On the Performance of Hybrid Digital-Analog. Coding for Broadcasting Correlated Gaussian Sources

Transcription

1 O the Performace of Hybrid Digital-Aalog Codig for Broadcastig Correlated Gaussia Sources Hamid Behroozi, Member, IEEE, Fady Alajaji, Seior Member, IEEE ad Tamás Lider, Seior Member, IEEE Abstract We cosider the problem of sedig a bivariate Gaussia source S = (S, S ) across a power-limited two-user Gaussia broadcast chael. User i (i =, ) observes the trasmitted sigal corrupted by Gaussia oise with power σi ad desires to estimate S i. We study hybrid digital-aalog (HDA) joit source-chael codig schemes ad aalyze the regio of (squared-error) distortio pairs that are simultaeously achievable. Two cases are cosidered: ) broadcastig with badwidth compressio, ad ) broadcastig with badwidth expasio. We modify ad adapt HDA schemes of Wilso et al. [] ad Prabhaara et al. [], origially proposed for broadcastig a sigle commo Gaussia source, i order to provide achievable distortio regios for broadcastig correlated Gaussia sources. For compariso, we also exted the outer boud of Soudararaja et al. [3] from the matched source-chael badwidth case to the badwidth mismatch case. Idex Terms Gaussia broadcast chael, badwidth compressio/expasio, joit source-chael codig, hybrid digital-aalog (HDA) codig, Costa codig, Wyer-Ziv codig, layered codig, ucoded trasmissio. This wor was supported i part by a Postdoctoral Fellowship from the Otario Miistry of Research ad Iovatio (MRI) ad by the Natural Scieces ad Egieerig Research Coucil (NSERC) of Caada. The material i this paper was preseted i part at the IEEE Iteratioal Symposium o Iformatio Theory (ISIT), Seoul, Korea, Jue 009 ad at the IEEE Iformatio Theory Worshop (ITW), Taormia, Italy, Oct H. Behroozi was with the Departmet of Mathematics ad Statistics, Quee s Uiversity, Kigsto, Otario, Caada. He is ow with the Electrical Egieerig Departmet, Sharif Uiversity of Techology, Tehra, Ira ( behroozi@sharif.edu). F. Alajaji ad T. Lider are with the Departmet of Mathematics ad Statistics, Quee s Uiversity, Kigsto, Otario, K7L 3N6, Caada ( {fady,lider}@mast.queesu.ca).

2 I. INTRODUCTION WE cosider the reliable trasmissio of a correlated bivariate Gaussia source S = (S, S ) across a power-limited two-user Gaussia broadcast chael. Oe motivatio of our study is the problem of sedig a correlated vector source such as the pair (temperature, pressure) of a reactor to moitorig sites. Differet compoets of the source could have their ow fidelity requiremets istead of a average or total distortio measure eve though they are joitly coded. First let us cosider the problem of broadcastig a sigle memoryless source to two destiatios. A Gaussia source sequece of mea zero ad variace σ S is to be trasmitted across a Gaussia two-user broadcast chael with power costrait P ad with respective oise variaces N ad N (N > N ) (e.g., see [4]). For this example, it is ow that ucoded trasmissio performs better tha the best separate source-chael code (see, e.g., [5] [8]). Let C = log(+ P N ) ad C = log(+ P N ) be, respectively, the capacities of the two uderlyig poit-to-poit chaels. If separate source ad chael codig is used, i.e., the Gaussia source is optimally quatized ad the quatizatio bits are ecoded with a capacity-achievig chael code (see Fig. ), the mea squared-error (MSE) pair of achievable distortios satisfies D = σ S + ( γ)p γp +N ; D = σ S ( ) ( ), () + ( γ)p γp +N + γp N where γ ca be chose i [0, ] to provide the desired tradeoff betwee D ad D. Sice the Gaussia problem we cosider is successively refiable [7], [9], this result follows from combiig R i = R(D) = log( σ S D ) with the pair of achievable rates for a broadcast chael as R = ( γ)p log(+ γp +N ) ad R = R + γp log(+ N ) [4], [7]. Note that for each value of γ, we ca desig a chael code that provides a particular achievable rate pair (which gives a specific distortio pair). However, applyig ucoded trasmissio yields the followig distortio pair: D = σ S + P N ; D = σ S + P N. () These distortios are clearly ot simultaeously achievable by separate source-chael codes. This simple example provides a multi-user sceario where aalog iformatio is more valuable tha digital iformatio. I a similar spirit, this paper cosiders broadcastig correlated Gaussia sources ad aims to characterize MSE distortio pairs that are simultaeously achievable at the two receivers usig hybrid digital-aalog

3 3 (HDA) codig schemes. Shao proved that the separate (idepedet) desig of source ad chael codig is a optimal strategy for a fixed chael sigal-to-oise ratio (CSNR) i ergodic poit-to-poit commuicatio systems (where optimality i terms of reproducig the source at the destiatio withi a prescribed fidelity is achieved asymptotically as the codig/decodig delay ad complexity icrease without boud) [0]. Such a scheme is ofte referred to as a digital tadem source-chael codig scheme. There are two iheret problems associated with the digital tadem scheme: the levelig-off effect ad the threshold effect [], []. Sice the system typically performs well at a certai desiged CSNR, the system performace does ot improve with icreased CSNR (levelig-off effect), ad it degrades drastically whe the true CSNR falls beeath the desiged CSNR (threshold effect). It is also ow that this coceptually simple codig scheme does ot i geeral lead to the optimal performace theoretically attaiable (OPTA) i etwors; see e.g. [4], [3]. O the other had, for the poit-to-poit trasmissio of a sigle Gaussia source through a additive white Gaussia oise (AWGN) chael, it is well ow (e.g., see [5], [3]) that if the chael ad source badwidths are equal, simple ucoded trasmissio achieves OPTA. Ucoded (or aalog) trasmissio i this case (ad i the rest of this paper) meas scalig the ecoder iput subject to the chael power costrait ad trasmittig without explicit chael codig. The optimality of ucoded trasmissio i some multi-user commuicatio systems was recetly show i [4] [6]. I order to exploit the advatages of both aalog trasmissio ad digital techiques, a family of HDA schemes were itroduced i the literature, see e.g., [], [], [7], [], [7] [6]. These methods usually offer better distortio performace tha the purely aalog or digital schemes; they do ot suffer from the levelig-off effect, have a less severe threshold effect [8] compared to digital tadem source-chael codig schemes, ad they ca asymptotically achieve Shao s OPTA limit at the desiged CSNR. The case of broadcastig a sigle memoryless Gaussia source with badwidth mismatch betwee the source ad the chael usig HDA schemes is cosidered i [8], [0]. Bross et al. [7] show that there exists a cotiuum of HDA schemes with optimal performace for the trasmissio of a Gaussia source over a average-power-limited Gaussia chael with matched badwidth. Tia ad Shamai [8] geeralize this result to the mismatched badwidth case. I [9] Gao ad Tucel propose two ew schemes for trasmittig

4 4 a Gaussia source over a Gaussia chael. These schemes directly geeralize previous result of [7] by maig better use of the dirty-paper codig auxiliary radom variable. A complete characterizatio of the set of achievable distortio pairs i trasmittig a Gaussia source with memory over a arbitrarily colored Gaussia broadcast chael with matched badwidth is preseted i []. I [30] ier ad outer bouds for the distortio regio i broadcastig a Gaussia mixture source is provided. Broadcastig a commo source to multiple receivers havig differet correlated side iformatio is ivestigated i [3] [34]. A HDA codig scheme for broadcastig a commo source to two receivers with matched badwidth havig differet correlated side iformatio is proposed i [35], where the authors show that uder certai coditios their scheme achieves the same performace as i poit-to-poit commuicatio simultaeously at both receivers ad is thus optimal. I [36], a HDA scheme is preseted for the problem of sedig a parallel Gaussia source over a white Gaussia broadcast chael. Related wor o broadcastig correlated sources ca be foud i [3], [6], [37] [44]. Lossless trasmissio of fiite alphabet sources is cosidered i [37] [4], [45], ad ucoded trasmissio for broadcastig correlated Gaussia sources is evaluated i [6]. It is show i [6] that the ucoded scheme is optimal below a certai CSNR-threshold. I [46], we itroduce a layered HDA scheme for broadcastig a bivariate Gaussia source with matched badwidth. A complete characterizatio of the achievable distortio regio i sedig a bivariate Gaussia source over badwidth-matched Gaussia broadcast chael was recetly derived i [44]. I a recet mauscript [47], the problem of broadcastig two correlated Gaussia sources usig optimal separate source ad chael codes is studied, where it is show that the proposed scheme is very competitive for ay badwidth compressio/expasio sceario. However, as metioed before, separatio based digital schemes suffer from the threshold effect while the HDA cosidered offer better performace i the presece of CSNR mismatch. The problem of sedig a pair of fiite alphabet correlated sources through a broadcast chael with correlated side iformatio at the receivers is studied i [4]. A lattice-based hybrid codig is proposed i [3] for broadcastig idepedet as well as correlated Gaussia sources i the case of matched badwidth. The authors i [3] show that their proposed scheme is optimal for broadcastig idepedet sources ad performs better tha separate source/chael codig for broadcastig correlated sources below a certai CSNR-threshold. Our system model is illustrated i Fig.. We aim to determie achievable distortio regios usig HDA

5 5 schemes for two cases: ) broadcastig with badwidth compressio, i.e., broadcastig with λ chael uses per source sample, where λ <, ad ) broadcastig with badwidth expasio, where λ >. To the best of our owledge, apart from [3], [6], [44] ad the recet result of [47], i which the problem of broadcastig correlated Gaussia sources is aalyzed, there are o explicit distortio-regios i the literature for broadcastig correlated Gaussia sources. We are also ot aware of ay prior wor discussig HDA schemes for broadcastig correlated Gaussia sources with badwidth mismatch. This paper reports o progress towards solvig this difficult problem. We evaluate the performace of layered codig schemes for broadcastig correlated Gaussia sources ad provide explicit expressios for the achievable distortio regios. Such schemes, which exted the HDA schemes of Wilso et al. [] ad Prabhaara et al. [] for the broadcastig of a sigle commo Gaussia source, judiciously mix various codig strategies, ragig from HDA joit source-chael codig, Costa dirty paper codig [48], ad Wyer-Ziv codig. Although the distortios are derived explicitly (i closed-form expressios) for all proposed schemes, a geeral ad aalytical performace compariso of those schemes is quite difficult. I fact, the problem of fidig a optimal power allocatio policy amog layers i order to optimize the achievable overall ed-to-ed distortio pairs is still ope. Istead, we umerically evaluate the achievable distortio regios of differet schemes ad oly preset the best scheme i each badwidth mismatch case. I additio, we provide a outer boud for the achievable distortio regio ad compare the achievable regios to that outer boud. I the case of badwidth compressio, a scheme combiig aalog trasmissio, superpositio ad Costa codig is preseted. For badwidth expasio, we itroduce a hybrid Wyer-Ziv (HWZ) scheme, which cosists of a aalog layer ad two layers each cosistig of a Wyer-Ziv coder followed by a chael coder. I [49] we showed that our HWZ scheme performs similarly to the adapted Rezic-Feder-Zamir scheme, origially proposed i [0] for broadcastig a commo Gaussia source to two users. Numerical examples idicate that there is a gap betwee the achievable distortio regios ad the outer regio for both badwidth mismatch cases ad the costructio of ew schemes that ca close or arrow this gap remais a iterestig ad challegig future directio. The remaider of this paper is orgaized as follows. I Sectio II, we preset the system model ad problem statemet. We derive the achievable distortio regios of HDA schemes with badwidth Although i geeral the compariso for few examples may ot provide a geeral isight ito optimality, a similar behavior was observed by evaluatig the achievable distortio regios i may other examples with differet system parameters.

6 6 compressio ad expasio i Sectios III ad IV, respectively. A outer regio for broadcastig correlated Gaussia sources with mismatched badwidth is provided i Sectio V. I Sectio VI, the boudaries of the distortio regios for the preseted HDA schemes as well as the outer boud i both badwidth mismatch cases are compared via umerical examples. A example ivolvig our layered scheme with aalog trasmissio ad Costa codig of [46] is also preseted; it is observed that the layered scheme s achievable regio matches the outer boud regio, idicatig its potetial optimality. Coclusios are give i Sectio VII. II. PROBLEM STATEMENT Cosider broadcastig correlated Gaussia sources (or equivaletly a bivariate Gaussia source) across a two-user power-limited Gaussia broadcast chael. User i (i =, ) receives the trasmitted sigal corrupted by Gaussia oise with power N i ad aims to estimate source S i. We assume that N > N ad hece call user the wea user ad user the strog user. Let S ad S be correlated Gaussia radom variables ad let {(S (t), S (t))} t= be a statioary Gaussia memoryless vector source with margial distributio that of (S, S ). We assume that S (t) ad S (t) have zero mea ad variace σs ad σs, respectively, ad correlatio coefficiet ρ (, ). We represet the first istaces of the first ad secod source compoets by the data sequeces S = (S (), S (),, S ()) ad S = (S (), S (),, S ()), respectively. The two-user Gaussia broadcast chael with receivers estimatig the bivariate source compoets is show i Fig.. Data sequeces S ad S are joitly ecoded to X = ϕ ( ) S, S, where the ecoder fuctio is of the form ϕ : R R R. (3) The badwidth compressio/expasio ratio is defied by λ = chael uses per source sample. We aim to fid achievable distortio regios of HDA schemes for broadcastig with badwidth compressio where λ < (we specifically cocetrate o λ = ) ad badwidth expasio where λ > (i particular we set λ = ). The trasmitted sequece X is average-power limited to P > 0, i.e., E [ X(t) ] P. (4) t= User i observes the trasmitted sigal X(t) corrupted by a Gaussia oise V i (t) with power (variace) N i, so that at time t the receiver observes

7 7 Y i (t) = X(t) + V i (t), i =, (5) where V i (t) N (0, N i ) are idepedetly distributed over i ad t, ad are idepedet of X(t). Based o the chael output Yi, receiver i provides a estimate Ŝi of the ith compoet of the source, S i. We cosider the average MSE distortio i = E[ S i (t) Ŝi(t) ]. The recostructed sigal at receiver i ca be described by Ŝi = ψi (Yi ), where decoder fuctios are mappigs t= ψ i : R R, i =,. (6) Let F (,) (P ) deote all ecoder ad decoder fuctios (ϕ, ψ, ψ ) that satisfy (3) (6). For a particular codig scheme (ϕ, ψ, ψ ), the performace is determied by the chael power costrait P ad icurred distortio pairs ad at both receivers. For ay give power costrait P, the distortio regio D is defied as the closure of the covex hull of the set of all distortio pairs (D, D ) for which (P, D, D ) is achievable, where a power-distortio pair (P, D, D ) is achievable if for ay δ > 0, there exist sufficietly large itegers ad = λ, ecodig ad decodig fuctios (ϕ, ψ, ψ ) F (,) (P ), such that i D i + δ (i =, ). III. DISTORTION REGION FOR BANDWIDTH COMPRESSION: LAYERING WITH ANALOG, SUPERPOSITION AND COSTA CODING We cosider the problem of broadcastig a bivariate Gaussia source with : badwidth compressio. We desire to trasmit = samples of a bivariate Gaussia source (S, S ) i uses of a power-limited broadcast chael to two users. The two-user broadcast chael has the power costrait P. We split both compoets of the bivariate Gaussia source ito two equal legth parts, i.e., we split samples of each source vector S i ito two vectors of legth : S i, ad S i,. I this scheme, we will closely follow the otatio ad code costructios i []. Here we oly give a high-level descriptio ad aalysis of the schemes without detailed proofs. I particular, i may steps of the aalysis we treat fiite-bloclegth codig schemes as idealized systems with asymptotically large bloclegths. Detailed proofs ca be give followig argumets i [], where a layerig structure is itroduced for broadcastig a memoryless Gaussia source. Here, we adapt this scheme for broadcastig a bivariate Gaussia source with a chage i the structure of the secod layer.

8 8 I the first (aalog) trasmissio layer, a liear combiatio of the first samples of the bivariate Gaussia source compoets are scaled such that the power of the trasmitted sigal i this layer Xa becomes P a. Here X a (t) = α a i S i, (t), where α = P a a σ S +a σ S +a a ρσ S σ S. This layer is meat for both strog ad i= wea users. Now fix P ad P to satisfy P = P a + P + P. I the secod ad the third layers, we wor o the remaiig samples of the source compoets, i.e., S, ad S,, respectively. I the secod layer, we use two merged streams, X ad X. The secod part of the first compoet of the source, S,, is broadcasted to two users. The first source ecoder is a optimal source ecoder with rate [4, Sectio 5..3] R = I(X ; Y ) = log( + ( γ)p γp +P a+p +N ), where I( ; ) deotes the mutual iformatio. The secod source ecoder is a optimal ecoder for the residual error of the first ecoder with rate R R = I(X ; Y X ) = log( + γp P a+p +N ). The, we ecode the quatizatio bits with capacity-achievig chael codes ad trasmit the resultig streams with powers ( γ)p ad γp, respectively. I the third layer, which is meat for the strog user, samples of the secod compoet of the source, S, are Wyer Ziv coded usig the estimate of S, at the receiver as side iformatio. The Wyer-Ziv idex, m {,,, R }, is the ecoded usig Costa s dirty paper codig that treats both X a ad X as iterferece ad uses power P = P P a P. Let U be a auxiliary radom variable give by U = X +α (X a +X ), where X N (0, P ), X ad X a are idepedet of each other ad α = P P +N. We geerate a legth i.i.d. Gaussia codeboo U with I(U ;Y ) codewords, where each compoet of the codeword is Gaussia with zero mea ad variace P + α (P a + P ), ad each codeword is the radomly placed ito oe of R bis with R = I(U ; Y ) I(U ; X a, X ) = log( + P N ). Let i(u ) be the idex of the bi cotaiig U. For a give m, we loo for a U such that i(u ) = m ad (U, X a, X ) are joitly typical. The, we trasmit X = U α (X a + X ). We liearly combie all three layers ad trasmit X = X a + X + X. A achievable distortio-regio ca be obtaied by varyig P a, P ad P subject to P = P a +P +P. For a give P a, P ad P, the achievable distortio pairs ca be computed as follows. At the decoder, we loo for a X that is joitly typical with Y. The wea user estimates S = (S,S,) by MMSE estimatio from the received sigal Y ad the decoded X. The decoder recostructs the sequece S, as Ŝ,(i) = X (i). The a estimate of the first compoet, S,, ca be obtaied as Ŝ,(i) = (Y (i) X (i))

9 9 where = α(a σ S +a ρσ S σ S ) γp +P a+p +N. Thus, the overall distortio see at the wea user is []: D = D + ( )D = D + D, (7) where D j (j =, ) is the MMSE distortio i estimatig S,j from Y ad U. Sice i the secod layer we require a rate of oe chael use per source symbol, ad the Gaussia source is successively refiable, by combiig the Gaussia rate-distortio fuctio with the pairs of achievable rates for a broadcast chael (R, R ), the correspodig achievable distortio pairs are: σ S R ad σ S R. The wea user forms a MMSE estimate of S with the followig distortio: D = (σ S α (a σs + a ρσ S σ S ) ) + σs γp + P a + P + N ( γ)p +. (8) γp +P a+p +N At the strog user, based o joit typicality, first a estimate of S, ca be obtaied as Ŝ,(i) = X (i) withi distortio D = + γp P a+p +N + σs ( γ)p. γp +P a+p +N This estimate acts as side iformatio for obtaiig the estimate of S, usig the decoded Wyer-Ziv bits. The resultig distortio for the strog user is thus give by D = (σ S α (a σs + a ρσ S σ S ) P a + P + N ) + σs ( ρ ( D σ S )) ( + P ). (9) N Fially, ote that if we set ρ = ad σ S = σ S, the the results of [], [], which curretly appear to be the best ow results for broadcastig a Gaussia source with badwidth compressio, are obtaied. IV. DISTORTION REGION FOR BANDWIDTH EXPANSION: LAYERING WITH ANALOG AND WYNER-ZIV CODING (HWZ SCHEME) We wat to trasmit samples of a bivariate Gaussia source S = (S, S ) i = λ uses of a power-limited broadcast chael to two users where λ > (we specifically cocetrate o λ = ). The two-user broadcast chael has the power costrait P. We propose a HDA scheme, which we refer to as the HWZ scheme, ad provide a achievable distortio regio. I [49] we also adapt the proposed HDA scheme for broadcastig a commo source by Rezic, Feder ad Zamir [0] to the problem of broadcastig

10 0 correlated sources. Numerical examples idicate that both schemes have similar performace. This scheme comprises three layers, a aalog layer ad two layers each cosistig of a Wyer-Ziv coder followed by a chael coder. The scheme is similar to the oe proposed i [] for broadcastig a sigle memoryless Gaussia source with badwidth compressio except for the followig: ) Here we cosider broadcastig correlated Gaussia sources. ) The secod layer i the scheme of [] is a HDA Costa codig while here it is a Wyer-Ziv coder followed by a chael coder. 3) Sice we cosider broadcastig with badwidth expasio, oly the codewords of the secod layer ad the third layer (digital layers) are merged together, ad the the trasmitted sequece is obtaied by multiplexig the codeword of the aalog layer with the codeword of the digital layer, while i [] the codewords of all three layers are merged as badwidth compressio is examied. Bloc diagrams of the ecoder ad the decoder are show i Fig. 3. I the first layer, the aalog trasmissio layer, a liear combiatio of the samples of the bivariate Gaussia source compoets are scaled such that the power of the trasmitted sigal, Xa, i this layer is P. Thus at time t we have X a (t) = α a i S i (t) where α = P a σ S +a σ S +a a ρσ S σ S. I the secod layer, = samples of the i= first compoet of the source, S are Wyer Ziv coded at rate R = I(X d ; Y d ) = log( + P P +N ) usig a estimate of S at the receiver as side iformatio. The Wyer-Ziv idex, m {,,, R } is the ecoded treatig the third layer message as a oise ad the codeword X d with power P is trasmitted. I the third layer, which is meat for the strog user, the secod compoet of the source, S, is also Wyer Ziv coded at rate R = I(X d ; Y d X d ) = log( + P N ) usig the estimate of S at the receiver as side iformatio. The Wyer-Ziv idex, m {,,, R }, is the ecoded that treats X d iterferece ad uses power P such that P + P = P. As show i Fig. 3, the trasmitted sequece is obtaied by multiplexig (i time) the codeword of the aalog layer X a with the codeword of the digital layer, X d = X d + X d. Thus, the trasmitted sequece ca be represeted as X = [Xa, X d ]. At the decoder, from the received first compoets of Y = [Y Ŝ a ca be obtaied with a average distortio a, Y d D = σ S Ŝa = σ S α (a σ S + a ρσ S σ S ) P + N, as ], a MMSE estimate of S as where Ŝa(i) = E[S (i) Y a (i)] = Y a () ad = α(a σ S +a ρσ S σ S ) P +N. Sice the Wyer-Ziv idex m

11 must be decoded by the wea user, it is imposed that ( ) log D = D ( log + P ). (0) P + N Therefore, the overall average distortio at the wea user ca be expressed as ( D = D + P ) λ. () P + N At the strog user we wat to mae use of all trasmitted layers. Sice the trasmitted sequece of the secod layer (which carries iformatio about S ) should be decoded by both the wea ad the strog users, we esure that we are able to obtai a estimate of S at the strog user as Ŝ. However, at the strog user, our aim is to obtai a estimate of the secod compoet of the source, S. Based o both the aalog ad the third layer trasmitted sequeces, ad also the available side iformatio at the strog user (i.e., Ŝ), we obtai a estimate of S. At first, from the aalog layer, the strog user forms a estimate of the first compoet of the source, S with MMSE distortio D = σ S α (a σ S + a ρσ S σ S ) P + N. () The, a estimate of the first compoet of the source ca be obtaied withi distortio D = D ( + P ) λ. (3) P + N This estimate acts as side iformatio that ca be used i obtaiig the estimate of S for the strog user usig the decoded Wyer-Ziv bits. Usig the decodig coditio for the Wyer-Ziv idex m, the overall distortio for the strog user i estimatig S ca be obtaied as ( D = D + P ) λ, (4) N where ( ( )) D = σs ρ D. (5) σs V. OUTER BOUND REGION I [3], [6], [50], by assumig the owledge of S at the receiver of the strog user, outer bouds for broadcastig correlated Gaussia sources with matched badwidth were developed. By maig mior

12 modificatios to the proof of Theorem i [3], the followig result ca be obtaied: Lemma : The distortio regio for broadcastig correlated Gaussia sources with badwidth mismatch ratio λ cosists of all pairs (D, D ) such that ( D σs where η [0, ]. + ( η)p D σ S ( ρ ) ηp +N ) λ ( + ηp N ) λ (6) Here, we have assumed that the receiver of the strog user has access to the other source compoet; this is a reasoable assumptio whe the correlatio coefficiet is small. However, this outer boud might ot be tight for high values of the correlatio coefficiet. To exted this outer boud, we assume that the decoder have access to a oisy versio of the other source compoet, S. Let S = γs + ν with ν beig idepedet of S, σ ν = σ S ( γ ) ad γ [0, ]. We obtai the followig boud which icludes (6) as a special case where γ = : ( D σs + ( η)p { D max γ ηp +N ) λ σ S ( γ ρ ) ( ) } λ + P( γ ( η)) N (7) VI. NUMERICAL RESULTS Example (Badwidth Compressio): We trasmit = samples of a bivariate Gaussia source (S, S ) with the covariace matrix Λ = ρ i uses of a power-limited broadcast chael to two ρ users (wea ad strog) with observatio oise variaces N = 5 db ad N = 0 db, respectively. The distortio regio for the scheme preseted i Sectio III is show i Fig. 4 for two differet correlatio coefficiets, ρ = 0. ad ρ = 0.8. For compariso, we also depict the outer boud give by (7) of Lemma for the set of all achievable distortio pairs i broadcastig correlated Gaussia sources. The outer boud is tight oly for small values of the correlatio coefficiet ad thus it is oly show for ρ = 0.. Example (Badwidth Expasio): We trasmit samples of a bivariate Gaussia source S = (S, S ) with the covariace matrix Λ = ρ i = uses of a power-limited broadcast chael to two ρ users with observatio oise variaces N = 5 db ad N = 0 db, respectively. The two-user broadcast

13 3 chael has the power costrait P = 3 db. The boudaries of the outer boud i (7) of Lemma ad of the distortio regio for the scheme of Sectio IV are show i Fig. 5(a)-(b) for two differet values of the correlatio coefficiet, ρ = 0. ad ρ = 0.8. We observe that there is a gap betwee the achievable distortio regio ad the outer regio. Example 3 (Matched Badwidth): We trasmit samples of a bivariate Gaussia source with covariace matrix Λ = 0. i uses of a power-limited broadcast chael to two users with observatio oise 0. variaces N = 5 db ad N = 0 db, respectively. The broadcast chael has the power costrait P = 0 db. The boudaries of the distortio regio for the layerig with aalog ad Costa codig scheme which we itroduced i [46, Sectio III.B] as well as the lattice-based codig scheme of [3] are show i Fig. 6. The outer boud i (6) of Lemma is also show. We observe that layerig with aalog trasmissio ad Costa codig outperforms both ucoded trasmissio ad lattice-based codig. Surprisigly, the outer boud is exactly o the boudary of our scheme. Based o several additioal umerical evaluatios ad also by comparig the distortio regio of our achievable scheme with the optimal distortio regio, recetly derived i [44], we cojecture that the proposed HDA JSCC scheme a optimal trasmissio scheme. VII. CONCLUSIONS We cosidered HDA codig schemes for the trasmissio of a bivariate correlated Gaussia source over a power-limited two-user Gaussia broadcast chael. I particular, layered JSCC schemes were aalyzed uder mismatched badwidth assumptios ad their achievable distortio regios were derived. Variatios of these schemes have previously bee used i the literature for broadcastig a sigle memoryless Gaussia source. We also adapted the distortio outer boud of [3] i broadcastig correlated Gaussia sources with matched badwidth to the badwidth mismatch case. Numerical examples reveal a gap betwee their achievable distortio regios ad the outer regio. Further research is eeded ito developig improved codig schemes to close this gap.

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17 7 S Source Chael X Ecoder Ecoder Source Chael X Ecoder Ecoder X V V Y Y Receiver Receiver ~ S ~ S Fig.. Broadcastig a sigle memoryless Gaussia source usig separate source-chael codes. Source ecoder is a optimal ecoder with rate R, source ecoder is a optimal ecoder for the residual error of ecoder with rate R R. The two codes are superpositioed ad trasmitted across a power-limited Gaussia two-user broadcast chael. S S Ecoder X V V Y Receiver S Y Receiver S Fig.. Broadcastig correlated Gaussia sources over a two-user power-limited Gaussia broadcast chael. Receiver i aims to obtai a estimate of its correspodig source compoet, Si, to withi fidelity D i (i =, ).

18 8 a a α X a a d X = [ X, X ] Bi variate S Wyer Ziv Ecoder Chael Ecoder X d Source = S Wyer Ziv Ecoder Chael Ecoder (a) Ecoder X d X V Y Y Y a Y d Y a X d MMSE Estimator Chael Decoder m Y m d Chael Wyer Ziv V Decoder Decoder Chael Ecoder Chael Decoder MMSE Estimator (b) Decoder Wyer Ziv Decoder m S Wyer Ziv Decoder S Fig. 3. Broadcastig a bivariate source S = (S, S ) with badwidth expasio: the HWZ scheme.

19 9 0 Aalog, Superpositio ad Costa Codig Outer boud 0 log 0 (D ) 0.5 ρ=0.8 ρ= log 0 (D ) Fig. 4. Distortio regio of a HDA codig scheme i broadcastig with badwidth compressio. System parameters are P = 0 db, N = 5 db ad N = 0 db.

20 Outer boud HWZ Scheme log 0 (D ) log 0 (D ) (a) ρ = Outer boud HWZ Scheme 0 log 0 (D ) log 0 (D ) (b) ρ = 0.8 Fig. 5. Achievable[ distortio ] regio of the HWZ scheme ad the outer boud regio i broadcastig with badwidth expasio. System ρ parameters are Λ =, P = 3 db, N ρ = 5 db ad N = 0 db.

21 0 log 0 (D ) Ucoded Trasmissio Lattice Based Codig Aalog ad Costa Codig Outer Boud log 0 (D ) Fig. 6. Distortio regios i broadcastig a bivariate Gaussia source across a power-limited two-user Gaussia broadcast chael with matched badwidth. System parameters are ρ = 0., P = 0 db, N = 5 db ad N = 0 db.