THE optimal detection of a coded signal in a complicated

Transcription

1 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 6, NO., NOVEMBER Achevable Rates of MIMO Systems Wth Lnear Precodng and Iteratve LMMSE Detecton Xaojun Yuan, Member, IEEE, L Png, Fellow, IEEE, Chongbn Xu, and Aleksandar Kavcc, Senor Member, IEEE Abstract We establsh area theorems for teratve detecton and decodng or smply, teratve detecton over coded lnear systems, ncludng multple-nput multple-output channels, ntersymbol nterference channels, and orthogonal frequency-dvson multplexng systems. We propose a lnear precodng technque that asymptotcally ensures the Gaussanness of the messages passed n teratve detecton, as the transmsson block length tends to nfnty. Area theorems are establshed to characterze the behavor of the teratve recever. We show that, for unconstraned sgnalng, the proposed sngle-code scheme wth lnear precodng and teratve lnear mnmum mean-square error LMMSE detecton s potentally nformaton lossless, under varous assumptons on the avalablty of the channel state nformaton at the transmtter. We further show that, for constraned sgnalng, our proposed sngle-code scheme consderably outperforms the conventonal multcode parallel transmsson scheme based on sngular value decomposton and water-fllng power allocaton. Numercal results are provded to verfy our analyss. Index Terms Lnear precodng, teratve LMMSE detecton, area theorem, superposton coded modulaton. I. INTRODUCTION A. Area Propertes THE optmal detecton of a coded sgnal n a complcated wreless envronment may ncur excessve computatonal complexty. Iteratve detecton provdes a low-cost soluton by decomposng the overall recever nto two or more local processors and conductng message-passng between the local processors to refne the detecton output see [] [] and the references theren. The analyss of an teratve detecton process s an ntrgung problem. The densty-evoluton technque [2] shows that carefully desgned low-densty partycheck LDPC codes can acheve near-capacty performance n addtve whte Gaussan nose AWGN channels wth teratve Manuscrpt receved March 8, 22; revsed August 6, 24; accepted August 6, 24. Date of publcaton August 9, 24; date of current verson October 6, 24. Ths work was supported by the Research Grant Councl of Hong Kong under Grant 4872 and Grant CtyU 83. Ths paper was presented at the 2 Internatonal Symposum on Informaton Theory. X. Yuan s wth the School of Informaton Scence and Technology, ShanghaTech Unversty, Shangha 23, Chna e-mal: yuanxj@ shanghatech.edu.cn. L. Png and C. Xu are wth the Department of Electrcal Engneerng, Cty Unversty of Hong Kong, Hong Kong e-mal: eelpng@ctyu.edu.hk; xchongbn2@ctyu.edu.hk. A. Kavcc s wth the Department of Electrcal Engneerng, Unversty of Hawa at Manoa, Honolulu, HI USA e-mal: kavcc@hawa.edu. Communcated by D. Guo, Assocate Edtor for Shannon Theory. Color versons of one or more of the fgures n ths paper are avalable onlne at Dgtal Object Identfer.9/TIT message-passng decodng algorthms. It was further shown n [2] that the achevable rate of an teratve scheme for an erasure channel can be measured by the area under the so-called extrnsc nformaton transfer EXIT curves [3] and the channel capacty s approachable when the two local processors have matched EXIT curves. Ths area property s extended n [4] to scalar AWGN channels or smply, AWGN channels usng the measure of mnmum mean-square error MMSE, n whch a suffcent condton s establshed to approach the capacty of a bnary-nput AWGN channel wth teratve detecton. It s commonly accepted that, wth random nterleavng, the extrnsc nformaton.e., the messages from an a posteror probablty APP decoder for a bnary forward-error-control code can be modeled as a sequence of observatons from an effectve scalar AWGN channel [3]. Thus, the authors n [4] made two basc assumptons: the messages passed between the local processors are modeled as the observatons from an effectve scalar AWGN channel referred to as the AWGN assumpton and the localprocessorsareoptmaln the sense of APP detecton/decodng. Assumpton ensures that each local processor can be characterzed by a snglevarable transfer functon nvolvng sgnal-to-nose rato SNR and MMSE. The mutual nformaton and MMSE relatonshp establshed n [5] s then appled to derve the area property, and capacty-approachng performance s proven when the transfer curves of the local processors are matched. B. MIMO Channels The work n [4] s lmted to scalar channels. A straghtforward extenson to multple-nput multple-output MIMO channels s to convert a MIMO channel nto a set of ndependent scalar sub-channels usng lnear processng based on sngular value decomposton SVD, to allocate power among these sub-channels usng the water-fllng WF prncple [44] or ts varaton mercury water-fllng [6], and then to apply varable-rate codng to these sub-channels wth one codebook assgned to one sub-channel. Hence the results n [4] can be appled to each sub-channel ndvdually. Ths mult-code SVD-WF approach wth varable-rate codng s conceptually smple, but may encounter the followng dffcultes n practce. Frst, the rate and power par obtaned by SVD-WF for each sub-channel vares and so a large number of encoders wth varous rates are requred. The related complexty can be IEEE. Personal use s permtted, but republcaton/redstrbuton requres IEEE permsson. See for more nformaton.

2 774 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 6, NO., NOVEMBER 24 reduced by rate quantzaton, but performance may deterorate notceably when quantzaton nterval ncreases. Second, ths SVD-WF approach requres accurate channel state nformaton at the transmtter CSIT. In practce, CSIT error can be caused by the Doppler effect or lmted backhaul capacty. In general, perfect orthogonalty between the sub-channels cannot be establshed n the presence of CSIT error. Then the optmal transcever desgn based on the SVD-WF prncple becomes a complcated ssue. The above dffcultes motvate us to study alternatve transmsson technques for MIMO systems wthout orthogonal channel decomposton. Lnear precodng LP [2] [26] can be used for ths purpose. Iteratve detecton can be appled to MIMO systems wth LP. However, the extenson of the area property n [4] to MIMO systems wthout orthogonal channel decomposton s not straghtforward. Ths s because a multvarate functon s requred n such a system to characterze the recever s behavor, but then assumpton mentoned earler may not hold. Also, optmal decodng for a MIMO system wthout orthogonal channel decomposton can be too costly, especally n the recent development of massve MIMO systems. Lnear MMSE LMMSE detecton [8] has the advantage of low complexty, but t s not optmal for non- Gaussan sgnalng, mplyng that assumpton may not hold ether. C. Contrbutons of Ths Paper In ths paper, we consder a jont LP and teratve LMMSE detecton scheme. We show that the proposed lnear precodng technque ensures that the AWGN assumpton asymptotcally holds for the output of the LMMSE detector, provded that the transmsson block length s suffcently large. Ths allows us to use a sngle par of nput and output parameters to characterze the behavor of the LMMSE detector. Also, we adopt superposton coded modulaton SCM [9], [2] to approxmate Gaussan sgnalng, for whch the local LMMSE detecton s near-optmal. These treatments ensure the valdty of the two assumptons n Secton I.A for MIMO setups. Based on ths, we establsh area theorems for the proposed LP and teratve LMMSE detecton LP-LMMSE scheme n MIMO channels, under the assumptons of perfect and partal CSIT, respectvely. We show that, for unconstraned sgnalng, the proposed LP-LMMSE scheme wth a sngle code s potentally nformaton lossless based on the curve-matchng prncple, even though the suboptmal teratve LMMSE detecton technque s employed. The above area theorems provde gudelnes n the practcal desgn of the LP-LMMSE scheme. Specfcally, we show that optmal power allocaton for non-gaussan nputs can be obtaned by solvng a convex optmzaton problem. Also, effcent practcal FEC codes, such as LDPC codes, can be desgned based on the curve-matchng prncple. Furthermore, we show that, for non-gaussan sgnalng, the proposed snglecode LP-LMMSE scheme acheves much hgher rates than the mult-code SVD-WF scheme when the encodng and decodng complextes of the two schemes are kept at a comparable level. From nformaton theory, the capacty of a MIMO channel can be acheved usng a sngle random code wth SVD and WF [58]. However, the realzaton of such a snglecode scheme wth affordable encodng and decodng complexty remans a challengng problem. As such, one of the major contrbutons of ths paper s to show a practcal sngle-code scheme wth LP and teratve LMMSE detecton that acheves the MIMO capacty. D. Comparsons Wth Exstng LP Technques It s nterestng to compare the proposed lnear precodng technque wth other alternatves see [2] [28] for full CSIT and [29] [34] for partal CSIT. From an nformaton-theoretc vewpont, the ultmate crteron for precoder desgn s to acheve the channel capacty see [23], [33], [34] and the references theren. It s well known that channel codng s requred to acheve the capacty. However, most exstng works on precoder desgn focus on un-coded MIMO systems equpped wth a sgnal detector at the recever performng far from optmal. Along ths lne, a varety of desgn crtera have been studed, such as par-wse error probablty mnmzaton [22], max-mn sgnal-to-nterference-plus-nose rato SINR [23], average SINR maxmzaton [23], SINR equalzaton [24], [25], and MMSE [26], [27], etc. The works for un-coded systems can be appled to coded systems by concatenatng the detector wth a decoder. Unfortunately, the resdual nterference at the output of the detector n general results n a consderable performance loss [36], [55], [56]. Iteratve detecton and decodng can effcently suppress the resdual nterference left by the detector. Then, a major challenge s to jontly desgn FEC codng and lnear precodng at the transmtter, takng nto account the effect of teratve detecton at the recever. Ths paper provdes a smple soluton to ths problem. Our analyss shows that, wth curve-matchng codes, the proposed LP-LMMSE scheme s capacty-achevng under varous assumptons on CSIT. E. Comparsons Wth the SVD-WF Approach As aforementoned, the conventonal SVD-WF approach s conceptually smple, provded that deal Gaussan sgnalng s used and perfect CSIT s avalable. For non-gaussan nputs, the mercury water-fllng MWF technque [6] can be used for power allocaton among dfferent egen-modes after SVD. However, t s shown n [4] and [42] that the optmal lnear precoder for non-gaussan nputs s n general non-dagonal even for ndependent parallel channels. To the best of our knowledge, t s computatonally ntensve to determne the optmal precoder n ths case. Furthermore, the SVD-WF approach requres perfect CSIT that s dffcult to acqure n practce, e.g., due to the Doppler effect or lmted backhaul capacty. CSIT uncertanty n general mpars the orthogonalty between the parallel sub-channels. Thus, effcent nterference cancellaton technques, such as teratve detecton and decodng, are requred at the recever [4], [5], [7], [].

3 YUAN et al.: ACHIEVABLE RATES OF MIMO SYSTEMS 775 The above dscussons mply that lnear precodng and teratve detecton are necessary n the practcal desgn of an effcent MIMO transcever. In ths regard, the LP-LMMSE scheme developed n ths paper provdes a low-cost soluton to the problem. In partcular, only one encoder s requred n the proposed LP-LMMSE scheme. Ths s much smpler than the SVD-WF scheme that requres multple encoders wth varous rates. Further, the proposed LP-LMMSE scheme can acheve hgher nformaton rates than SVD-WF and ts varaton SVD-MWF for practcal systems wth non-gaussan nputs, as demonstrated by the numercal results provded later n Secton VI. F. Outlne of the Remander of the Paper Secton II descrbes our system model and teratve LMMSE detecton under consderaton. Secton III descrbes lnear precodng for MIMO systems wth perfect CSIT, and characterzes the correspondng teratve recever usng SINR-varance evoluton. In Secton IV, we establsh the area theorems and consder the precoder desgn. Secton V extends our results to MIMO channels wth CSIT uncertanty. Secton VI provdes numercal results, and Secton VII offers some concludng remarks. II. SYSTEM MODEL AND ITERATIVE LMMSE DETECTION A. Generc Lnear System A generc complex-valued lnear system s modeled as y Ax + η where y s a J -by- receved sgnal vector, A s a J -by-j transfer matrx, x [x, x 2,,x J ] T s a J-by- transmt sgnal vector, and η CN,σ 2 I a J -by- addtve nose vector. Notatons and I, respectvely, represent an all-zero vector and an dentty matrx wth a proper sze. Note that both J and J are ntegers, wth the meanngs revealed later. Throughout ths paper, we assume full channel state nformaton at the recever,.e., the recever perfectly knows the transfer matrx A. We wll dscuss the stuatonsof both perfect and partal CSIT. B. Messages The transmtter structure for the system n s shown n the upper part of Fg.. The forward-error-correcton FEC encoder generates a frame of JK coded symbols, denoted by x [x, x 2,...,x JK ]T where K s the number of the uses of system n a codeword frame. These coded symbols are randomly nterleaved by the nterleaver and then parttoned nto K segments wth equal length J. Each segment serves as an nput x to the system. We emphasze that x and x are not of the same length, for that x represents the symbols n an overall codeword frame consstng of K segments, whle x only represents the symbols n any one of these segments. The teratve recever, as llustrated n the lower half of Fg., conssts of two local processors, namely, the detector Fg.. The transcever structure of the proposed scheme over the generc lnear system n. s the nterleaver and s the correspondng de-nterleaver. and the decoder, nter-connected by the nterleaver and the de-nterleaver. Partcularly, the detector consdered n ths paper follows the LMMSE prncple, hence the name LMMSE estmator. The LMMSE estmator estmates x based on the channel observaton y and the messages from the decoder denoted by α. The outputs of the estmator are extrnsc messages denoted by β. The message set α s defned as follows. Denote by x the th entry of x. Each x s constraned on a dscrete sgnalng constellaton S {s, s 2,...,s S },where S represents the sze of S. We always assume that each x s randomly and unformly taken over the constellaton ponts n S. Ths assumpton does not lose any generalty snce sgnal shapng e.g., to approach Gaussan sgnalng can be realzed by properly desgnng the constellaton ponts of S. For each use of system, let α be a vector of J messages as α [α,α 2,...,α J ] T, and each message α s a set of S lkelhood values for x,.e. α {α, α 2,...,α S }, where α k represents the lkelhood of x s k S pror to the processng of the detector, and S α k. k Smlar notaton apples to the message vector β [β, β 2,...,β J ] T. The decoder decodes x based on the nput message set β, and outputs the extrnsc message set α. Decodng s performed on the overall frame, and so both α and β contan JK messages. Let x be the th entry of x,andβ k be the lkelhood of x s k S pror to decodng. Then, we can express β [β,β 2,...,β JK ]T where β {β, β 2,...,β S } for,...,jk. Messages n α allow smlar expressons. After decodng, α are nterleaved and parttoned to form the nput of the LMMSE estmator, whch completes one round of teraton.

4 776 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 6, NO., NOVEMBER 24 The LMMSE estmator and the decoder are executed teratvely untl convergence. C. Basc Assumptons Here we dscuss some basc assumptons used throughout ths paper. For notatonal convenence, we ntroduce an auxlary random varable, denoted by a, to represent the nformaton carred by each α. The condtonal probablty of a gven x s defned as pa x s k α k. Smlarly, an auxlary random varable b s defned for each β. The condtonal probablty of b gven x s gven by pb x s k β k. Denote a [a,...,a J ] T and b [b,...,b JK ]T. We make the followng assumptons on the nputs of the local processors. Assumpton : For the detector, each x s ndependently drawn from S wth equal probablty,.e., px s k / S, forany and k; the messages n a are condtonally ndependent and dentcally dstrbuted..d. gven x,.e. J pa x pa x, and for any dummy varable t and k, 2,..., S, pa t x s k pa j t x j s k. Assumpton 2: For the decoder, the messages n b are condtonally..d. gven x,.e. pb x JK pb x, and for any dummy varable t and k, 2,..., S, pb t x s k pb j t x j s k. Wth Assumpton, the jont probablty space of x, a, andy seen by the detector can be expressed as J px, a, y px pa x py x, 2a where py x s determned by. Smlarly, wth Assumpton 2, the jont probablty space of x and b seen by the decoder s px, b px JK pb x 2b where x s assumed to be unformly taken over the codebook,.e., px s a unform dstrbuton over the codebook. Remark : Assumptons and 2 decouple the probablty space seen by the detector and decoder, whch smplfes the analyss of the teratve process. Smlar assumptons have The auxlary random varables are ntroduced for notatonal convenence. Usually, an teratve recever drectly evaluates β wthout explctly calculatng {b }. However, we wll see later that {b } are ndeed calculated n the teratve LMMSE detecton, based on whch β s determned. been wdely used n teratve decodng and turbo equalzaton algorthms [2], [3], [7], [37]. We emphasze that, n fact, Assumptons and 2 are requred only for those nvolved symbols n the computaton tree of the teratve algorthm at a depth of the maxmum number of teratons. Assumpton only nvolves J symbols, and thus can be made vald by random nterleavng as K tends to nfnty. However, strctly speakng, Assumpton 2 cannot be justfed n ths way, for that ths assumpton nvolves JK symbols and the correspondng messages are generally correlated due to the LMMSE detecton operaton, no matter what nterleaver s used. A remedy to ths ssue s to requre Assumpton 2 hold only for the neghborhood of the computaton tree, nstead of for all the JK symbols n the codeword. Ths remedy wll make the dervaton of the results n ths paper more rgorous, but wll complcate the notaton and the related dscussons. For ease of presentaton, we do not adopt ths remedy and keep Assumpton 2 as t s. D. LMMSE Estmaton The detector delvers the extrnsc messages defned as β k p x s k a, y, for, 2,...,J and k,..., S where the jont probablty space of x, a,andy s gven n 2a, and a represents the vector obtaned by deletng the th entry of a. Here, extrnsc means that the contrbuton of the aprormessage a s excluded n calculatng each β. For system, a drect evaluaton of β k may be excessvely complcated. The LMMSE estmaton s a low cost alternatve. It s suboptmal n general, but t s optmal f x are generated usng Gaussan sgnalng. Denote the mean and covarance of x seen by the detector by x [ x,..., x J ] T and vi, respectvely, wth S x E[x a ] α ks k 3a and k [ v E x x 2], for any ndex, 3b where the expectaton s taken over the jont dstrbuton of a and x. From Assumpton, v s nvarant wth respect to the ndex. In practce, v s approxmated by the sample varance as J S v J α k s k x 2, k whch converges to the true varance when J. The LMMSE estmator of x gven y s [38] ˆx x + va H R y A x where R s the covarance matrx of y gven as R vaa H + σ 2 I. 4a 4b Recall that the extrnsc message β should be ndependent of a. To meet ths requrement, we calculate the extrnsc mean and varance for x denoted by b and u, respectvely

5 YUAN et al.: ACHIEVABLE RATES OF MIMO SYSTEMS 777 by excludng the contrbuton of a accordng to the Gaussan message combnng rule cf., 54 and 55 n [39] as u M, v 5a and b ˆx u M, x v, 5b where M, represents the, th entry of the MMSE matrx M vi v 2 A H R A. 5c Fnally, the output messages of the detector can be calculated as pb x s k β k S k pb, for any and k. 5d x s k In the above, β k can be readly calculated by assumng that b s an observaton of x over an effectve AWGN channel wth nose power u. The justfcaton of ths assumpton can be found n Lemma n Secton III.C. We note that b n 5b s just the auxlary varable of β k defned n Secton II.C; see Footnote. E. APP Decodng The decoder decodes x based on the messages β followng the APP decodng prncple. The extrnsc output of the decoder for each x s defned as α k p x s k b 6 for, 2,..., JK and k,..., S, where the jont probablty space of x and b s gven n 2b. Assumpton 3: The local decoder performs APP decodng. In practce, APP decodng s usually computatonally expensve. Low-complexty message-passng algorthms can be used to acheve near-optmal performance, provded that the code structure allows a sparse graphc descrpton [37]. Messagepassng decodng s well-studed n the lterature, and thus the detals are omtted here. Compared wth APP decodng, the performance loss of message-passng decodng s usually margnal. Ths loss s not of concern n ths paper. Therefore, we ntroduce Assumpton 3 to smplfy our analyss. III. LINEAR PRECODING WITH PERFECT CSIT In ths secton, we assume perfect CSIT. The case of mperfect CSIT wll be dscussed n Secton V. We propose a lnear precodng technque to equalze the output SINRs of the detector. We then establsh an SINR-varance transfer chart technque to analyze the performance of the proposed LP-LMMSE scheme. A. MIMO Channels The teratve LMMSE detecton prncple descrbed n the prevous secton can be appled to any coded lnear systems, ncludng MIMO channels, nter-symbol nterference ISI channels, and orthogonal frequency-dvson multplexng OFDM systems, etc. In the followng, as an example, we establsh a connecton between a MIMO channel and the system n. A Gaussan MIMO channel wth N transmt and M receve antennas can be modeled as ỹ l H x l + η l 7 where l represents the l-th channel use, ỹ l s an M-by- receved sgnal vector, H s the M-by-N channel transfer matrx known at both the transmtter and the recever, x l s an N-by- transmt sgnal vector, and η l CN, σ 2 Isan M-by- addtve nose vector. A transmsson block.e., one use of system nvolves J/N uses of the channel 7, where J s the block length defned n Secton II.A. We assume that J and J are properly chosen so that J /M J/N s an nteger representng the number of channel uses n system. Denote ỹ [ỹ T,...,ỹT J/N ]T x [ x T,..., xt J/N ]T η [ η T,..., ηt J/N ]T. Combnng J/N channel uses, we wrte an extended system as ỹ H x + η where the extended channel s gven by H I J/N H 8a 8b wth beng the Kronecker product and I n beng an n-by-n dentty matrx. The sgnal power s constraned as E [ x H x ] /J P. 8c The SVD of H s gven by H UΛV H 9 where Λ s an JM/N-by-J dagonal matrx wth non-negatve dagonal elements, and U and V are untary matrces. We assume that U and V are chosen such that the dagonal entres of Λ are asymptotcally uncorrelated as J tends to nfnty. 2 Ths property s useful n establshng Lemma n Secton III.C. B. Lnear Precodng We focus on the followng lnear precodng operaton: x VW /2 Fx a where x and x are gven n 8a and, respectvely, V s defned n 9, W s a dagonal matrx wth non-negatve dagonal elements for power allocaton, and F s the normalzed DFT matrx wth the, kth entry gven by wth j. F, k J /2 exp j2π k /J b 2 Although H s determnstc, the orderng of ts sngular values can be arbtrarly chosen. Here we choose such a random orderng that the dagonal of Λ s an asymptotcally uncorrelated sequence as J tends to nfnty.

6 778 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 6, NO., NOVEMBER 24 From Assumpton, the entres of x are uncorrelated. Wthout loss of generalty, we further assume that the entres of x have normalzed power. Denote by W, the, -th dagonal entry of W. Then the power constrant n 8c becomes J J W, P. The optmzaton of W wll be detaled n Secton IV. The use of F n a s to ensure that the SINRs are equal for all symbols after LMMSE detecton; see Secton III.C. Incdentally, the choce of the DFT matrx for F also allows the fast Fourer transform FFT algorthm n the mplementaton of LMMSE detecton []. We note that smlar DFT-based precoders have been used n MIMO systems for other purposes, e.g., for harnessng dversty n [46]. At the recever sde, the receved vector ỹ s post-processed by the matrx U H : Combnng 8 2 and lettng y U H ỹ. 2 D ΛW /2, we obtan an equvalent channel as y DFx + U H η Ax + η 3a 3b where A DF and η U H η CN,σ 2 I snce U s untary. Clearly, 3b has the same form as. The teratve detecton and decodng procedure outlned n Secton II can be drectly appled to 3b. C. Characterzaton of the Estmator We now am to characterze the behavor of the estmator. It was shown n [23] that, wth the precoder n a, the SINR becomes unform for non-teratve LMMSE estmaton. We next show that a smlar property holds n our proposed teratve system. We further show that, for the extended system n 8a, the resdual nterference n the output of the estmator s asymptotcally Gaussan, followng the central lmt theorem. To start wth, we establsh the fact that M,, the th dagonal element of M n 5c, s not a functon of. Tosee ths, we substtute 4b and A DF nto 5c: Then M vi v 2 F H D H vdd H + σ 2 I DF. M, v v 2 f H DH vdd H + σ 2 I Df J v J v 2 Dk, k 2 vdk, k 2 + σ 2 J k J v + k 4a 4b Dk, k2 σ 2 4c where f s the th column of F, and 4b utlzes b. Clearly, M, s the same for all. We now ntroduce a defnton: φv M, v. Note that φv defned above s not a functon of the ndex,as M, s nvarant wth respect to. Recall that {Dk, k Λk, kwk, k /2, k, 2,...,J} andthat{λk, k} are the sngular values of H n 8b and so {Λk, k} contans J/N copes of the sngular values of H denoted by λ,...,λ N. We assume that a same amount of power s allocated to any two egen-modes wth the same channel gans,.e., f Λk, k Λ j, j, thenwk, k W j, j. 3 Then, {Dk, k} haveatmostn dfferent values, denoted by d,...,d N. Together wth 4c and the defnton of φv above, we obtan N φv N v + d2 n σ 2 v. 5 n Remark 2: We descrbe some useful propertes of φv. Frst, as the sgnal power.e., the power of each entry of x s normalzed to, the varance v n 5 vares from to. Second, φv s monotoncally decreasng n v, mplyng that for the LMMSE estmator the hgher the nput varance, the lower the output SINR. Thrd, as v,φv tends to the average SNR gven by φ N dn 2 N σ 2. n Remark 3: We emphasze that F s chosen to meet the followng crtera: a F s untary; b Fx allows fast computaton wth complexty OJlogJ; c the dagonal elements of FTF H s nvarant for any real dagonal matrx T and so are the dagonal elements of M. The choce of F to meet these crtera s not unque. Besdes the DFT matrx, we may alternatvely choose F to be, e.g., the Hadamard matrx. We next show that the outputs of the LMMSE estmator can be modeled as the observatons from an AWGN channel provded that J s suffcently large, and the related SINR s gven by ρ φv. Defne v n M, ρ f H DH v DD H +σ 2 I Ax \ x \ +η 6 where x \ or x \ represents the vector obtaned by settng the th entry of x or x to zero. It s clear that n s ndependent of x. Moreover, we have the followng result wth the proof gven n Appendx A. Lemma : For any ndex, b n 5d can be expressed as b x + n, 7 where n s ndependent of x and ts dstrbuton converges to CN, /φv as J. In the above, n represents the resdual nterference plus nose at the output of the LMMSE estmator n teraton. Hence, ρ φv n 5 represents the related SINR. It s well-known that the resdue error of the LMMSE estmaton s approxmately Gaussan [38], [52]. Lemma reveals that 3 Ths treatment doesn t ncur any nformaton loss, as seen from Theorem.

7 YUAN et al.: ACHIEVABLE RATES OF MIMO SYSTEMS 779 ths approxmaton becomes exact for the precoder n a over the extended channel n 8a for a suffcently large J. We henceforth always assume that J s suffcently large, so that each b can be modeled as an observaton of x from an effectve AWGN channel wth SNR φv. D. Characterzaton of the Decoder We now consder the characterzaton of the decoder s behavor. The decoder performs APP decodng upon recevng the messages b modeled as ndependent observatons of x over an AWGN channel wth SNR ρ cf., Lemma. Defne the extrnsc varance of each x as v MMSEx b E [ x E[x b ] 2], 8a where the expectaton s taken over the jont probablty space of x and b. We denote by v the average of v over the ndex. Clearly, the average varance v s a functon of ρ, denoted as v ψρ. 8b Note that 8b s referred to as the SINR-varance transfer functon of the decoder. Remark 4: For practcal FEC codes, the output varances {v } generally vary wth the ndex. Then, the queston s whether the average varance v s a good performance measure of the decoder output, or equvalently, whether the behavor of the estmator solely depends on v. To answer ths queston, we recall the estmator s sgnal model n 3b: y DFx + η. Letv be the aprorvarance of x. Note that {v } are obtaned by nterleavng {v }. From Assumpton, the messages from the decoder are uncorrelated. Thus, the apror covarance matrx of x s gven by dag {v, v 2,...,v J }. Then, the a pror covarance of the vector Fx s gven by Fdag{v, v 2,...,v J } F H. Clearly, Fdag{v, v 2,...,v J } F H s a crculant matrx, and ts dagonal s a constant gven by the average varance v. Further, due to random nterleavng, {v, v 2,...,v J } s a random sequence. Then, t can be shown that the off-dagonals of Fdag{v, v 2,...,v J } F H tend to zero as J tends to nfnty, or equvalently, the aprorcovarance of Fx s approxmately a scaled dentty matrx gven by vi. Ths mples that, provded that J s suffcently large, the performance of the LMMSE detector only depends on the average output varance v of the decoder. Ths justfes the use of v to characterze the output of the decoder. We next establsh a relaton between the rate of the FEC code per symbol of x and the transfer functon ψρ.our result s based on the SCM [9], [2]. The detal can be found n the lemma below, wth the proof gven n Appendx B. Lemma 2: Assume ψρ satsfes the followng regularty condtons: ψ andψρ, for ρ [, ; 9a monotoncally decreasng n ρ [, ; 9b contnuous and dfferentable everywhere n [, except for a countable set of values of ρ; v lm ρψρ. 9d ρ 9c Let Ɣ n be an n-layer SCM code wth the SINR-varance transfer functon ψ n ρ and the rate R n. Then, there exst {Ɣ n } such that: ψ n ρ ψρ, foranyρ andany nteger n; as n, R n R ρ + ψρ dρ. 2 We now gve an ntutve explanaton of Lemma 2. Let the MMSE of each x after decodng be mmseρ MMSEx b E [ x E[x b ] 2], 2a where the expectaton s taken over the jont probablty space of x and b. Compared wth 8, the MMSE n 2a s obtaned by ncludng the contrbuton of b n estmatng x. We next establsh a connecton between ψρ and mmseρ based on the followng two hypotheses: x s Gaussan, and the extrnsc message of x s also Gaussan. Wth these hypotheses, we conclude that the condtonal dstrbuton of x gven b s also Gaussan and gven by px b CNE[x b ], ψρ. 2b Then, mmseρ and ψρ are related as mmseρ MMSEx b MMSEx b, b MMSEx b, x px b MMSEx b, x CNE[x b ], ψρ ρ + ψρ. where the thrd equalty follows from the fact that b x x b forms a Markov chan from Assumpton 2 n Secton II.C, the fourth equalty from 2b, and the last one utlzes the fact that b can be modeled as an AWGN observaton n Lemma. From [4] see [4, Lemma ], we express the code rate as R mmseρdρ ρ + ψρ dρ 22 whch s exactly the rate lmt gven n Lemma 2. Recall that the above dervaton s based on two Gaussan hypotheses. These hypotheses are dffcult to meet exactly, as practcal systems mostly employ dscrete sgnalng. However, the concdence of 2 and 22 mples that SCM can approxmate Gaussan sgnalng as the number of SCM layers tends to nfnty, and therefore, the rate n 22 s ndeed approachable by properly constructng an SCM-based FEC code. E. SINR-Varance Transfer Chart From the prevous dscussons, the LMMSE estmator can be characterzed by ρ φv; and smlarly, the decoder can be characterzed by v ψρ. The teratve process of the estmator and decoder can be tracked by the recurson of ρ and v. Letq be the teraton number. We have ρ q φv q and v q ψρ q, q, 2,...

8 78 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 6, NO., NOVEMBER 24 Fg. 2. An llustraton of the SINR-varance transfer chart. The recurson contnues and converges to a pont v satsfyng φv ψ v and φv >ψ v, for v v, ] where ψ s the nverse of ψ whch exsts snce ψ s contnuous and monotonc [43]. Note that v snce the sgnal power s normalzed; v mples that x can be perfectly recovered. The above recursve process s llustrated by the SINR-varance transfer chart n Fg. 2. We say that the estmator and the decoder are matched f φv ψ v, for v, ]. 23 Note that: φ > snce the estmator s output always contans the nformaton from the channel even f there s no nformaton from the decoder; and φ < snce the estmator s extrnsc output cannot resolve the uncertanty ntroduced by the channel nose even f the messages from the decoder are perfectly relable. Then, we equvalently express the curve-matchng condton 23 as ψρ φ φ, for ρ<φ; ψρ φ ρ, for φ ρ<φ; ψρ, for φ ρ<. 24a 24b 24c The above curve-matchng prncple plays an mportant role n establshng the area theorems, as seen n the next secton. IV. AREA THEOREMS AND PRECODER OPTIMIZATION In ths secton, we establsh area theorems for the proposed LP-LMMSE scheme. We further dscuss the optmzaton of the power matrx W. A. Area Property for Unconstraned Sgnalng The man theorem of ths paper s presented below, wth the proof gven n Appendx C. Theorem : As J, f the detector and the decoder are matched, then an achevable rate per antenna per channel use of the LP-LMMSE scheme s gven by R N N n log + d2 n σ Remark 5: The curve-matchng condton 23 s assumed n achevng the rate n 25; see the proof n Appendx C. Then, a fundamental queston s whether ths curve-matchng condton can be fulflled for any channel realzaton. The answer s affrmatve. To see ths, we recall from Lemma 2 that an SCM code can be constructed to match an arbtrary functon ψρ satsfyng the regularty condtons n 9 and to acheve the rate n 2. Then, wth 23, the queston can be rephrased as: for any channel realzaton, does the matched ψρ n 24 satsfy the regularty condtons n 9? Wth φv gven by 5 and the propertes of φv n Remark 2, t s straghtforward to verfy that the regularty condtons n 9 are ndeed satsfed for any channel realzaton. Note that a typcal realzaton of φ ρ s llustrated n Fg. 2. Remark 6: We remark that 25 s the nput-output mutual nformaton of the channel n 3b wth Gaussan nputs. Ths observaton mples that our proposed scheme s potentally nformaton lossless. In addton, the rate R n 25 s acheved usng the curve-matchng SCM code wth the number of SCM layers tends to nfnty. Later, we wll show by numercal results that, for the proposed LP-LMMSE scheme, practcal values of R can be closely approached usng a small number of SCM layers. B. Water-Fllng Precodng Now we consder the optmzaton of the power matrx W. Recall from 3a that D ΛW /2, or equvalently, d λ w /2,for,...,N. Then, 25 becomes R N log + λ2 n w n N σ n We am at maxmzng the above rate over W subject to the power constrant n. Clearly, the soluton concdes wth the renowned water-fllng power allocaton [44]. Corollary : Under the condtons of Theorem, f W follows water-fllng power allocaton, then the LP-LMMSE scheme acheves the water-fllng capacty of the channel n 8a. Corollary shows that lnear precodng, together wth proper error-control codng and teratve LMMSE detecton, can potentally acheve the water-fllng channel capacty. Ths achevablty s based on a sngle code wth a matched decodng transfer functon. C. Area Property for Constraned Sgnalng So far, we have shown that the proposed lnear precodng scheme s asymptotcally optmal n the sense of achevng the water-fllng capacty as the sgnalng approaches Gaussan. However, n practce, the sgnalng s usually constraned on a fnte-sze dscrete constellaton. We next establsh an area theorem for constraned sgnalng.

9 YUAN et al.: ACHIEVABLE RATES OF MIMO SYSTEMS 78 Defne the γ -functon as [ γρ E x E[x x + η] 2] 27 where x s unformly taken over S, andη s ndependently drawn from CN, /ρ. We have the followng result wth the proof gven n Appendx D. Theorem 2: Suppose that the detector s nputs {a } are modeled as ndependent observatons of {x } from an effectve AWGN channel. Then, as J, f the detector and the decoder are matched, an achevable rate of the LP-LMMSE scheme s R log S + γρ + φγ ρdρ 28 where the φ-functon s gven by 5, and γ s defned n 27. Remark 7: Theorem 2 holds under the AWGN assumpton on {a }. It was observed that ths assumpton s emprcally true for BPSK, QPSK, and SCM wth BPSK/QPSK layers. However, ths assumpton may be far from true f other modulaton technques, such as bt-nterleaved coded modulaton [47], are employed. In ths later case, the area theorem based on the measure of mean-square error MSE establshed n [48] can be used for performance evaluaton. We refer nterested readers to [48] for detals. Remark 8: Theorem 2 can be readly extended to the case of dscrete non-unform nput of system. The only dfference s that, n the defnton of the γ -functon n 27, the dstrbuton of x s replaced by a non-unform dstrbuton over S. Correspondngly, log S n 28 s replaced by the entropy of x calculated based on the non-unform dstrbuton. Remark 9: Theorem and Corollary reveal that our proposed lnear precodng and teratve LMMSE detecton scheme s capacty-achevng. It s noteworthy that there are other approaches to acheve the channel capacty. One smple approach s to use SVD to convert the MIMO channel nto a set of ndependent parallel channels, and then to conduct water-fllng power allocaton over the parallel channels. It s desrable to compare the performance and complexty of our proposed scheme wth the SVD-MWF scheme n practcal systems wth non-gaussan nputs. Theorem 2 can be used n ths comparson. Later, we wll numercally demonstrate that, n ths case, our proposed scheme consderably outperforms the SVD-MWF scheme. D. Precoder Optmzaton for Constraned Sgnalng Recall that, for Gaussan sgnalng, the optmal power matrx W s the water-fllng soluton. However, water-fllng s not necessarly optmal for dscrete sgnalng. Based on the area property n Theorem 2, we can optmze W to maxmze the achevable rate of the LP-LMMSE scheme. Ths problem can be formulated as: maxmze log S + γρ+ φγ ρdρ 29a subject to N N w n P. 29b n In the above, P s the maxmum sgnal power gven n 8c. The followng result s useful n solvng 29. Lemma 3: The rate R n 28 s a concave functon of {w,...,w N }, provded that γρ s a convex functon of ρ. The proof of Lemma 3 can be found n Appendx E. The γ -functon s convex for most commonly used sgnalng constellatons [43]. From Lemma 3, the problem n 29 can be solved usng standard convex programmng [49]. V. EXTENSIONS TO MIMO CHANNELS WITH CSIT UNCERTAINTY In ths secton, we extend the results n Sectons III and IV to the general case that channel state nformaton s not perfectly known at transmtter. A. Lnear Precodng Wth Imperfect CSIT We now consder MIMO systems wth partal CSIT. Here, partal CSIT means that the channel matrces are not exactly known, nstead, only the statstcs of the channel s known at the transmtter sde. The precodng technque n Secton III.B requres perfect CSIT, and so cannot be drectly appled here. The followng s a modfed soluton. Return to the extended system n 8a ỹ H x + η, 3a where the extended channel contans J/N channel realzatons: H H 2 H.... 3b H J/N We focus on the ergodc case, n whch {H } are ndependent realzatons of an M-by-N random matrx, wth abuse of notaton, denoted by H. Ths model also ncludes orthogonal frequency-dvson multplexng OFDM systems [4] wth ndependent fadng over dfferent sub-carrers. The channel nput x s related to x as x P Π Fx 3a wth P dag{p, P,...,P} F dag{f, F,...,F}. 3b 3c where P s a J-by-J block-dagonal matrx wth P of sze N-by-N, Π s a J-by-J permutaton matrx, and F s a J-by-J block- dagonal matrx wth each block F beng the normalzed DFT matrx of sze L-by-L. Combnng 3a and 3 and lettng y ỹ and η η, we obtan an equvalent channel as wth y Ax + η A H P Π F. The power constrant now becomes 32a 32b N tr{pp H } P. 33

10 782 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 6, NO., NOVEMBER 24 Wth abuse of notaton, defne [ { φv N E tr v I + σ 2 HQHH }] v. 34 where Q PP H, and the expectaton s taken over the dstrbuton of H. We am to show that ρ /u φv, for,...,j. 35 Wth 32b, we can rewrte 5c as Fg. 3. An llustraton of the proposed precoder for mperfect CSIT wth J 6 and N 4. The precodng matrx P dag{p,...,p} allows the precoder to explot the beneft provded by the avalable CSIT. 4 The optmzaton of P s brefly dscussed n Secton V.C. The precodng matrx F ensures that every coded symbol n x s suffcently dspersve over tme and space. Thus, t s requred that the DFT sze L s suffcently large. A convenent choce of L s L J/N, and then there are N length-l DFT matrces n F. Now we consder the desgn of the permutaton matrx Π. Let q [q T,...,qT N ]T Fx and x [x T,...,xT N ]T,where each DFT block q Fx s of sze L-by-. Also denote c [c T,...,cT J/N ]T Πq, where each c s an N-by- vector. Thetwocrterafor Π are lsted below. For each, the entres of q are transmtted at dfferent channel uses,.e., no two entres of q are connected to asamec j for any ndex j,...,j/n; and Treat PH k as the kth realzaton of the equvalent channel PH. Then, any N consecutve entres of q are transmtted at dfferent transmt-antennas of the equvalent channel PH. As a result, for each ndex j, the set of {the jth entry of c k k,...,j/n} contans J/N 2 entres of q,,...,n. 5 The choce of Π satsfyng the above crtera s not unque. A smple choce s as follows: for each ndex k, thekth entry of q s connected to the k mod Nth nput of the equvalent channel PH k ; then, for j 2,...,J/N, the connecton pattern of each q j s just a one-entry cyclc shft of the prevous one. An example for J 6 and N 4 s llustrated n Fg. 3. The above choce of F and Π ensures that the behavor of the teratve recever can be characterzed by a sngle-varable recurson, as wll be detaled n the next subsecton. B. SINR-Varance Transfer Chart Iteratve LMMSE detecton descrbed n Secton II s appled to the system n 32. We next show that the outputs of the detector can stll be characterzed by a sngle SINR value. 4 We emphasze that P s desgned to be adaptve to the avalable channel state nformaton. Here, P remans constant for dfferent channel uses, as the channel statstcs does not change. However, f the channel statstcs vares, P should vary accordngly. 5 Here we assume that J s properly chosen such that J /N 2 s an nteger. M F H Π H vi v 2 P H H H v H P P H H H + σ 2 I H P Π F F H B F where B Π H vi v 2 P H H H v H P P H H H + σ 2 I H P Π. 36 We can express M and B n a block-wse form wth each block of sze J/N-by-J/N. Let M and B be the th dagonal block of M and B, respectvely. Then, we obtan M dag FH B F dag... F H B N F dag, where dag returns a dagonal matrx specfed by the dagonal of the matrx n the parenthess. Recall that both H and P are block dagonal, and so s P H H H v H P P H H H +σ 2 I H P. Thus, from the crteron of Π n Secton V.A, every B s dagonal. Therefore, as J M k, k N J tr{b } 37a J tr{b} 37b {vi J tr v 2 P H H H v H P P H H H } + σ 2 I H P 37c { J tr v I + } σ H P P H H H 2 37d { J/N tr J v I + } σ 2 H PP H H H 37e { [tr N E v I + }] σ 2 HQHH 37f where 37a follows from the fact that, for each F H N N B F dag B n, n I N J J tr{ B }I, n 37b from the crteron of Π, 37c from 36, and 37e by substtutng H dag{h,...,h J/N }, P dag{p,...,p}, and Q PP H. Substtutng 37 nto 5a, we arrve at 35. Now, smlar to Lemma, we have the followng result. Lemma 4: The detector s output b can be expressed as b x + n,wheren s ndependent of x and converges to CN, /φv as J.

11 YUAN et al.: ACHIEVABLE RATES OF MIMO SYSTEMS 783 The above lemma s lterally the same as Lemma, except that φv here s gven by 34. The proof mostly follows that of Lemma. We omt the detals for brevty. Wth Lemma 4, we can stll characterze the behavor of the teratve recever usng the SINR-varance transfer functons, namely, ρ φv and v ψρ, as descrbed n Secton III.E. C. Area Theorem and Precoder Optmzaton Now we are ready to present the followng result. Theorem 3: As J, f the detector and the decoder are matched, an achevable nformaton rate per transmt antenna per channel use of the LP-LMMSE scheme s gven by R [log N E det I + ] σ 2 HQHH. 38 The proof of Theorem 3 s gven n Appendx F. Wth Theorem 3, we can formulate the followng rate-maxmzaton problem: [ maxmze E log det I + ] σ 2 HQHH subject to N tr{q} P. 39a 39b The above optmzaton problem s convex, and thus can be solved numercally usng standard convex programmng tools. Moreover, the explct solutons to 39 n a varety of CSIT settngs have been studed n the lterature. We refer nterested readers to [34], [5], and [5] for more detals. For constraned sgnalng, t s straghtforward to show that Theorem 2 holds lterally for the partal CSIT case, except that φv s replaced by 34. Then, we can formulate an optmzaton problem smlar to 29, whch s solvable usng standard convex programmng. Detals are omtted for smplcty. VI. NUMERICAL RESULTS In ths secton, we provde numercal examples to demonstrate the achevable rates of the proposed scheme. Note that the channel SNR s defned as SNR PN/σ 2. We frst consder the case of full CSIT n a randomly generated 2 2 MIMO 3-tap ISI channel wth the tap coeffcents [ ] j j.648, 4a.3347 j j.242 [ ].582 j j.7565, 4b.4968 j j.595 and [ ].5262 j j c.672 j j.2695 The DFT s appled to convert the above MIMO ISI channel to a set of J by-2 parallel MIMO channels. Note that the effect of cyclc prefx s gnored here. Fg. 4 shows the achevable rates of the proposed LP-LMMSE scheme wth varous power allocaton strateges. Varous power allocaton polces are consdered, ncludng equal power EP allocaton, water-fllng WF, and optmzed power OP allocaton. For EP, W εi where Fg. 4. The achevable rates of the proposed LP-LMMSE scheme and the SVD-based parallel transmsson scheme wth QPSK modulaton n the MIMO-ISI channel n 4. The water-fllng capacty and Gaussan EP capacty are also ncluded for comparson. ε s a scalng factor to meet the power budget; for WF, W s determned by the water-fllng soluton; and for OP, W s obtaned by solvng 29. QPSK modulaton s employed. From Fg. 4, we see that the LP-LMMSE schemes wth OP and WF power allocaton have smlar performance, and both consderably outperform the flat precoder n the low SNR regon. Ths mples that the water-fllng precoder provdes an attractve low-complexty near-optmum opton for dscrete sgnalng. Fg. 4 also ncludes the Gaussan water-fllng capacty and Gaussan EP.e.,..d. capacty as references. We see that the LP-LMMSE schemes wth OP and WP approach the water-fllng capacty n the low SNR regon. Further, we compare the performance of the LP-LMMSE scheme wth that of the parallel transmsson scheme n whch the MIMO channels are converted nto parallel ndependent scalar channels usng SVD wth varous power allocaton strateges. We see that the LP-LMMSE scheme outperforms the parallel transmsson scheme wth MWF consderably n the medum SNR range. Ths ndcates the advantage of lnear precodng over parallel ndependent transmsson for MIMO channels. In Fg. 5, we employ both QPSK modulaton and standard 6-QAM see [57, SCM-] for sgnalng. Fg. 5 demonstrates that the achevable rate of the LP-LMMSE scheme s sgnfcantly ncreased by changng the sgnalng constellaton from QPSK to 6-QAM. We see that the gap between the achevable rate of LP-LMMSE wth 6-QAM and the waterfllng capacty s not sgnfcant for a rate up to 4 bts. For a hgher transmsson rate, a larger sgnalng constellaton s necessary. We now provde code desgn examples to verfy that the theoretcal lmts gven by our analyss are ndeed approachable. We bascally follow the desgn approach descrbed n [45]. Here, we only provde the desgn results. We consder standard 6-QAM sgnalng wth 2-layer SCM. Each SCM

12 784 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 6, NO., NOVEMBER 24 Fg. 5. The achevable rates of the proposed LP-LMMSE scheme n the MIMO-ISI channel n 4. Both QPSK and standard 6-QAM are consdered. The water-fllng capacty s also ncluded for reference. Fg. 7. The BER performance of the LP-LMMSE scheme wth 2-layer SCM over the MIMO ISI channel n 4. Water-fllng precodng and standard 6-QAM constellaton are used. Some smulaton settngs are M N 2, J 256, and K 256. Fg. 6. The SINR-varance transfer functons of the detector and the decoder for the LP-LMMSE scheme wth 2-layer SCM over the MIMO ISI channel n 4. Water-fllng precodng and standard 6-QAM constellaton are used. The transfer functon of the detector s gven by 5 at the channel SNR 6.7 db. layer s QPSK modulated wth Gray mappng. Note: for standard 6-QAM sgnalng, the power rato of the two SCM layers s 4:. Two rregular LDPC codes are desgned based on the curve-matchng prncple. For the SCM layer wth hgher power, the edge degree dstrbutons of the LDPC code are gven by {λ x.3495x +.242x x x x 7 +.x 2,ρ x x 6 }; for the other layer, the edge degree dstrbutons of the LDPC code are {λ 2 x.397x +.854x x +.877x +.394x 33,ρ 2 x x 6 }. The system throughput s 4 bts per channel use. The transfer curves of the detector and the decoder are llustrated n Fg. 6, and the smulated system performance s gven n Fg. 7. From Fg. 7, the desgn threshold s 6.7 db,. db away from the water-fllng capacty,.5 db away from the performance lmt of the LP-LMMSE scheme wth standard 6-QAM. The smulated system performance wth code length 5 at BER 4 s about 7. db, only.3 db away from the desgn threshold. Fg. 8. The achevable rates of the LP-LMMSE scheme n the flat-fadng 2-by-2 MIMO channel wth partal CSIT of θ.5. The water-fllng capacty s also ncluded for reference. It s nterestng to pay specal attenton to pont A n Fg. 6. Ths s the transton pont where, for the two layers of the LDPC codes, the one wth hgher power s nearly fully decoded.e., wth very relable outputs and the one wth lower power starts to be decoded. Now consder the case of partal CSIT. We assume the mean- feedback model [53],.e., each H s a random realzaton of H E[H]+ H where the entres of E[H] are ndependently drawn from CN,θwth θ [, ], and those of H from CN, θ. In smulaton, θ s set to.5. The channel mean E[H] remans constant for each frame, but vares ndependently from frame to frame. Each frame contans 5 ndependently generated H. The achevable rates of the LP-LMMSE scheme averaged over 2 frames are llustrated n Fg. 8. We see that smlar trends as n Fg. 5 have been observed.