Linear Precoding for Large MIMO Configurations and QAM Constellations

Transcription

1 1 Linear Precoding for Large MIMO Configurations and QAM Constellations Abstract In this paper, the proble of designing a linear precoder for Multiple-Input Multiple-Output (MIMO systes in conjunction with Quadrature Aplitude Modulation (QAM is addressed. First, a novel and efficient ethodology to evaluate the input-output utual inforation for a general Multiple-Input Multiple-Output (MIMO syste as well as its corresponding gradients is presented, based on the Gauss-Herite quadrature rule. Then, the ethod is exploited in a block 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 coplexity-reducing per-group processing (PGP technique, which is sei-optial, to both perfect channel state inforation at the transitter (CSIT as well as statistical channel state inforation (SCSI scenarios, with high transitting and receiving antenna size, and for constellation size up to M = 64. We show by nuerical results that the precoders developed offer significantly better perforance than the configuration with no precoder, and the axiu diversity precoder for QAM with constellation sizes M = 16, 32, and 64 and for MIMO channel size 0 0. 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, BICMB [1] [3] has shown great potential for practical application, due to its excellent diversity gains and its siplicity. For exaple, BICMB in conjunction with convolutional coding offers axiu diversity and axiu spatial ultiplexing siultaneously [1], thus it represents an optial technique for this type of Forward Error Correction (FEC. In addition, there are any past works available which investigated linear precoding through exploitation of a unitary precoding atrix with success, ainly fro a diversity axiization point of view [4], [5]. On the other hand, LDPC coding is the currently prevailing, near-capacity achieving error-correction technique that operates based on input-output utual inforation [6], [7]. The proble of designing an optial linear precoder toward axiizing the utual inforation between the input and output was first considered in [8], [9] where the first optial power allocation strategies are presented (e.g., Mercury Waterfilling (MWF, together with general equations for the optial precoder design. In addition, [] also considered precoders for utual inforation axiization and showed that the left eigenvectors of the optial precoder can be set equal to the right eigenvectors of the channel. Finally, in [11], a utual inforation axiizing precoder for a parallel layer MIMO detection syste is presented reducing the perforance gap between axiu likelihood and parallel layer detection. Recently, globally optial linear precoding techniques were presented [12], [13] for perfect channel state inforation available at the transitter (CSIT 1 scenarios with finite alphabet inputs, capable of achieving utual inforation rates uch higher than the previously presented MWF [8] techniques by introducing input sybol correlation through a unitary input transforation atrix in conjunction with channel weight adjustent (power allocation. Furtherore, ore recent work has shown that when only statistical channel state inforation (SCSI 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 looking at an equivalent constant channel and its corresponding adjustents as per the pertinent theory [16], and applying a odified expression for the corresponding ergodic utual inforation evaluation over all channel realizations. This developent allows for a precoder optiization under SCSI in a uch easier way [16]. However, existing research in the area does not provide any results of optial linear precoders in the case of QAM with constellation size M 16, with the exception of [17]. In past research work, a ajor ipedient toward developing optial precoders for QAM has been a lack of an accurate and efficient technique toward input-output utual inforation evaluation, its gradients, and evaluation of the input-output iniu ean square error (MMSE error covariance atrix, as required by the precoder optiization algorith and other algoriths involved, e.g., the equivalent channel deterination in the SCSI case [16]. In this paper, we propose optial linear precoding techniques for MIMO with LDPC coding, suitable for QAM with constellation size M 16. Carrying out this caculation has been very difficult to do until now due to the coplexity involved in tackling this proble. Our approach entails a novel application of the Herite-Gauss quadrature rule [18] 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 block gradient ascent ethod [19] in order to deterine the globally optial linear precoder for MIMO systes, in a siilar fashion to [12], 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 1 Under CSIT the transitter has perfect knowledge of the MIMO channel realization at each transission. 2 SCSI pertains to the case in which the transitter has knowledge of only the MIMO channel correlation atrices [14], [15] and the theral noise variance.

2 2 precoder (MDP of [4] and the no precoder case, at low signalto-noise ratio (SNR for a standard 2 2 MIMO channel, however the absolute utilization gain achieved is lower than 1 b/s/hz. We then proceed to show that significantly higher gains are available for different channels, e.g., a utilization gain of 1.30 b/s/hz at SNR = db, when M = 16. We then eploy higher antenna configurations, e.g., up to with CSIT and M = 16, 32, and 64 together with the coplexity reducing technique of per-group processing (PGP which was originally presented in [20], and show very high gains available with reduced syste coplexity. Finally, we also eploy SCSI scenarios in conjunction with PGP and show very significant gains for high antennas sizes and M = 16, 32, 64. Our ain advantages copared with [17] lie over three ain directions: a It offers a globally optial precoder solution for each subgroup, instead of a locally optial one, b It is faster, and c It allows for higher constellation size, e.g., M = 32, 64 results with ease. The paper is organized as follows: Section II presents the syste odel and proble stateent. Then, in Section III, we present a novel Gauss-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 the globally optial linear precoder for LDPC MIMO. In Section IV, we present nuerical results for the globally optial precoder that ipleents the Gauss-Herite approxiation in the block coordinate gradient ascent ethod. Finally, our conclusions are presented in Section V. II. SYSTEM MODEL AND PROBLEM STATEMENT The N t transit antenna, N r receive antenna MIMO odel is described by the following equation y = HGx + n, (1 where y is the N r 1 received vector, H is the N r N t MIMO channel atrix, G 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, and n represents the circularly syetric coplex Additive White Gaussian Noise (AWGN of size N r 1, with ean zero and covariance atrix K n = σni 2 Nr, where I 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 will be considered, e.g., channels coprising independent coplex Gaussian coponents or spatially correlated Kronecker-type channels [14] (including siilar to the 3GPP spatial correlation odel (SCM ones [21] used in [17] or [15]. The precoding atrix G needs to satisfy the following power constraint tr(gg 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 [12] y = Σ H Σ G V h Gx + n, (3 where Σ H and Σ G are diagonal atrices containing the singular values of H, G, respectively and V G is the atrix of the right singular vectors of G. When LDPC is eployed in this MIMO syste, the overall utilization in b/s/hz is deterined by the utual inforation between the transitting branches x and the receiving ones, y [6], [7]. It is shown [12] that the utual inforation between x and y, I(x; y, is only a function of W = V G Σ 2 HΣ 2 GVG h. The optial precoder, G is found by solving: axiize G I(x; y subject to tr(gg h = N t, (4 called the original proble, and axiize V G,Σ G I(x; y subject to tr(σ 2 G = N t, 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. A ajor difficulty in the QAM case stes fro the fact that there are ultiple evaluations of I(x; y in the block coordinate ascent ethod eployed for deterining the globally optial precoder. More specifically, for each block coordinate gradient ascent iteration, there are two line backtracking searches required [12], which deand one I(x; y plus its gradient evaluations per search trial, and one additional evaluation at the end of a successful search per backtracking 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 LDPC MIMO. III. ACCURATE APPROXIMATION TO I(x; y FOR MIMO SYSTEMS BASED ON GAUSS-HERMITE QUADRATURE In Appendix A we prove that by applying the Gauss- Herite quadrature theory for approxiating the integral of a Gaussian function ultiplied with an arbitrary real function f(x, i.e. F. = (5 exp( x 2 f(xdx, (6 which is approxiated in the Gauss-Herite approxiation with L weights and nodes as F L c(lf(v l = c t f, (7 l=1 with c = [c(1 c(l] t, {v l } L l=1, and f = [f(v 1 f(v L ] t, being the vector of the weights, the nodes, and function node values, respectively (see Appendix A, the following 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 ake the overall understanding easier. Let n e denote the equivalent to n, real vector of length

3 3 2N r derived fro n by separating its real, iaginary parts as follows n e = [n r1 n i1 n rnr n inr ] t, (8 with n rv, n iv being the values of the real, iaginary part of the v (1 v N r eleent of n, respectively. Let us also define the real vector v({k rv, k iv } Nr v=1 of length 2N r defined as follows v({k rv, k iv } Nr v=1 = [v kr1 v ki1,, v krnr v kinr ] t, (9 with k rv, k iv (1 v N r being perutations of indexes in the set {1, 2,, L}. Then the following lea is true concerning the Gauss-Herite approxiation for I(x; y. Lea 1. For the MIMO channel odel presented in (1, the Gauss-Herite approxiation for I(x; y with L nodes per receiving antenna is given as where ˆf k = I(x; y N t log 2 (M ( Nr 1 L π L k r1=1 k i1=1 Nr log(2 1 M N t ˆf M N k, t k=1 L L k rnr =1 k inr =1 ( c(k r1 c(k i1 c(k rnr c(k inr g k (σn kr1, σn ki1,, σn krnr, σn kinr, with (11 g k (σv kr1, σv ki1,, σv krnr, σv kinr (12 being the value of the function log 2 ( exp( 1 σ 2 n HG(x k x 2 evaluated at n e = σv({k rv, k iv } Nr v=1. The proof of this lea is presented in Appendix A. IV. GLOBAL OPTIMIZATION OVER G TOWARD MAXIMUM I(x; y FOR QAM A. Description of the Globally Optial Precoder Method (13 Siilarly to [12], we follow a block 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 [12] that I(x; y is a concave function over W and Σ 2 G. It thus becoes efficient to eploy two different gradient ascent ethods, one for W, and another one for Σ 2 G. We eploy Θ and Σ to denote VG h and Σ2 G, respectively, evaluated during the optiization algorith s execution. There will in general be ultiple evaluations of I(, 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 herein the possibility of finding the globally optial precoder for QAM with M 16 becoes reality, as our results deonstrate. B. Deterination of W I, Σ 2 G I We first set M = W 1 2. Then, it is easy to see that I is a function of M (see, e.g., [22] 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 G Σ H Σ G V h G = W 1 2 for HG in ( results in the sae value of I(x; y. In other words, since M is a function of H, G, 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 G on the left, resulting in ỹ = V G y = V G Σ H Σ G V h Gx + V G n, (14 where the odified noise ter V G n has the sae statistics with n, because V G is unitary. By applying the Gauss-Herite approxiation to (14, we see that we get the desired result, i.e., the value of I 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 siple theore, we can prove easily part of Theore 1 in [12], naely the fact that I(x; y is only a function of W, as My a sufficient statistic for y and M is a function of W. Assue without loss of generality that N t = N r. The gradient of I with respect to M can be found (see Appendix B for the derivation fro the Gauss-Herite expression presented in ( as follows M I = 1 ( Nr 1 1 L L L L log(2 M Nt π c(k r1 c(k i1 c(k rnr c(k inr k r1=1 k i1=1 R(σv kr1, σv ki1,, σv krnr, σv kinr, k rnr =1 k inr =1 (15 3 The theore applies without loss of generality to the N t = N r case. If N t N r, then Σ H needs to be either shrunk, or extended in size, by eliination or addition of zeros, respectively.

4 4 where R(σv kr1, σv ki1,, σv krnr, σv kinr is the value of the N t N t atrix 1 exp( 1 σ n M(x 2 k x 2 k exp( 1 σ 2 n M(x k x 2 ((n M(x k x (x k x h + ((n M(x k x (x k x h h (16 evaluated at n e = σv({k rv, k iv } Nr 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 [23], we get the desired result as follows (see Appendix B. Lea 2. For the MIMO channel odel presented in (1, the Gauss-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, (17 where reshape(a, k, n is the reshape of atrix A (with total nuber of eleents kn to a atrix with k rows, n coluns and where denotes Kronecker product of atrices. Then, as I is a concave function of W [22], we can axiize over W in a straightforward way using closed for expressions. This is based on the fact that the approxiated I through the Gauss-Herite approxiation is very accurate, as shown in the next section. Finally, since fro [24] we have that Σ 2 G I = diag(v h G WIV G Σ 2 H, we can easily evaluate it through the above presented procedure. V. NUMERICAL RESULTS The results presented herein eploy QAM with 16, 32 or 64 constellation sizes. We eploy MIMO systes with N t = N r = 2 when global precoding optiization is perfored. We have used an L = 3 Gauss-Herite approxiation which results in 3 2Nr total nodes due to MIMO. We apply the coplexity reducing ethod of PGP [20] which offers sei-optial results under exponentially lower transitter and receiver coplexity [20]. PGP 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 SCSI cases, as they both present significant application scenarios for 5G. A. Results for SCSI in conjunction with PGP In Fig. 7 we present results for PGP versus a no precoding SCM channel with N t = N r = 0 and M = 16. To the best of our knowledge, results for such high size antenna configurations were not available in the literature. Siilar to PGP No precoding SNR b, db Fig. 7. Results for PGP and no precoding cases for a 0 0 H SCSI MIMO syste and QAM M = 16 odulation. the previous results, we observe high inforation rate gains in the high SNR b regie as the PGP syste achieves the full capacity of 400 b/s/hz while the no precoding schee saturates at 320 b/s/hz. The PGP syste eployed uses 50 groups of size 2 2 each. B. Results for CSIT in conjunction with PGP For a 4X4 channel H = i i i i i i i i i i i i used with M = 64 we get the no precoding and the PGP results using 2 groups of size 2 2 each depicted in Fig. 8. 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 PGP over the no precoding No precoding PGP SNR, db b Fig. 8. Results for PGP and no precoding cases for a 4 4 H CSIT MIMO syste and QAM M = 64 odulation. case in the high SNR b regie. To the best of our knowledge this type of results for optial precoding in conjunction with,

5 5 M = 64 were not available in the literature. Finally, in Fig. 9 we present results for an asyetric randoly generated MIMO channel with N t = 4, N r =, and M = 16. PGP eploys 2 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 No precoding PGP SNR b, db Fig. 9. Results for PGP and no precoding cases for a randoly generated 4 H CSIT MIMO syste and QAM M = 16 odulation. C. Results for Massive MIMO Massive MIMO [25] [27] has attracted uch interest recently, due to its potential to offer high data rates with low signal power. We present results for the uplink, and downlink of a Massive MIMO syste based on 0 base station, 4 user antennas, respectively, with M = 16, 64, and for a Kroneckerbased 3GPP SCM urban channel in a CSIT scenario. Fig. shows results for the 4 0 uplink of the syste. We eploy PGP to draatically reduce the syste coplexity at the transitter and receiver sites. Under no precoding, the channel achieves the axiu utual inforation rate, thus achieving high gains on the uplink 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 uplink significantly outperfors the PGP M = 16 uplink. Second, the PGP M = 6 uplink offers further gains by, e.g., achieving the axiu possible rate of 24 b/s/hz. For the downlink, in Fig. 11 we show results where the no precoding case operates under 0 antenna inputs all correlated through the right eigenvectors of the channel, thus creating a very deanding environent at the user, due to the exponentially increasing axiu a posteriori (MAP detector coplexity [20]. On the other hand, eploying PGP with only two input sybols per receiving antenna, i.e., with draatically reduced decoding coplexity, the PGP syste achieves uch higher throughput in the lower SNR regie, with SNR gain on the order of db, albeit achieving a axiu of 32, 48 b/s/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 PGP schee achieves 45% higher throughput that the M = 16 one, a significant iproveent. We also like to ephasize that the no precoding schee requires a very high exponential MAP detector coplexity, on the order of M 0, while for the low-snr-superior PGP, 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 PGP on the Massive MIMO downlink. On the other hand, in lower SNR, the PGP schee achieves both uch higher throughput with siultaneously exponentially lower MAP detector coplexity at the user site detector PGP, M=64 No precoding, M=64 PGP, M=16 No precoding, M= No precoding, N t =0, M=16, PGP with two sybols per receiving antenna, M=16 PGP with two sybols per receiving antenna, M= SNR b, db SNR, db b Fig. 11. Results for PGP and no precoding cases for a randoly generated 4 0 H downlink CSIT MIMO syste and QAM M = 16 odulation. Fig.. Results for PGP and no precoding cases for a randoly generated 0 4 uplink H CSIT MIMO syste and QAM M = 16 odulation. saturates and fails to eet the axiu possible utual inforation of 16 b/s/hz, while with PGP the syste clearly VI. CONCLUSIONS In this paper, the proble of designing a linear precoder for MIMO systes eploying, e.g., LDPC codes toward utual inforation axiization is addressed for QAM with M 16

6 6 and in conjunction with high MIMO syste size. A ajor obstacle toward this goal is a lack of efficient techniques for evaluating I(x, y and its derivatives. We have presented a novel solution to this proble based on the Gauss-Herite quadrature. I(x; y = N t log 2 (M at high SNR. Furtherore, we apply the sae Gauss-Herite approxiation approach to CSIT high antenna size channels with the sae success. We show that for specific type of channels siilar to urban 3GPP SCM [21], the PGP approach offers very high gains over the no precoding case in the high SNR b regie. Finally, we consider a Massive MIMO scenario in conjunction with CSIT and show that by carefully designing the downlink, uplink precoders, the ethodology shows very high gains, especially on the downlink, although it eploys an expenentially sipler MAP detector at the user site. Based on the evidence presented, the novel application of the Gauss-Herite quadrature rule in the MIMO scenario allows for generalizing the interesting results presented in [12], [16] to the QAM case with ease. Because of the siplification achieved by the cobination of PGP and the Gauss-Herite approxiation, we were able to derive results with, e.g., N t = N r = 0 as well as with M = 64 efficiently. The novel cobination of the Gauss-Herite approxiation with PGP allows for application of optiized procoding to 5G oriented assive MIMO systes with ease. In addition, the presented Gauss-Herite approxiation offers iportant siplification in the evaluation of the MMSE error covariance atrix of the MIMO channel which is required in, aong other areas, the SCSI equivalent channel deterination as per [16]. Finally, copared with other interesting proposals for large MIMO sizes, e.g., [17], which also eploys PGP, the presented approach has the following advantages: a It offers a globally optial precoder solution for each subgroup, instead of a locally optial one, b It is faster, and c It allows for higher constellation size, e.g., M = 32, 64 results with ease, thus it sees to be ore efficient in these regards. APPENDIX A GAUSS-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 [12] H(x y = Nr log(2 + 1 M N t k ( ( E n log 2 exp( 1 σ n HG(x 2 k x 2 = N r log(2 + 1 (18 N M N t c(n 0, σ 2 I k log 2 ( exp( 1 σ 2 n HG(x k x 2 dn, where N c (n 0, σ 2 I represents the probability density function (pdf of the circularly syetric coplex AWGN. Let us define. f k = N c (n 0, σ 2 I log 2 ( x exp( 1 σ 2 n HG(x k x 2 dn. (19 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, the vth receiving antenna real and iaginary noise coponent, respectively. Also define by (HG(x k x rv and (HG(x k x iv, the vth receiving antenna real and iaginary coponent of (HG(x k x, respectively. We then have N c (n 0, σ 2 1 I = π Nr σ exp( l n2 rv + n 2 iv 2Nr σ 2, (20 N r dn = dn rv dn iv, (21 v=1 and exp( 1 σ 2 n HG(x k x 2 = + v exp( 1 σ 2 ( v (n rv (HG(x k x rv 2 (n iv (HG(x k x iv 2. The Gauss-Herite quadrature is as follows: exp( x 2 f(xdx (22 L c(lf(v l, (23 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 [18] c(l = l=1 2L 1 L! 2π L 2 (H L 1 (v l 2 (24 where H L 1 (x = ( 1 L 1 exp(x 2 dl 1 dx L 1 (exp( x 2 is the (L 1th order Heritian polynoial, and the value of the node v l equals the root of H L (x for l = 1, 2,, L. Applying the Gauss-Herite quadrature 2N r ties in tande, to the integral in (19, and after changing variables, we get that ( f k ˆf Nr 1 L L L L k = c(k r1 c(k i1 π k r1=1 k i1=1 k rnr =1 k inr =1 c(k rnr c(k inr g k (σn kr1, σn ki1,, σn krnr, σn kinr, (25 where g k (σn kr1, σn ki1,, σn krnr, σn kinr (26

7 7 is the value of the function (fro (25 log 2 ( exp( 1 σ 2 n HG(x k x 2 evaluated at n e = σv({k rv, k iv } Nr v=1. APPENDIX B DERIVATION OF W I THROUGH THE GAUSS-HERMITE APPROXIMATION (27 Without loss of generality, let s assue that N t = N r. Using Theore 1, we can write by using the Gauss-Herite approxiation with M instead of HG, I(x; y N t log 2 (M N r log(2 1 M Nt ˆf k. (28 In order to derive the gradient of I with respect to W, we first derive the gradient of I with respect to M. Start with the differential of I with respect to M in (28 and approxiate the f k by ˆf k, to get for the differential of I(x; y over M. The full details can be found in [28], but are oitted fro here, due to lack of space. We can then eploy identities fro [23] to get the desired result. REFERENCES [1] E. Akay, E. Sengul, and E. Ayanoglu, Bit-Interleaved Coded Multiple Beaforing, IEEE Transactions on Counications, vol. 55, pp , Septeber [2] B. Li, H. J. Park, and E. Ayanoglu, Constellation Precoded Multiple Beaforing, IEEE Transactions on Counications, vol. 59, pp , May [3] B. Li and E. Ayanoglu, Diversity Analysis of Bit-Interleaved Coded Multiple Beaforing with Orthogonal Frequency Division Multiplexing, IEEE Transactions on Counications, vol. 61, pp , Septeber [4] Y. Xin, Z. Wang, and G. Giannakis, Space-Tie Diversity Systes Based on Linear Constellation Precoding, IEEE Transactions on Wireless Counications, vol. 2, pp , March [5] V. Tarokh, N. Seshadri, and A. Calderbank, Space-Tie Codes for High Data Rate Counications: Perforance Criterion and Code Construction, IEEE Transactions on Inforation Theory, vol. 44, pp , March [6] S. ten Brink, G. Kraer, and A. Ashihkin, Design of Low-Density Parity-Check Codes for Modulation and Detection, IEEE Transactions on Counications, vol. 52, pp , April [7] Y. Jian, A. Ashihkin, and N. Shara, LDPC Codes for Flat Rayleigh Fading Channels with Channel Side Inforation, IEEE Transactions on Counications, vol. 56, pp , August [8] F. Perez-Cruz, M. Rodriguez, and S. Verdu, MIMO Gaussian Channels with Arbitrary Inputs: Optial Precoding and Power Allocation, IEEE Transactions on Inforation Theory, vol. 56, pp , March 20. [9] A. Lozano, A. Tulino, and S. Verdu, Optial Power Allocation for Parallel Gaussian Channels With Arbitrary Input Distributions, IEEE Transactions on Inforation Theory, vol. 52, pp , July [] M. Payaro and D. Paloar, On Optial Precoding in Linear Vector Gaussian Channels with Arbitrary Input Distribution, in Proceedings IEEE International Syposiu on Inforation Theory, [11] E. Ohler, U. Wachsann, and G. Fettweis, Mutual Inforation Maxiizing Linear Precoding for Parallel Layer MIMO Detection, in Proceedings 12th International Workshop on Signal Processing Advances in Wireless Counications, [12] C. Xiao, Y. Zheng, and Z. Ding, Globally Optial Linear Precoders for Finite Alphabet Signals Over Coplex Vector Gaussian Channels, IEEE Transactions on Signal Processing, vol. 59, pp , July [13] M. Laarca, Linear Precoding for Mutual Inforation Maxiization in MIMO Systes, in Proceedings International Syposiu of Wireless Counication Systes k [14] D. Shiu, G. Foschini, M. Gans, and J. Kahn, Fading Correlation and Its Effect on the Capacity of Multieleent Antenna Systes, IEEE Transactions on Counications, vol. 48, pp , March [15] W. Weichselberger, M. Herdin, H. Ozcelik, and E. Bonek, A Stochastic MIMO Channel Model with Joint Correlation of Both Links, IEEE Transactions on Wireless Counications, vol. 5, pp. 90 0, January [16] Y. Wu, C.-K. Wen, C. Xiao, X. Gao, and R. Schober, Linear precoding for the MIMO Multiple Access Channel With Finite Alphabet Inputs and Statistical CSI, IEEE Transactions on Wireless Counications, pp , February [17] Y. Wu, C.-K. Wen, D. Ng, R. Schober, and A. Lozano, Low-Coplexity MIMO Precoding with Discrete Signals and Statistical CSI, anuscript subitted for publication. [18] M. Abraowitz and I. Stegun, Handbookof Matheatical Functions with Forulas, Graphs, and Matheatical Tables. Washington D.C.: U.S. Governent Printing Office, [19] S. Boyd and L. Vandenberghe, Convex Optiization. Cabridge, UK: Cabridge University Press, [20] T. Ketseoglou and E. Ayanoglu, Linear precoding for MIMO with LDPC Coding and Reduced Coplexity, IEEE Transactions on Wireless Counications, pp , April [21] J. Salo, G. D. Galdo, J. Sali, P. Kyosti, M. Milojevic, D. Laselva, and C. Schneider, MATLAB Ipleentaion of the 3GPP Spatial Channel Model, Available at [22] Y. Wu, C. Wen, C. Xiao, X. Gao, and R. Schober, Linear Precoding for the MIMO Multiple Access Channel with Finite Alphabet Inputs and Statistical CSI, IEEE Transactions on Signal Processing, vol. 60, pp , February [23] A. Hj rugnes, Coplex-Valued Matrix Derivatives With Applications in Signal Processing and Counications. Cabridge, UK: Cabridge University Press, [24] W. Zeng, C. Xiao, and J. Lu, A Low Coplexity Design of Linear Precoding for MIMO Channels with Finite Alphabet Inputs, IEEE Wireless Counications Letters, vol. 1, pp , February [25] T. Marzetta, Noncooperative Cellular Wireless with Unliited Nubers of Base Station Antennas, IEEE Transactions on Wireless Counications, vol. 9, pp , Noveber 20. [26] J. Jose, A. Ashikhin, T. Marzetta, and S. Vishwanath, Pilot Containation and Precoding in Multi-Cell TDD Systes, IEEE Transactions on Wireless Counications, vol., pp , August [27] H. Ngo, E. Larsson, and T. Marzetta, Energy and Spectral Efficiency of Very Large Multiuser MIMO Systes, IEEE Transactions on Counications, vol. 61, pp , April [28] T. Ketseoglou and E. Ayanoglu, Linear Precoding for MIMO Channels with QAM Constellations and Reduced Coplexity, 2016, eprint: arxiv: v1 [cs.it]. View publication stats