Linear Precoding Gain for Large MIMO Configurations with QAM and Reduced Complexity

Transcription

1 Linear Precoding Gain 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 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 to both perfect channel state inforation at the transitter (CSIT) as well as statistical channel state inforation (SCSI) scenarios, with large 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 as well as the axiu diversity precoder for QAM with constellation sizes M = 16, 32, and 64 and for MIMO channel size up to 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 designing an optial linear precoder toward axiizing the utual inforation between the input and output has been extensively considered. For exaple, in [1], [2] the optial power allocation strategies are presented (e.g., Mercury Waterfilling (MWF)), together with general equations for the optial precoder design. In addition, [3] 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. Recently, globally optial linear precoding techniques were presented in [4], [5] 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 [1] T. Ketseoglou is with the Electrical and Coputer Engineering Departent, California State Polytechnic University, Poona, California (e-ail: tketseoglou@csupoona.edu). E. Ayanoglu is with the Center for Pervasive Counications and Coputing, Departent of Electrical Engineering and Coputer Science, University of California, Irvine (e-ail: ayanoglu@uci.edu). 1 Under CSIT the transitter has perfect knowledge of the MIMO channel realization at each transission. techniques by introducing input sybol correlation through a unitary input transforation atrix in conjunction with channel weight adjustent (power allocation). In addition, ore recently, [6] has presented an iterative algorith for precoder optiization for su rate axiization of Multiple Access Channels (MAC) with Kronecker MIMO channels. 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 [9], 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 [9]. 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 [10]. 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 and its gradients. Traditionally, MIMO linear precoding has been studied in conjunction with either a) Gaussian, or b) finite alphabet inputs. The latter are ore realistic and also present special issues, e.g., their offered capacity reaches saturation at high signal-to-noise ratio (SNR) [4]. Futherore, QAM odulation with M 16 is also instruental in achieving higher data rates. In addition, as MIMO systes of larger transitting and receiving sizes are becoing popular for 5G [11], linear precoding ethods capable of offering high gains with low coplexity for large MIMO transitting and receiving antenna sizes and QAM alphabet size M 16 prevail as necessary. This proble reains open, although its significance is high toward practical applications. In this paper, we propose optial linear precoding techniques for MIMO, suitable for QAM with constellation size M 16 which solve the above open proble. The proposed techniques can accoodate MIMO configurations with very large antenna sizes, e.g., while their coplexity grows linearly with the antenna size. The only related work in this area is [10] which has antenna sizes up to Our approach entails a novel application of the Herite-Gauss quadrature rule [12] which offers a very 2 SCSI pertains to the case in which the transitter has knowledge of only the MIMO channel correlation atrices [7], [8] and the theral noise variance /16/$ IEEE

2 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 [13] in order to deterine the globally optial linear precoder for MIMO systes, in a siilar fashion to [4], 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 [14] and the no-precoder case, at low signal-to-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 = 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 (PGP) which was originally presented in [15], 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 large antenna sizes, e.g., and M = 16, 32, 64. The ain advantages of our work copared with other interesting proposals for large MIMO sizes, e.g., [10], lie over four ain directions: a) It offers a globally optial precoder solution 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 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 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 instantaneous 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, 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, eing the (coded) sybol signal-to-noise 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 [7] (including those siilar to the 3GPP spatial correlation odel (SCM) [16]), or ore generally Weichselberger channels [8]. 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 [4] 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 a capacity-approaching Forward Error Correction (FEC) code 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 [17], [18]. It is shown [4] that the utual inforation between x and y, for channel realization H, I(x; y), is only a function of W = V G Σ 2 HΣ 2 GV h G. The optial CSIT precoder G is found by solving: axiize I(x; y) G 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 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 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 [4], 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 CSIT. In the SCSI case, the corresponding optiization proble becoes (5) axiize E H {I(x; y)} G subject to tr(gg h ) = N t, (6) where the expectation is perfored over all the channels H. The ground-breaking work of [9] 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 G through well-

3 deterined paraeters of a deterinistic channel, including the utual inforation of the corresponding deterinistic channel, i.e., a CSIT scenario. Thus, ethods that offer siplification of CSIT utual inforation evaluation, I(x; y), are also iportant in the SCSI case toward deterining the globally optial linear precoder for 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. = + exp( x 2 )f(x)dx, (7) which is approxiated in the Gauss-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, 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), 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 ake 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 n e = [n r1 n i1 n rnr n inr ] t, 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({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 k r1=1 k i1=1 Nr log(2) 1 M N t ˆf M N k, t k=1 k rnr =1 k inr =1 (10) 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 ). ) (13) The proof of this lea is presented in Appendix A, together with a siplification available for this expression in the N t = N r = 2 case. Let us stress that, the presented novel application of the Gauss-Herite quadrature in the MIMO odel allows for efficient evaluation of I(x; y) for any channel atrix H, and precoder G, as required in the precoder optiization process, as explained below. IV. GLOBAL OPTIMIZATION OVER G TOWARD MAXIMUM I(x; y) FOR QAM A. Description of the Globally Optial Precoder Method Siilarly to [4], 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 [4] 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 V h G and Σ2 G, respectively, evaluated during the execution of the optiization algorith.the algorith s pseudocode for a nuber of iterations t is presented under the heading Algorith 1. Algorith 1 Global precoder optiization algorith with t iterations 1: procedure PRECODER(Σ H) 2: while i t do 3: Deterine W NEW = Θ h Σ 2 Θ through backtracking line search 4: Set VG h = Θ 5: Deterine W NEW = V GΣ 2 GΣ 2 HVG h 6: Deterine Σ 2 NEW,G through backtracking line search 7: Set negative entries on the diagonal of Σ 2 NEW,G to zero 8: Noralize Σ 2 NEW,G to a trace equal to N t 9: Set Σ G = Σ NEW,G 10: Deterine W NEW = V GΣ 2 GΣ 2 HVG h 11: Set W = W NEW 12: Evaluate I(W) 13: end while 14: return I(W) 15: end procedure 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., [4] 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 (10) results in the sae value of I(x; y). In other 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 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(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. 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 (10) as follows M I = 1 ( ) Nr 1 1 L log(2) M Nt π k r1=1 k i1=1 k rnr =1 k inr =1 c(k r1 ) c(k inr )R(σv kr1, σv ki1,, σv krnr, σv kinr ), (15) where R(σv kr1, σv ki1,, σv krnr, σv kinr ) is the value of the N t N t atrix 1 exp( 1 n M(x σ 2 k x ) 2 ) k exp( 1 σ n M(x 2 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 [19]), 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 standard reshape of a atrix A (with total nuber of eleents kn) to a atrix with k rows, n coluns, and where denotes Kronecker 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 [4], 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 Gauss-Herite approxiation is very accurate, as shown in the next section. 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 perfored. We have used an L = 3 Gauss-Herite approxiation which results in 3 2Nr total nodes due to MIMO. The ipleentation of the globally optiizing ethodology is perfored by eploying two backtracking line searches, one for W and another one for Σ 2 G at each iteration, in a fashion siilar to [4]. 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 PGP [15] which offers sei-optial results under exponentially lower transitter and receiver coplexity [15]. For all cases presented here, we use PGP with a group size of 2 2. Based on [15] this ethod achieves a coplexity reduction at both the transitter precoding design and the receiver axiu a posteriori (MAP) detector on the order of 2 N t M 2(Nt 2). We divide this section into four subsections. In the first subsection, we exaine the accuracy of the proposed Gauss- Herite approxiation and provide a coparison with the lower bound technique presented in [20]. In the second subsection we present results for SCSI channels siilar to the ones in 3GPP SCM [16], with antenna size up to with PGP and odulation size M = 16. In the third subsection, we present results for CSIT jointly with PGP and high 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 Gauss-Herite approxiation technique In Fig. 1 we present results for I through siulation, the Gauss-Herite approxiation (GH) with L = 3, and the lower bound developed in [20] versus the signal-to-noise ratio per bit ( ) in db, for QAM with M = 16, and for the coonly used channel [4], [14], H 1 = [ In the sae figure, we also show a lower bound for I(x; y) which appeared in [20]. We see excellent accuracy for the approxiation, i.e., no observable difference between the Gauss-Herite approxiation and the siulations, over all values. ]. B. Results for SCSI in Conjunction with PGP In Fig. 2 we present results for PGP versus a no-precoding urban SCM channel with half-wavelength antenna eleent spacing [16] and with N t = N r = 100 and M = 16.

5 I bps/hz 5 3 Siulation & GH Lower bound [20] 10 8 No precoding 6 PGP Fig. 1. Results for I(x; y) without precoding for the H 1 channel and QAM M = 16 odulation. Fig. 3. I(x; y) results for PGP and no-precoding cases for a randoly generated 10 4 H CSIT MIMO syste and QAM M = 16 odulation. To the best of our knowledge, results for such large MIMO configurations are not available in the literature. Siilar to present results for the uplink, and downlink of a Massive MIMO syste based on 100 base station, 4 user antennas, respectively, with M = 16, 64, and for a Kronecker-based 3GPP SCM urban channel in a CSIT scenario in a single user configuration. Fig. 4 shows results for the uplink of the syste. We eploy PGP to draatically reduce the syste coplexity at the transitter and receiver sites. Under PGP No precoding SNR b Fig. 2. I(x; y) results for PGP and no-precoding cases for a H SCSI MIMO syste and QAM M = 16 odulation. the previous results, we observe high inforation rate gains in the high 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. C. Results for CSIT in Conjunction with PGP In Fig. 3 we present results for an asyetric randoly generated MIMO channel with N t = 4, N r = 10, and M = 16. PGP 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 regie, e.g., around 3 db in lower than 7 db. D. Results for Massive MIMO Massive MIMO [11], [21], [22] has attracted uch interest recently, due to its potential to offer high data rates. We PGP, M=64 No precoding, M=64 PGP, M=16 No precoding, M= Fig. 4. I(x; y) results for PGP and no-precoding cases for a randoly generated uplink H CSIT MIMO syste and QAM M = 16, 64 odulation. no precoding, the channel saturates and fails to eet the axiu possible utual inforation of 16 b/s/hz, while with PGP the syste clearly 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 = 64 uplink offers further gains by, e.g., achieving the axiu possible rate of 24 b/s/hz. For the downlink, in Fig. 5 we show results where the no-precoding case operates under 100 antenna inputs all correlated through the right eigenvectors of the channel, thus creating a very deanding environent at the user, due to the

6 No precoding, N =100, M=16, 64 t 15 PGP with two sybols per receiving antenna, M=16 PGP with two sybols per receiving antenna, M= Fig. 5. I(x; y) results for PGP and no-precoding cases for a randoly generated H downlink CSIT MIMO syste and QAM M = 16, 64 odulation. exponentially increasing MAP detector coplexity [15]. 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 10 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, e.g., = 4, the M = 64 PGP schee achieves 45% higher throughput that the M = 16 one, a significant iproveent. We would also like to ephasize that the no-precoding schee requires a very high exponential MAP detector coplexity, on the order of M 200, while for the low-snr-superior PGP, this coplexity is on the order of M 4 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. Note that the perforance provided by PGP on the downlink 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 uplink 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 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. We then applied a global optiization fraework to derive the 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 b/s/hz. However, we deonstrated that by eploying another 2 2 channel gains as high as 1.4 b/s/hz are possible. For systes of large MIMO configurations, we applied the coplexity-siplifying PGP concept [15] to derive sei-optial precoding results. Under SCSI, we showed that for urban 3GPP SCM channels, an interesting saturation effect in the no-precoding case takes place, while the sei-optial PGP precoder offers draatically better results in this case while 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 Gauss-Herite approxiation approach to CSIT with a large nuber of antenna with the sae success. We showed that for specific type of channels siilar to urban 3GPP SCM [16], the PGP approach offers very high gains over the noprecoding case in the high regie. Finally, we considered a Massive MIMO scenario in conjunction with CSIT and showed that by carefully designing the downlink and uplink precoders, the ethodology shows very high gains, especially on the downlink, although it eploys an exponentially 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 [4], [9] 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 = 100 as well as with M = 64 efficiently. 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 [4] 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) (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

7 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) 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. The Gauss-Herite quadrature is + exp( x2 )f(x)dx L l=1 c(l)f(v l), 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 [12] c(l) = 2L 1 L! 2π L 2 (H L 1 (v l )) 2 (20) where H L 1 (x) = ( 1) L 1 exp(x 2 ) dl 1 (exp( x 2 )) is the dx L 1 (L 1)th 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 k = c(k r1 )c(k i1 ) π k r1=1 k i1=1 k rnr =1 k inr =1 c(k inr )g k (σn kr1, σn ki1,, σn krnr, σn kinr ), (21) where g k (σn kr1, σn ki1,, σn krnr, σn kinr ) is the value of the function (fro (21)) log 2 ( exp( 1 σ n HG(x 2 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 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. (22) 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 (22) and approxiate the f k by ˆf k, to get for the differential of I(x; y) over M. 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