Joint Source-Channel Coding for the MIMO Broadcast Channel

Transcription

1 Jont Source-Channel Codng for the MIMO Broadcast Channel Danel Persson, Johannes Kron, Mkael Skoglund and Erk G. Larsson Lnköpng Unversty Post Prnt N.B.: When ctng ths work, cte the orgnal artcle IEEE. Personal use of ths materal s permtted. However, permsson to reprnt/republsh ths materal for advertsng or promotonal purposes or for creatng new collectve works for resale or redstrbuton to servers or lsts, or to reuse any copyrghted component of ths work n other works must be obtaned from the IEEE. Danel Persson, Johannes Kron, Mkael Skoglund and Erk G. Larsson, Jont Source-Channel Codng for the MIMO Broadcast Channel, 2012, accepted IEEE Transactons on Sgnal Processng. Postprnt avalable at: Lnköpng Unversty Electronc Press

2 Manuscrpt for IEEE Transactons on Sgnal Processng, (Correspondence Item) Jont Source-Channel Codng for the MIMO Broadcast Channel Danel Persson *, Johannes Kron, Mkael Skoglund, and Erk G. Larsson Malng address: Danel Persson, Erk G. Larsson, Department of Electrcal Engneerng, Lnköpng Unversty, SE Lnköpng, Sweden Emal: Phone: , Fax: Malng address: Johannes Kron, Mkael Skoglund, School of Electrcal Engneerng, Royal Insttute of Technology, Osquldas väg 10, SE Stockholm, Sweden Emal: Phone: , Fax: * Correspondng author SOFTWARE: Ubuntu Lnux, TeX, BbTeX, Kle COPYRIGHT: Copyrght (c) 2011 IEEE. Personal use of ths materal s permtted. However, permsson to use ths materal for any other purposes must be obtaned from the IEEE by sendng a request to pubs-permssons@eee.org. Abstract: We nvestgate the problem of broadcastng analog sources to several users usng short codes, employng several antennas at both the transmtter and the recever, and channel-optmzed quantzaton. Our man objectve s to mnmze the sum mean square error dstorton. A jont mult-user encoder, as well as a structured encoder wth separate encoders for the dfferent users, are proposed. The frst encoder outperforms the latter, whch n turn offers large mprovements compared to state-of-the-art, over a wde range of channel sgnal-to-nose ratos. Our proposed methods handle bandwdth expanson,.e., usage of more channel than source dmensons, automatcally. We also derve a lower bound on the dstorton.

3 Jont Source-Channel Codng for the MIMO Broadcast Channel Danel Persson, Johannes Kron, Mkael Skoglund, and Erk G. Larsson Abstract We nvestgate the problem of broadcastng analog sources to several users usng short codes, employng several antennas at both the transmtter and the recever, and channel-optmzed quantzaton. Our man objectve s to mnmze the sum mean square error dstorton. A jont mult-user encoder, as well as a structured encoder wth separate encoders for the dfferent users, are proposed. The frst encoder outperforms the latter, whch n turn offers large mprovements compared to state-of-the-art, over a wde range of channel sgnal-to-nose ratos. Our proposed methods handle bandwdth expanson,.e., usage of more channel than source dmensons, automatcally. We also derve a lower bound on the dstorton. Index Terms Short codes, multple-nput multple-output (MIMO), mult-user, broadcast, downlnk, jont sourcechannel codng (JSCC), low latency. I. INTRODUCTION We study the multple-nput multple-output (MIMO) broadcast problem wth short codes, wth the end-to-end sum mean square error (MSE) dstorton as performance measure. Ths nvestgaton can be motvated by low-latency applcatons, where long codes are not sutable. Our study s both of fundamental nterest, and relevant for applcatons. D. Persson and E. G. Larsson are wth the Department of Electrcal Engneerng, Lnköpng Unversty, SE Lnköpng, Sweden (e-mal: danelp@sy.lu.se; egl@sy.lu.se). J. Kron and M. Skoglund are wth the School of Electrcal Engneerng, Royal Insttute of Technology, Osquldas väg 10, SE Stockholm, Sweden (e-mal: johk@ee.kth.se; skoglund@ee.kth.se). Ths work has been supported n part by the Swedsh Research Councl (VR) and the Swedsh Foundaton for Strategc Research (SSF). E.G. Larsson s a Royal Swedsh Academy of Scences (KVA) Research Fellow supported by a grant from the Knut and Alce Wallenberg Foundaton.

4 2 A. Background MIMO technology mproves both capacty and robustness n tradtonal communcatons [1]. Recently, Wengarten et al. [2] found the capacty regon for the MIMO broadcast channel, and t was establshed that drty-paper codng (DPC) s optmal n the sense that technques based on DPC can acheve any rate-tuple n ths regon. However, DPC reles on nfntely long codewords, and s therefore not sutable for a low-delay scenaro. In a scenaro wth short codes, the source-channel separaton theorem n general does not apply, and the optmal soluton conssts of jont optmzaton of the source and the channel code. Tradtonal dgtal communcaton systems consst of several separate unts: quantzer, channel encoder, precoder, and modulator. Jont source-channel codng (JSCC) strateges, where these unts are co-desgned, should be consdered for the case of short codes. Many treatments of JSCC exst, e.g., characterzaton of the dstorton regons for the problems of sendng a bvarate Gaussan source over bandwdth-matched Gaussan broadcast channels [3] and multple-access channels [4], nformaton-theoretc analyss of separaton of source and channel codng over several multuser channels [5], transmsson over non-ergodc channels based on nstantaneous capacty-achevng codes [6], dstorton-exponents-based analyss for hgh channel sgnal-to-nose rato (CSNR) [7], lnear source-channel mappngs for the broadcast channel [8], and other parametrc mappngs for Gaussan multple-access relay channels [9]. Channel-optmzed quantzaton s a way to perform JSCC, that does not depend on the assumptons of nstantaneous capacty-achevng codes, hgh CSNR approxmatons, and lnear and other mappng parametrzatons. Ths s a tandem approach where channel code usage and modulaton format are frst decded, and where the quantzer, and the mappng of quantzer ndexes to channel codewords, are optmzed n a second step dependng on the channel transton probabltes determned after the frst step. The algorthm n [10] determnes channel-optmzed vector quantzer reconstructon vectors and encodng regons, for sngle-user communcatons. Ths algorthm can be seen as an extenson to nosy channels of the generalzed Lloyd algorthm (GLA) [11]. A beneft of the algorthm n [10] s that t chooses the tradeoff between quantzaton dstorton and channel-nduced dstorton automatcally, and t can thus be seen as an adaptve quantzaton and channel codng scheme. Numerous extensons of the scheme n [10] have been found. Some recent developments are, e.g., the study of mappngs of a scalar source to several channel dmensons,.e., bandwdth expanson [12], channel-optmzed source mappngs for the relay channel [13], and channel-optmzed quantzaton for

5 3 sources wth memory [14], [15]. Some of these schemes were ntended for tradtonal dgtal communcatons [12], [14], [15], whle [13] focuses on analog communcaton. Ths s however only a matter of presentaton of the works, and all schemes based on [10] are vald for both dgtal and analog domanbased communcatons. More modulaton ponts result n a lower source reconstructon dstorton, but also a hgher computatonal complexty. B. Our scope and contrbuton In contrast to the channel-optmzed quantzaton approaches stemmng from [10] and descrbed n Sec. I-A, we study channel-optmzed quantzaton for nterferng mult-user channels n ths paper. More precsely, we nvestgate the MIMO broadcast scenaro, where dfferent streams are smultaneously sent to dfferent users, and where all streams nterfere at each user. The employed codes are short, and we utlze several antennas at both the transmtter and the recever sdes. Our man objectve s to mnmze the sum end-to-end dstorton. Our JSCC schemes employ the vector perturbaton precoder (VPP) scheme [16] as a buldng block. The basc dea of VPP s to reduce power consumpton through expandng the orgnal sgnalng constellaton. VPP has been documented to reduce the error probablty sgnfcantly compared to lnear methods and scalar Tomlnson-Harashma precodng [17], see [16]. However, our proposed JSCC schemes can deal wth MIMO nterference by proper modelng of the transton probabltes between codewords, for any precoder, modulaton, and detector scheme. Our proposed methods handle bandwdth expanson,.e., usage of more channel than source dmensons, automatcally. Our man contrbutons are: We propose a jont mult-user encoder whch outperforms the methods of comparson. Ths encoder has exponental complexty n the number of users. A transmtter wth separate encoders for the dfferent users, and lnear complexty n the number of users, s developed. Despte beng based on the separate user approxmaton, ths encoder lowers dstorton sgnfcantly compared to state-of-the-art. In our experments we further note that the proposed JSCC approach contrbutes more to lowerng the dstorton than the VPP approach. We derve a lower bound on the dstorton. II. MIMO BROADCAST TRANSMISSION WITH SHORT CODES In ths secton, we present two new optmzed low-latency codng strateges for communcaton over a MIMO broadcast channel, as well as a dervaton of a lower bound on the dstorton. Real-valued

6 4 H 1 u 1 τˆl 1 y 1 τl B ỹ 1 β 1 ˆx 1 x 1 α 1 x α y 2 ỹ 2 β 2 ˆx 2 x 2 α1 H 2 u 2 (a) τˆl 2 (b) Fg. 1. In (a), the proposed JSCC system s pctured. In (b), the proposed smplfed encoder n Sec. II-B s shown. sgnalng per antenna s consdered n ths paper for smplcty. However, complex-valued sgnalng can be represented n ths format, by approprately treatng a complex number as a vector of two real numbers. Our goal s to mnmze the sum end-to-end MSE dstorton. We assume perfect channel state nformaton (CSI) at the transmtter and at the recevers. For each channel realzaton, a new coder has to be desgned. In order to keep the notaton smple, we consder two users, see Fg. 1(a). It s conceptually straghtforward to generalze the developments to any number of termnals, though no fundamental aspects are added by such an extenson. The source varables x 1 R and x 2 R are ndependent and dentcally dstrbuted (..d.) zeromean Gaussan wth unt varance (a non-zero mean s handled by smply removng the mean at the transmtter and addng t back at the recever), and wll be delvered to users 1 and 2 respectvely. We defne x = [x 1,x 2 ] T. User s channel conssts of a mxng matrx H R NR NT, and addtve nose u R NR whose components are..d. Gaussan wth zero mean and unt varance, for = 1,2, where N T s the total number of transmt antennas, and N R s the number of receve antennas per user. We also wrte H = [H T 1 HT 2 ]T and u = [u T 1 ut 2 ]T. We defne our analog communcaton system as follows. We use VPP [16] wth the precodng matrx B = [B 1 B 2 ], where B R NT NR, for = 1,2. Followng the orgnal development n [16], we constran B to beng of regularzed nverse-type B = H T (HH T + η NT P I) 1, where P s the mean total transmt power per realzaton of x, and η [0, ). The encoder mappng s α(x) : R 2 R 2NR Ω 2NR, where Ω 2NR s a 2N R -dmensonal cube wth sde less than τ, centered at orgo. The perturbaton vectors are τl(α), wth l(α) L, and L = {l : l k { L, L + 1,...,L} for k = 1,...,2N R }, obtaned by calculatng l (α) = argmn B(α(x) + τl ) 2. Fnally, we wrte the receved sgnals as l L y = HB(α(x)+τl(α))+u, where y = [y T 1 yt 2 ]T, and y R NR, for = 1,2. Estmatesˆl 1 = [ˆl 1,...,ˆl NR ] T of the elements l 1,...,l NR are retreved and subtracted at user 1 s decoder,

7 5 and estmates ˆl 2 = [ˆl NR+1,...,ˆl 2NR ] T of the elements l NR+1,...,l 2NR are retreved and subtracted at user 2 s decoder. The standard estmates [16] are y + (HB) k,kτ 2 ˆl k =, k = 1,...,2N R, (1) (HB) k,k τ where s the floorng operaton. The decoder mappngs are β (y ) : R NR Ω NR R, for = 1,2. Transmttng x 1 over H 1 B 1 s motvated by B beng of regularzed nverse-type, so that H 1 B 1 contans more energy than H 1 B 2 (ths reasonng s smlar for x 2 ). It s dffcult to optmze B and τ n order to mnmze the sum end-to-end dstorton, and smple closed form expressons for B and τ may not exst. We defer a more detaled dscusson on how we set B, τ, and L to Sec. III-A. We further assume a mean power constrant E x [ B(α(x)+τl(α)) 2 ] P, (2) whch s fulflled by assumng α to be unformly dstrbuted over Ω 2NR, and adjustng the sze of Ω 2NR, pror to optmzng the mappngs α(x), β 1 (y 1 ), and β 2 (y 2 ). The rest of ths secton wll be dedcated to the desgn of optmzed source-channel mappngs α(x), β 1 (y 1 ), and β 2 (y 2 ), gven ths channel-codng scheme. We defne a cost functon We thus want to solve J(α(x),β 1 (y 1 ),β 2 (y 2 )) [ ] 1 2 = E x,u1,u 2 (x β (y 2 )) 2 (3) =1 [ ] 1 2 = E x,y1,y 2 (x β (y 2 )) 2. (4) =1 {α (x), β 1 (y 1), β 2 (y 2)} = arg mn J(α(x),β 1 (y 1 ),β 2 (y 2 )). (5) {α(x), β 1(y 1 ), β 2(y 2 )} Throughout our treatment, we wll contnue to use ( ) to denote optmal mappngs. The optmzaton problem n (5) s very complcated. More precsely, the optmal mappngs could be general non-lnear mappngs wth no closed-form soluton, and t s dffcult to acheve both the encoder and the decoders smultaneously. We therefore employ a sub-optmal strategy, namely, we ntroduce a dscretzaton of the

8 6 channel space, and solve teratvely for the optmal encoders gven the decoders, and for the optmal decoders gven the encoders, n Sec. II-A and II-B. We frst dscretzeω 2NR wthm ponts along each dmensons = {( m M 1 2 and restrct α(x) : R 2 S 2NR. Moreover, y s least squares-decoded to yeld ỹ ), m = 0,...,M 1 }, ỹ = argmn y H B (s+τˆl ) 2, = 1,2. (6) s S N R The decoder (6) s a maxmum lkelhood (ML) decoder n the lmt where the non-gaussan MIMO nterference s weak, and where τ s bg enough compared to the sze of S and the nose power, so that l s seldom ms-detected. The standard tandem approach from [16] that we have appled here, where ˆl s frst estmated by (1), and where ỹ s subsequently obtaned by (6), could be replaced by a jont estmaton of ˆl and ỹ, for = 1,2. The decoder mappngs β now operate on ỹ, β (ỹ ) : S NR R for = 1,2. As long as 2 s small compared to the nose varance, and M s bg enough so that S 2NR spans over the whole volume of Ω 2NR, the dscretzaton s a good approxmaton [13]. We descrbe how to choose n Sec. III-A n order to meet the mean power constrant (2). We now rewrte (5) as {α (x), β 1(ỹ 1 ), β 2(ỹ 2 )} = arg mn J(α(x),β 1 (ỹ 1 ),β 2 (ỹ 2 )). (7) {α(x), β 1(ỹ 1 ), β 2(ỹ 2 )} We note that whle the state-of-the-art VPP detector wth (1) and (6) s sub-optmal, the optmzaton problem (7) takes transtons probabltes between codewords, whch ncorporate the MIMO nterference, as well as artfacts from the channel decodng by means of (1) and (6), nto consderaton. We can vew our dscretzaton approach as a dgtal communcaton system. As already stated n the dscusson followng (5), wth the dscretzaton n (7), t s stll dffcult to obtan both the encoder and the decoders smultaneously. Therefore, we further ntroduce two suboptmal strateges. In Sec. II-A, we present a jont consderaton of the two users at the encoder, and no other approxmatons than the teratve Lloyd approach that conssts of encoder optmzaton for a gven decoder, and decoder optmzaton for a gven encoder. In Sec. II-B, we further approxmate by separatng the jont mult-user encoder from Sec. II-A nto ndvdual encoders for the users n order to lower the computatonal complexty.

9 7 A. Jont mult-user encoder Channel transton probabltes between transmt and receve constellaton ponts are frst determned by means of Monte Carlo smulaton, and the decoders are ntalzed by β (0) at teraton step 0, for = 1,2. Thereafter, at teraton step k, k = 1,...,K, the optmal encoder mappng α (k) for gven decoder mappngs β (k 1), and optmal decoder mappngs β (k) decded by means of α (k) (x) = argmn s S 2N R 2 =1 Eỹ s for gven encoder mappngs α (k), are [ ( ) ] x β (k 1) 2 (ỹ ), (8) β (k) (ỹ ) = E x ỹ [x ], = 1,2. (9) The update equatons (8) and (9) yeld lower dstorton at each teraton step k, k = 1,...,K. The teratve approach may however end up at a local mnmum, and does not necessarly solve (7). When solvng (9) n practce, the expectaton s substtuted by the emprcal mean. Encodng by means of (8) n the general mult-user case wth N U users requres on the order of N U M (NU+1)NR operatons. Snce ths number does not scale well wth N U, we wll propose a low-complexty method n the next secton. B. Separate encoders for the dfferent users We separate the problem (7) by constranng the source mappng α(x) to have two ndependent parts α(x) = [α T 1 (x 1) α T 2 (x 2)] T where α (x ) : R S NR for = 1,2, see Fg. 1(b). The receve mappngs are defned as before. Ideally, we would now lke to solve the problem {α 1 (x 1 ), α 2 (x 2 ), β 1(ỹ 1 ), β 2(ỹ 2 )} = arg mn {α 1(x 1), α 2(x 2), β 1(ỹ 1 ), β 2(ỹ 2 )} J([α T 1 (x 1) α T 2 (x 2)] T,β 1 (ỹ 1 ),β 2 (ỹ 2 )), (10) but agan, as wth (7), the global optmal soluton s hard to acheve. Instead, we solve (10) teratvely for the optmal encoders gven the decoders, and for the optmal decoders gven the encoders, by use of the equatons [ ( ) ] α (k) (x) = arg mn Eỹ s x β (k 1) 2 (ỹ ), (11) s S N R β (k) (ỹ ) = E x ỹ [x ], (12)

10 8 for =1,2, where (11) and (12) reduces dstorton at each teraton step k, for k = 1,...,K. Smlarly to the algorthm n Sec. II-A, the teratve approach may however end up n a local mnmum, and does not necessarly solve (10). For encodng by means of (11) n the general mult-user case wth N U users, the computatonal complexty ncreases as N U M 2NR, whch s lnear n N U nstead of exponental, cf. the encoder (8) n Sec. II-A. It should also be noted that both proposed encoders have an extra complexty term (2L + 1) NUNR stemmng from the VPP encodng, as well as a complexty contrbuton stemmng from the calculaton of transton probabltes, whch s exponental n the number of users. Dfferently from the encoder, whch s defned for an analog x, these terms are straghtforward to pre-process n slow fadng. C. A lower bound on the dstorton In order to obtan a lower bound on the dstorton, we employ the source-channel separaton theorem, calculate the maxmum sum-rate, and dstrbute ths rate among the sources n an optmal manner. 1 The concept of uplnk-downlnk dualty [20], [21] s employed, and the convex optmzaton problem R 1 = max S 1,S 2 4 log 2 I +HT S 1 0 H, 0 S 2 S 1, S 2 symmetrc and postve defnte, Tr(S 1 )+Tr(S 2 ) P, (13) s solved n order to fnd the maxmum rate R per user, n bts per channel use. Observng that the optmal rate allocaton to two..d. sources s equal, and that the maxmum rate that can be conveyed over the channel s 2R, we have the bound J 2 2R. Ths bound s not achevable n general, snce we are not able to dstrbute the rate n (13) equally to the two users. A related result, a characterzaton of the achevable dstorton regon for the problem of sendng two correlated scalar Gaussan sources over a bandwdth-matched Gaussan broadcast channel, was recently presented n [3]. 1 The source-channel separaton theorem does not hold when we have correlated sources. Ths s so because the nterference becomes lower when the correlaton between the sources ncreases [18, p. 71], [19, p. 448]. In our case, we have uncorrelated sources.

11 9 III. EXPERIMENTS A. Expermental prerequstes The experment parameters are chosen as follows, n order to supply relevant expermental settngs, and not favor any scheme: Proposed systems: Assumng that α s equdstrbuted over the constellaton ponts, the dstance between the constellaton ponts n one dmenson,, s adjusted by bsecton so that the mean energy of the VPP constellaton s P ±0.005P, cf. (2). As already dscussed n Sec. II, smple closed form expressons for B and τ that mnmze the sum end-to-end dstorton may not exst. The MMSE precodng matrx,.e., η = 1, was chosen for the experments presented n the fgures, cf. [22]. Wth ZF precodng,.e., η = 0, our proposed JSCC schemes n Sec. II-A and Sec. II-B performed the same snce there s no mult-user nterference, and they stll ncreased performance compared to the methods of comparson, though the MMSE precoder results were better. For smplcty, we chose τ by assumng that α s equdstrbuted over the constellaton ponts, that only α on the edges of the constellaton contrbutes to msdetecton of the perturbaton vector, and that the nterference s Gaussan and ndependent for each dmenson, ( ( ))) and we obtan τ = max 2σ (HB), Q 1 M 2 (1 (1 P Err ) 1 2N R + (M 1), where σ 2 = P 2NR 2N R j=1,j (HB)2,j +1. The error probablty assocated wth a msdetected perturbaton vector s set to be less than P Err = Prelmnary nvestgatons showed that ncreasng L beyond 1 dd not lower dstorton sgnfcantly. Channel transtons probabltes for use wth the channel-optmzed coders are calculated by means of Monte Carlo smulatons, takng MIMO nterference as well as artfacts from the channel decodng by means of (1) and (6), nto account. Monte Carlo smulatons are run wth vectors per possble realzaton of α for the case of N T = 2 and N R = 1; and wth 300 vectors per possble realzaton of α for the case of N T = 4 and N R = 2. Intal β 1 and β 2 are chosen by generatng decoder source codebooks wth GLA, and mappng these codewords to constellaton ponts ordered lexcographcally accordng to ther dmensons. For coder optmzaton wth (8) and (9), as well as wth (11) and (12), a tranng set that s dfferent from the evaluaton set and consstng of realzatons, s used. The number of teratons K = 15. In the experments wth a MIMO channel of dmenson N T = 2 and N R = 1, we use M = 32; and for the experments wth N T = 4 and N R = 2, we use M = 16. Source and channel evaluaton data: The elements of H are..d. Gaussan wth zero mean and

12 10 unt varance. We consder slow fadng channels. In partcular, H s constant for consecutve realzatons of x. Results are calculated as means over channel realzatons, whch are evaluated n the Monte-Carlo sense usng 90 realzatons of H. The channel sgnal-to-nose rato (CSNR) s defned as the mean transmtted power P dvded by the mean nose power. The code used for generatng the results can be found at [23]. B. Results The man purpose of Fg. 2(a) s to show a comparson between the proposed jont mult-user coder n Sec. II-A and the proposed low-complexty system wth separate coders for all users n Sec. II-B for L = 0 and L = 1. We use a MIMO channel wth N T = 2, and N R = 1. A lnear ZF-based analog,.e., quantzaton-free, coder wth MMSE recevers and ts theoretcal performance predcton, and the lower bound from Sec. II-C, are added as well for comparson. We see that the mult-user coder n Sec. II-A wth L = 1 outperforms the other coders, for a large range of CSNR values. We can also conclude that the mprovement gven by the mult-user coder n Sec. II-A compared to the system wth separate coders n Sec. II-B s bgger than the mprovement from usng VPP wth L = 1. The relatvely large gap to the lower bound s explaned by the nablty of the short codes to acheve capacty and rate-dstorton optmalty. The purpose of Fg. 2(b) s to show how the system wth separate coders for all users n Sec. II-B compares wth state-of-the-art systems n a scenaro wth more antennas, N T = 4, and N R = 2. A comparson s gven between the system n Sec. II-B for L = 0 and L = 1; a standard separate source-channel codng (SSCC) approach wth GLA-based vector quantzaton, largest mnmum Hammng dstance-optmal lnear block codes wth dfferent rates (the number of nformaton and channel bts are gven for each method n the fgure), Gray mappng of bts to symbols, VPP wth L = 1, and 16 constellaton ponts per real dmenson; analog ZF-precoder-based repetton codng wth MMSE recevers, and ts theoretcal performance predcton; as well as the lower bound from Sec. II-C. One should further note that the range of optmal separate codng schemes wth varyng block code length consttutes an adaptve codng scheme. The proposed low-complexty system wth separate coders for all users from Sec. II-B and L = 1 outperforms the separate codng approaches and the repetton coder, for a wde range of CSNR values. By comparng the proposed JSCC system wth L = 0 and the SSCC systems, we note that JSCC contrbutes more than VPP to mprovng the soluton.

13 11 Source reconstructon MSE per user 10 0 Analog ln. Analog ln., theory Sec. II-B, L = 0 Sec. II-B, L = 1 Sec. II-A, L = 0 Sec. II-A, L = Lower bound CSNR (db) Source reconstructon MSE per user 10 0 SSCC, (2,8) SSCC, (4,8) SSCC, (6,8) SSCC, (8,8) Analog rep. Analog rep., theory Sec. II-B, L = 0 Sec. II-B, L = 1 Lower bound CSNR (db) (a) N T = 2, and N R = 1. (b) N T = 4, and N R = 2. Fg. 2. Comparsons between proposed schemes n Sec. II-A, Sec. II-B, and other methods, for dfferent channel sgnal-to-nose ratos (CSNR). Detals of the compared methods are gven n the man text. The repetton coder does not use a quantzer, and thus, t beats all other methods at extremely hgh CSNR. At extremely hgh CSNR, the separate source and channel coder system wthout a block-code, and the JSCC system wth L = 0, perform as well as the JSCC system wth L = 1. Ths should be so, snce only the source encoder matters, and our proposed scheme becomes the tradtonal GLA algorthm, n ths regme. We observe that the measured CSNR may devate from the target CSNR related to P. The reason for ths s that some transmt constellaton ponts are chosen more often than others. Ths practcal ssue could be taken care of by feedng a modfed mean energy target to the -optmzaton. IV. CONCLUSION We have studed source transmsson over the MIMO broadcast channel, usng short codes and wth end-to-end sum-mse as the performance measure. Two channel-optmzed quantzaton-type systems, one wth a jont mult-user encoder, and one wth separate encoders for the dfferent users, were proposed. The frst encoder, whch has exponental computatonal complexty n the number of users, outperforms the latter, whch has lnear computatonal complexty n the number of users. The latter encoder n turn has been shown to outperform state-of-the-art. REFERENCES [1] D. Tse and P. Vswanath, Fundamentals of Wreless Communcaton. Cambrdge, U.K.: Cambrdge Unv. Press, 2005.

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15 13 [23] Code repostory, Dvson of Communcaton Systems, Lnköpng Unversty. [Onlne]. Avalable: sy.lu.se/en/publcatons