; y) which appeared in [32]. We see excellent accuracy for the Fig. 2. Results for the average percent absolute noralized error in the auss Herite I (x; y) using 1000 generated rando H channels and QAM M = 16 odulation. Fig. 3. Results for the average percent absolute noralized error in the auss Herite I (x; y) for the H 1 channel and QAM M = 16 odulation. Fig. 4. Results for I (x; y) without precoding for the H 2 channel and QAM M = 32 odulation. approxiation, i.e., no observable difference between the auss-herite approxiation and the siulations, over all SNR b values. In Fig. 2 we show results on the auss-herite approxiation percent average absolute error noralized by the actual I (x; y) for L = 2, 3, 4, for 1000 randoly generated H N t = N r = 2 channels, and M = 16, with respect to sybol SNR in db. We see that L = 2hasa axiu noralized error of 6.25%. The average noralized error drops to less than 1% with L = 3forSNR > 5 db.

7 4202 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 64, NO. 10, OCTOBER 2016 Fig. 5. Results for achievable I (x; y) with channel estiation errors and M = 16 odulation. Furtherore, the results for L = 4 give little iproveent over the ones with L = 3. Notice that the sae type of behavior was found for M = 32, 64 over a wide variety of channels. We thus selected L = 3 in our wor as the best tradeoff value between low approxiation error and low syste coplexity. Furterore, it is iportant to stress that the variance of the approxiation error was found to be less than for L 2andSNR > 15 db. We thus see the auss-herite approxiation offers excellent accuracy when L = 3andfor awidesnr range. In Fig. 3 we present percent noralized absolute error results for H 1 and L = 1, 2,, 5inorderto assess the ipact of the value of L to the results, especially for higher L. We clearly see that by going fro L = 2 to L = 3, a significant reduction on the error is achieved. Further increase of L offers an additional reduction to the error, however it requires significant additional coplexity and processing delay. We thus see that L = 3 coes as the natural candidate due to its capability to attain percent error lower than 2% for a wide range of SNR. In Fig. 4 we present the corresponding results to Fig. 1 for M = 32 QAM in conjunction with the randoly generated channel [ H 2 = j j j j0.1 with the sae type of behavior as before, i.e., the auss- Herite approxiation offers excellent accuracy and that the lower bound is lagging behind in perforance, albeit by less than N r (1/ log(2) 1) 0.88, which is the shift introduced in [32] in order to approxiate I very closely for QPSK odulation. In Fig. 5 we present results with non-perfect channel estiation, i.e., channel estiation with errors for N t = N r = 2 [33]. To that end, we eploy two channel estiators and use their iperfect estiation to odel errors. We use two possible Least Squares Estiation (LSE) ethods in conjunction with 2 orthogonal pilots, each of length 2. In the first ethod, a siple LSE with pilot power equal to the 7 db independently of the essage signal power, called Constant Pilot Power (CPP), while in the second one we use pilots with two ties the essage signal power, called Variable Pilot Power (VPP). We generate 50 rando channels, eploy LSE with both CPP and VCC, and then average the corresponding I (x; y) on these channels. We observe that both proposed ], Fig. 6. I (x; y) results for optial precoding, MDP, and no-precoding cases for a 2 2 MIMO syste and QAM M = 16 odulation. Fig. 7. I (x; y) results for optial precoding, MDP, and no-precoding cases for a 2 2 H 1 MIMO syste and QAM M = 32 odulation. LSE ethods attain siilar results which show little ipact of channel estiation errors to the overall syste perforance, e.g., at SNR = 4 db there is degradation of 3.8% in the precoder perforance. In addition, since the two ethods proposed are siple, in reality, with ore sophisticated channel estiation, we expect a saller ipact of channel estiation errors to the precoder design and perforance. B. Results for lobally Optial Precoding for aussian Channels and CSIT In Fig. 6 we show results for the globally optial precoder based on the auss-herite approxiation with L = 3, and the Maxiu Diversity Precoder (MDP) presented in [25], for M = 16, for H 1. We observe that the optial precoder offers significant utilization gains in the low SNR region, while its gain diinishes in the higher SNR region. For exaple, at SNR b = 4dB, there is about 0.6 bps/hz gain attainable with the globally optial precoder over its MDP and no-precoding counterparts, which represents a 30% gain. Also, contrary to the BPSK/QPSK odulation case presented in [16] and [32], the MDP precoder offers no significant gain over the no-precoding case, a very iportant difference. In Fig. 7 we present results for I (x; y), for the sae channel, MDP, and no precoding, N t = N r = 2, and M = 32. There is no noticeable iproveent offered by MDP over the no-precoding case, siilarly to the M = 16 QAM case

8 KETSEOLOU AND AYANOLU: LINEAR PRECODIN AIN FOR LARE MIMO CONFIURATIONS 4203 Fig. 8. I (x; y) results for optial precoding, MDP, and no-precoding cases for a 2 2 H 2 MIMO syste and QAM M = 32 odulation. Fig. 10. I (x; y) results for PP and no-precoding cases for a H Statistical CSI MIMO syste and QAM M = 16 odulation. Fig. 9. I (x; y) results for PP and no-precoding cases for a H Statistical CSI MIMO syste and QAM M = 32 odulation. presented above. Clearly, the sae fundaental conclusions as in the M = 16 case hold true. The offered gain of the globally optial precoder in low SNR is still around 50% over its no- precoding and MDP counterparts. We next present sae type of results for H 2. In Fig. 8 we observe that for this type of channel, the gains achieved by the globally optial precoder are significantly higher and that in the high SNR regie the no-precoding case cannot achieve the axiu utual inforation given by N t log 2 (M) = 10, i.e., it becoes saturated. We will see that for certain channels, which we call saturated channels, this type of behavior is also observed for the large MIMO channel configurations in both CSIT and Statistical CSI channel cases. C. Results for Statistical CSI in Conjunction With PP We apply the asyptotic approxiation to ergodic utual inforation precoder optiization, e.g., by deterining the paraeters eq, eq,γ eq,ψ eq, R eq in [22] for Kronecer channels. We first consider an N t = N r = 20 SCM urban Fig. 11. I (x; y) results for PP and no-precoding cases for the H Statistical CSI MIMO channel eployed in [22] in conjunction with QAM M = 16 odulation. channel with M = 32. We create the large size asyptotic approxiation no-precoding results for a Statistical CSI channel based on [21]. For precoding to be efficient, but otherwise realistic, we eploy PP [26] with 10 groups of size 2 2 each. Due to the nature of this channel, we observe in Fig. 9 that the no-precoding case saturates at I (x; y) = 70 bps/hz. We see very significant gains offered by PP in the high SNR b regie. The PP syste attains the full capacity available in high SNR b. In Fig. 10 we present results for PP versus a no-precoding SCM channel with N t = N r = 100 and M = 16. To the best of our nowledge, results for such large MIMO configurations are not available in the literature. Siilar to the previous results, we observe high inforation rate gains in the high SNR b regie as the PP syste achieves the full capacity of 400 bps/hz while the noprecoding schee saturates at 320 bps/hz. The PP syste eployed uses 50 groups of size 2 2 each. In Fig. 11 we present results for PP versus a no-precoding SCM channel with N t = N r = 32 and M = 16, using the sae SCM i i i i H = i i i i i i i i, i i i i

9 4204 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 64, NO. 10, OCTOBER 2016 Fig. 12. I (x; y) results for PP and no-precoding cases for a 4 4 H CSIT MIMO syste and QAM M = 64 odulation. Fig. 14. I (x; y) results for PP and no-precoding cases for a randoly generated uplin H CSIT MIMO syste and QAM M = 16 odulation. Fig. 13. I (x; y) results for PP and no-precoding cases for a randoly generated 10 4 H CSIT MIMO syste and QAM M = 16 odulation. channel as in [22]. We observe that our proposed approach offers slightly better perforance than the one eployed in [22], although it is faster, as it is explained in Table 1. D. Results for CSIT in Conjunction With PP For a 4 4 channel H, as shown at the botto of the previous page, used with M = 64 we get the no-precoding and the PP results using 2 groups of size 2 2 each depicted in Fig. 12. This exaple represents the corresponding CSIT case exaple that is siilar to the SCM channels used in the previous subsection. We observe very high gains of PP over the no-precoding case in the high SNR b regie. To the best of our nowledge, these type of results for optial precoding in conjunction with M = 64 are not available in the literature. Finally, in Fig. 13 we present results for an asyetric randoly generated MIMO channel with N t = 4, N r = 10, and M = 16. PP eploys two groups of size N r = 5, N t = 2 each. In the current scenario, we observe that significant gains are shown in the low SNR b regie, e.g., around 3 db in SNR b lower than 7 db. E. Results for Massive MIMO Massive MIMO [34] [36] has attracted uch interest recently, due to its potential to offer high data rates. We present results for the uplin and downlin of a Massive MIMO syste based on 100 base station, 4 user antennas, respectively, with M = 16, 64, and for a Kronecer-based 3PP SCM urban channel in a CSIT scenario in a single user configuration. Fig. 14 shows results for the uplin of the syste. We eploy PP to draatically reduce the syste coplexity at the transitter and receiver sites. Under no precoding, the channel saturates and fails to eet the axiu possible utual inforation of 16 bps/hz, while with PP the syste clearly achieves the axiu utual inforation rate, thus achieving high gains on the uplin in the high SNR regie. We stress the uch higher throughput possible with M = 64 over the M = 16 case. For exaple, the no-precoding M = 16 uplin significantly outperfors the PP M = 16 uplin. Second, the PP M = 64 uplin offers further gains by, e.g., achieving the axiu possible rate of 24 bps/hz. For the downlin, in Fig. 15 we show results where the no-precoding case operates under 100 antenna inputs all correlated through the right singular vectors of the channel, thus creating a very deanding environent at the user, due to the exponentially increasing axiu a posteriori (MAP) detector coplexity [26]. On the other hand, eploying PP with only two input sybols per receiving antenna, i.e., with draatically reduced decoding coplexity, the PP syste achieves uch higher throughput in the lower SNR regie, with SNR gain on the order of 10 db, albeit achieving a axiu of 32, 48 bps/hz as there are a total of 8 M = 16, 64 QAM data sybols eployed, respectively. We observe the superior perforance of M = 64 over its M = 16 counterpart due to its increased constellation size. For exaple, at ediu SNR b, e.g., SNR b = 4, the M = 64 PP schee achieves 45% higher throughput that the M = 16 one, a significant iproveent. We would also lie to ephasize that the no-precoding schee requires a very high exponential MAP detector coplexity, on the order of M 100, while for the low-snr-superior PP, this coplexity is on the order of M 2 only. Thus, even in the higher SNR region where the no-precoding schee can achieve a higher throughput, the coplexity required at the user site becoes prohibitive. This deonstrates the superiority of PP on the Massive MIMO downlin. On the other hand,

10 KETSEOLOU AND AYANOLU: LINEAR PRECODIN AIN FOR LARE MIMO CONFIURATIONS 4205 Fig. 15. I (x; y) results for PP and no-precoding cases for a randoly generated H downlin CSIT MIMO syste and QAM M = 16, 64 odulation. in lower SNR, the PP schee achieves both uch higher throughput with siultaneously exponentially lower MAP detector coplexity at the user site detector. Note that the perforance provided by PP on the downlin depends on the nuber of sybols processed jointly. This nuber is currently liited by the coputational coplexity available. This liitation does not occur on the uplin since in that case N t < N r. VI. CONCLUSIONS In this paper, the proble of designing a linear precoder for MIMO systes toward utual inforation axiization is addressed for QAM with M 16 and in conjunction with large MIMO syste size. A ajor obstacle toward this goal is a lac of efficient techniques for evaluating I (x, y) and its derivatives. We have presented a novel solution to this proble based on the auss-herite quadrature. We then applied a global optiization fraewor to derive an alost globally optial precoder for the case of QAM with M = 16 and 32 and sall antenna size configurations. We showed that under CSIT in this case, significant gains are available for the lower SNR range over no precoding, or MDP. We showed that for the standard 2 2 channel, although the globally optial precoder offers significant gains over MDP and the noprecoder configurations in the low SNR region, it fails to offer gains as high as 1 bps/hz. However, we deonstrated that by eploying another 2 2 channel gains as high as 1.4 bps/hz are possible. For systes of large MIMO configurations, we applied the coplexity-siplifying PP concept [26] to derive sei-optial precoding results. Under Statistical CSI, we showed that for urban 3PP SCM channels, an interesting saturation effect in the no-precoding case taes place, while the sei-optial PP precoder offers draatically better results in this case and it does not experience any saturation as the SNR increases, e.g., it achieves the axiu inforation rate, I (x; y) = N t log 2 (M) at high SNR. Furtherore, we applied the sae auss-herite approxiation approach to CSIT with a large nuber of antennas with the sae success. We showed that for specific type of channels siilar to urban 3PP SCM [27], the PP approach offers very high gains over the no-precoding case in the high SNR b regie. Finally, we considered a Massive MIMO scenario in conjunction with CSIT and showed that by carefully designing the downlin and uplin precoders, the ethodology shows very high gains, especially on the downlin, although it eploys an exponentially sipler MAP detector at the user site. Based on the evidence presented, the novel application of the auss-herite quadrature rule in the MIMO scenario allows for generalizing the interesting results presented in [16] and [21] to the QAM case with ease. Because of the siplification achieved by the cobination of PP and the auss-herite approxiation, we were able to derive results with, e.g., N t = N r = 100 as well as with M = 64 efficiently. In addition, the presented auss-herite approxiation offers iportant siplification in the evaluation of the MMSE covariance atrix of the MIMO channel which is required in, aong other areas, the Statistical CSI equivalent channel deterination [21]. APPENDIX A AUSS-HERMITE QUADRATURE APPROXIMATION IN MIMO INPUT OUTPUT MUTUAL INFORMATION I (x; y) = H (x) H (x y) = N t log 2 (M) H (x y), where the conditional entropy, H (x y) can be written as [32] H (x y) = N r log(2) + 1 ( ( E n log 2 = N r log(2) + 1 log 2 ( exp( 1 )) σ 2 n H(x x ) 2 ) + N c (n 0,σ 2 I) exp( 1 σ 2 n H(x x ) 2 ) ) dn, (22) where N c (n 0,σ 2 I) represents the probability density function (pdf) of the circularly syetric coplex rando vector due to AWN. Let us define. + f = N c (n 0,σ 2 I) ( log 2 exp( 1 ) σ x 2 n H(x x ) 2 ) dn. (23) There is an intiate connection between f and the paraeter. { } H (y) = E y x log2 (p(y)) x = x called the Input- Dependent Output Entropy (IDOE) herein. Note that IDOE represents the entropy at the receiver output when the input is x. This paraeter is different that the conditional entropy. By using standard entropic identities, we can easily see that f = H (y) + 2N r log 2 (σ ) + N r log 2 () + N t log 2 (M). (24)

11 4206 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 64, NO. 10, OCTOBER 2016 Thus, since our auss-herite approxiation focuses on finding approxiations to each of the f ters, one per input sybol, it equivalently offers estiates of the IDOE ters H (y). This gives a physical eaning to the estiated ters fˆ presented below. Since n has independent coponents over the different receiving antennas, and over the real and iaginary diensions, the integral above can be partitioned into 2N r real integrals in tande, in the following anner: Define by n rv, n iv, with v = 1,, N r,thevth receiving antenna real and iaginary noise coponent, respectively. Also define by (H(x x )) rv and (H(x x )) iv, the vth receiving antenna real and iaginary coponent of (H(x x )), respectively. We then have N c (n 0,σ 2 1 I) = N r σ 2N exp( l n2 rv + n2 iv r σ 2 ), (25) N r dn = dn rv dn iv, (26) v=1 and exp( 1 σ 2 n H(x x ) 2 ) = exp( 1 σ 2 ( v (n rv (H(x x )) rv ) 2 + (n iv (H(x x )) iv ) 2 )). (27) v The auss-herite quadrature is as follows: + exp( x 2 ) f (x)dx c(l) f (v l ), (28) for any real function f (x), and with vector c = [c(1) c(l)] T being the weights, and v l are the nodes of the approxiation. The approxiation is based on the following weights and nodes [23] c(l) = l=1 2L 1 L! 2 L 2 (H L 1 (v l )) 2 (29) where H L 1 (x) = ( 1) L 1 exp(x 2 ) d L 1 (exp( x 2 )) is the dx (L 1)-th order Heritian polynoial, L 1 and the value of the node v l equals the root of H L (x) for l = 1, 2,, L. Applying the auss-herite quadrature 2N r ties in tande to the integral in (23), and after changing variables, we get that f ˆ f = where ( ) 1 Nr r1 =1 i1 =1 rnr =1 inr =1 c( r1 )c( i1 ) c( inr )g (σ n r1,σn i1,,σn rnr,σn inr ), (30) g (σ n r1,σn i1,,σn rnr,σn inr ) (31) is the value of the function (fro (30)) ( ) log 2 exp( 1 σ 2 n H(x x ) 2 ) evaluated at n e = σ v({ rv, iv } N r v=1 ). For the special case N r = 2, (30) becoes ˆ f = ( ) 1 2 r1 =1 i1 =1 r2 =1 i2 =1 c( r1 )c( i1 ) (32) c( r2 )c( i2 )g (σ n r1,σn i1,σn r2,σn i2 ), (33) which using basic properties of bilinear fors can be rewritten as ( ) fˆ 1 2 = c( r1 )c( i1 )c( r2 ) r1 =1 i1 =1 r2 =1 i2 =1 c( i2 )g (σ n r1,σn i1,σn r2,σn i2 ) ( ) 1 2 = c t F,1c, (34) where F,1 is an L L atrix with r1, i1 eleent equal to F,1 [ r1, i1 ]=c t V,1 c, (35) with V,1 being an L L atrix with r2, i2 eleent equal to V,1 [ r2, i2 ]=g (σ n r1,σn i1,σn r2,σn i2 ), (36) so that the different f can be approxiated efficiently, and then suing the over the different x, as per (22), we get an approxiation for H (x y), H (x y) N r log(2) + 1 = N r log(2) + 1 ˆ f c t V c = N r log(2) + 1 2M N ct ( V )c t = N r log(2) + 1 2M N ct Vc, t (37) where V =. V. The procedure presented above for the N t = N r = 2 case can be directly generalized to any N t = N r by following a recursive procedure as follows. We repeat the above recursive procedure until either F,l or V,l have included all the indices rj, ij. The procedure will end at a V,l if N r is even and an F,l if N r is odd. APPENDIX B DERIVATION OF W I THROUH THE AUSS-HERMITE APPROXIMATION Without loss of generality, let s assue that N t = N r. Using Theore 1, we can write by using the auss-herite approxiation with M instead of H, I (x; y) N t log 2 (M) N r log(2) 1 ˆ f. (38)

12 KETSEOLOU AND AYANOLU: LINEAR PRECODIN AIN FOR LARE MIMO CONFIURATIONS 4207 In order to derive the gradient of I (x; y) with respect to W, we first derive the gradient of I (x; y) with respect to M.Start with the differential of I (x; y) with respect to M in (38) and approxiate the f by fˆ, d M I (x; y) 1 d M r1 =1 i1 =1 ( ) fˆ = 1 rnr =1 inr =1 ( ) 1 Nr c( r1 )c( i1 ) c( rnr ) c( inr ) d M ( g (σ n r1,σn i1,,σn rnr,σn inr ) ). (39) Taing into account that g (σ n r1,σn i1,,σn rnr,σn inr ) is the value of ( ) log 2 exp( 1 σ 2 n M(x x ) 2 ) evaluated at n e = σ v({ rv, iv } N r v=1 ), we can develop (39) further, by using well-nown results [31], as follows d M I (x; y) 1 ( ) 1 1 Nr log(2) rnr =1 inr =1 r1 =1 i1 =1 c( r1 )c( i1 ) c( rnr )c( inr ) tr ( {R(σ n r1,r1,σn i1,i1,,σn rnr, rnr,σn inr, inr )} t dm ), (40) where R(σ n r1,σn i1,,σn rnr,σn inr ) is the value of 1 l exp( 1 n M(x σ 2 x l ) 2 ) ( exp( 1 σ 2 n M(x x ) 2 ) ( (n M(x x ))(x x ) h) ) (41) evaluated at n e = σ v({ rv, iv } N r v=1 ). In this derivation we have used the fact that ( d M exp( n M(x x ) 2 ) σ 2 ) = 1 σ 2 exp( n M(x x ) 2 σ 2 )(n M(x x )) (x x ) h. (42) Using the fact that for a Heritian atrix such as M, we need to add the Heritian of the differential above in order to evaluate the actual gradient (see [31]), we get (19). Now, since W = M 2, by taing vectors of the atrices on each side of this equation and applying soe identities fro [31], we get that dvec(w) = ( (M ) I + I M ) dvec(m), (43) where the notation vec(a) denotes the vector found fro atrix A by taing its coluns one at a tie, starting fro the leftost one. Since M = W 1 2, it is straightforward to derive (23) by eploying (43) as follows. The relationship between the gradient and the derivative is through reshaping the derivative row vector [31], we can thus write W I reshape((vec( M I) T ( (M ) I + I M ) 1 ), N t, N t ), (44) where reshape(a,, n) is the standard reshape of atrix A with total eleents n to a atrix with rows, n coluns. APPENDIX C EVALUATION OF COMPUTATIONAL COMPLEXITY IN CALCULATIN I (x; y) THROUH AUSS-HERMITE QUADRATURE Let s start with N r = 2. Then, the coplexity involved in the auss-herite quadrature approxiation of I (x; y) is deterined by the one required in the calculation of fˆ in (34). Fro (35), this is equal to (2L 1) 2 ties the nuber of operations, including suations and products, required in evaluating each eleent of the atrix V.SinceV is a size L L atrix with eleents given in (36), it can be seen fro (32) that the coplexity of each eleent V [ r2, i2 ] is L 2 N r (2N t + N r 1). Since there are suation ters over (the size of the overall ultiple input constellation), the total coplexity becoes M 2N t (2L 1) 4 L 4 N r (2N t + 1). For a general value of N r, in a siilar fashion, the corresponding coplexity becoes M 2N t (2L 1) (2N r ) L 4 N r (2N t + N r 1). REFERENCES [1] K.-H. Lee and D. P. Petersen, Optial linear coding for vector channels, IEEE Trans. Coun., vol. 24, no. 12, pp , Dec [2] J. Salz, Digital transission over cross-coupled linear channels, AT&T Tech. J., vol. 64, no. 6, pp , Jul [3] J. Yang and S. Roy, On joint transitter and receiver optiization for ultiple-input-ultiple-output (MIMO) transission systes, IEEE Trans. Coun., vol. 42, no. 12, pp , Dec [4] A. Scaglione,. B. iannais, and S. Barbarossa, Redundant filterban precoders and equalizers. I. Unification and optial designs, IEEE Trans. Signal Process., vol. 47, no. 7, pp , Jul [5] A. Scaglione, P. Stoica, S. Barbarossa,. B. iannais, and H. Sapath, Optial designs for space-tie linear precoders and decoders, IEEE Trans. Signal Process., vol. 50, no. 5, pp , May [6] H. Sapath, P. Stoica, and A. Paulraj, eneralized linear precoder and decoder design for MIMO channels using the weighted MMSE criterion, IEEE Trans. Coun., vol. 49, no. 12, pp , Dec [7] J. Yang and S. Roy, Joint transitter-receiver optiization for ultiinput ulti-output systes with decision feedbac, IEEE Trans. Inf. Theory, vol. 40, no. 5, pp , Sep [8] H. Sapath and A. Paulraj, Linear precoding for space-tie coded systes with nown fading correlations, IEEE Coun. Lett., vol. 6, no. 6, pp , Jun [9] D. P. Paloar, J. M. Cioffi, and M. A. Lagunas, Joint Tx-Rx beaforing design for ulticarrier MIMO channels: A unified fraewor for convex optiization, IEEE Trans. Signal Process., vol. 51, no. 9, pp , Sep [10] V. Raghavan, A. M. Sayeed, and V. V. Veeravalli, Seiunitary precoding for spatially correlated MIMO channels, IEEE Trans. Inf. Theory, vol. 57, no. 3, pp , Mar [11] F. Perez-Cruz, M. R. D. Rodrigues, and S. Verdu, MIMO aussian channels with arbitrary inputs: Optial precoding and power allocation, IEEE Trans. Inf. Theory, vol. 56, no. 3, pp , Mar

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Marzetta, and S. Vishwanath, Pilot containation and precoding in ulti-cell TDD systes, IEEE Trans. Wireless Coun., vol. 10, no. 8, pp , Aug [36] H. Q. Ngo, E.. Larsson, and T. L. Marzetta, Energy and spectral efficiency of very large ultiuser MIMO systes, IEEE Trans. Coun., vol. 61, no. 4, pp , Apr Thoas Ketseoglou (S 85 M 91 SM 96) received the B.S. degree fro the University of Patras, Patras, reece, in 1982, the M.S. degree fro the University of Maryland, College Par, MD, USA, in 1986, and the Ph.D. degree fro the University of Southern California, Los Angeles, CA, USA, in 1990, all in electrical engineering. He wored in the wireless counications industry, including senior level positions with Sieens, Ericsson, Rocwell, and Onipoint. Fro 1996 to 1998, he participated in TIA TR45.5 (now 3PP2) 3 standardization, aing significant contributions to the cda2000 standard. He holds several essential patents in wireless counications. Since 2003, he has been with the Electrical and Coputer Engineering Departent, California State Polytechnic University, Poona, CA, USA, where he is a Professor. He spent his sabbatical leave in 2011 with the Digital Technology Center, University of Minnesota, Minneapolis, MN, USA, where he taught digital counications and perfored research on networ data and achine learning techniques. He is a part-tie Lecturer with the University of California at Irvine. His teaching and research interests are in wireless counications, signal processing, and achine learning, with current ephasis on MIMO, optiization, localization, and lin prediction. Ender Ayanoglu (S 82 M 85 SM 90 F 98) received the B.S. degree fro Middle East Technical University, Anara, Turey, in 1980, and the M.S. and Ph.D. degrees fro Stanford University, Stanford, CA, USA, in 1982 and 1986, respectively, all in electrical engineering. He was with the Counications Systes Research Laboratory, part of AT&T Bell Laboratories, Holdel, NJ, USA, until 1996, and Bell Labs, Lucent Technologies, until Fro 1999 to 2002, he was a Systes Architect with Cisco Systes, Inc. Since 2002, he has been a Professor with the Departent of Electrical Engineering and Coputer Science, University of California at Irvine, CA, USA, where he served as the Director of the Center for Pervasive Counications and Coputing and held the Conexant-Broadco Endowed Chair fro 2002 to He was a recipient of the IEEE Counications Society Stephen O. Rice Prize Paper Award in 1995 and the IEEE Counications Society Best Tutorial Paper Award in Fro 1993 to 2014, he was an Editor of the IEEE TRANSACTIONS ON COMMUNICATIONS and served as its Editor-in-Chief fro 2004 to Currently, he is serving as a Senior Editor of the IEEE TRANSACTIONS ON COMMUNICATIONS. Fro 1990 to 2002, he served on the Executive Coittee of the IEEE Counications Society Counication Theory Coittee, and fro 1999 to 2001, he was its Chair.