Hybrid Coding for Gaussian Broadcast Channels with Gaussian Sources

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1 Hybrid Codig for Gaussia Broadcast Chaels with Gaussia Sources Rajiv Soudararaja Departmet of Electrical & Computer Egieerig Uiversity of Texas at Austi Austi, TX 7871, USA Sriram Vishwaath Departmet of Electrical & Computer Egieerig Uiversity of Texas at Austi Austi, TX 7871, USA arxiv: v1 [csit] 15 Ju 009 Abstract This paper cosiders a degraded Gaussia broadcast chael over which Gaussia sources are to be commuicated Whe the sources are idepedet, this paper shows that hybrid codig achieves the optimal distortio regio, the same as that of separate source ad chael codig It also shows that ucoded trasmissio is ot optimal for this settig For correlated sources, the paper shows that a hybrid codig strategy has a better distortio regio tha separate source-chael codig below a certai sigal to oise ratio threshold Thus, hybrid codig is a good choice for Gaussia broadcast chaels with correlated Gaussia sources 1 I INTRODUCTION The trasmissio of sources over a Gaussia broadcast chael [1] is a fudametal problem i iformatio theory ad arguably oe of the better uderstood questios I the case of idepedet sources over this degraded chael, the capacity regio is characterized i [], [3] The achievable strategy for this chael i [] is superpositio codig, but dirty paper codig [4] ca also be used to the same effect I cotrast, the existig body of work o correlated sources over a broadcast chael is somewhat limited [5] As sourcechael separatio does ot hold, it is difficult to costruct codig strategies ad establish their optimality Recetly i [6], ucoded trasmissio of correlated Gaussias over a Gaussia broadcast chael was show to be optimal below a sigal to oise ratio (SNR) threshold I related work, the trasmissio of a commo source over a Gaussia broadcast chael with receiver side iformatio was studied i [7] I this work, we cosider hybrid codig as a strategy for the Gaussia broadcast chael with or without correlated sources By hybrid codig, we mea strategies that superimpose ucoded ad coded trasmissio i commuicatig the sources to the destiatios Our hybrid codig strategy bears close resemblace to the dirty-paper-codig strategy usig lattices, as developed i [8] We show that this hybrid strategy is optimal for the Gaussia broadcast chael with idepedet Gaussia sources Extedig it to the correlated case, we fid that the strategy performs better (i terms of the distortio regio achieved) tha separate source ad chael codig for SNRs below a certai threshold 1 This work is supported by a grat from AT&T Labs Austi, a grat from the Army Research Office ad a grat from the Air Force Office of Sposored Research I the ext sectio, we preset the system model I Sectio III, we preset a outer boud o the distortio regio for this chael We preset the achievable scheme ad the resultig distortio regio for this chael i Sectio IV ad coclude with Sectio V (S 1,S ) ENCODER X II SYSTEM MODEL Fig 1 Z 1 Z System Model Y 1 Y DECODER 1 DECODER The system model is depicted i Fig 1 Cosider a sequece of idepedet ad idetically distributed (iid) pair of correlated Gaussias (S 1 (i),s (i)) i=1 with mea zero ad covariace matrix, [ ] σ ρσ Σ(i) = ρ The goal is to trasmit the pair over a degraded Gaussia broadcast chael to Receivers 1 ad respectively at the smallest possible distortio We assume, with loss of geerality, that ρ > 0 ad that the variaces of S 1 (i) ad S (i) are equal The trasmitter applies a ecodig fuctio o the observed source sequece ad trasmits it over the chael Mathematically, X = f(s 1,S ) where S1 ad S deote -legth vectors The trasmitter is limited by a average secod momet costrait o the chael iput give by 1 E[(X(i)) ] P i=1 Ŝ 1 Ŝ

2 The chael outputs at the two receivers are give by Y 1 (i) = X(i)Z 1 (i) Y (i) = X(i)Z (i) for i = 1,,, where Z 1 (i) ad Z (i) form a iid sequece, idepedet of each other ad are Gaussia distributed with mea zero ad variace ad N Further, we assume that the broadcast chael is physically degraded with N > The receivers obtai estimates of the sources by applyig a fuctio o the received outputs This is represeted mathematically as Ŝk = φ k (Yk ) for k = 1, The goal is to obtai estimates Ŝ 1 ad Ŝ withi the miimum possible mea squared error Therefore, we wish to obtai the smallest possible D 1 ad D where D k = 1 E[(S k (i) Ŝk(i)) ] i=1 for k = 1, The distortio regio D(,ρ,P,,N ) is defied as the set of all pairs (D 1,D ) such that there exist ecodig ad decodig fuctios f, φ 1 ad φ resultig i distortios D 1 ad D at Receivers 1 ad respectively Note that all logarithms are with respect to base throughout the paper ad E deotes the expected value of a radom variable III OUTER BOUND ON DISTORTION REGION We ow preset a outer boud o the coditioal distortio regio for the trasmissio of correlated Gaussia sources over a degraded broadcast chael Let φ 1 be the decodig fuctio give the kowledge of both Y1 ad S at Receiver 1 The coditioal distortio regio D c (,ρ,p,,n ), is defied as the set of all pairs (D 1,D ) such that there exist ecodig fuctio f ad decodig fuctios φ 1 ad φ resultig i distortios D 1 ad D at Receivers 1 ad respectively The regio described by Theorem 1 below is a alterative way of describig the outer boud regio for the same chael preseted i [6] Theorem 1: The coditioal distortio regio for trasmissio of correlated Gaussia sources over a degraded broadcast chael, D c (,ρ,p,,n ), cosists of all pairs (D 1,D ) such that D 1 σ (1 ρ ) ad D where α 1 [0,1] Proof: We first obtai a boud o the distortio D 1 By the data processig iequality (DPI), we have I(S 1;Ŝ 1 S ) I(S 1;Y 1 S ) (1) The distortio i S1 at Receiver 1 give that it kows S ad Y1 is D 1 ad the variace of S1 give S is σ (1 ρ ) Sice S is kow at both the trasmitter ad Receiver 1, I(S 1;Ŝ 1 S ) log σ (1 ρ ) D 1 () by defiitio of the rate distortio fuctio for Gaussia sources [9] Now, I(S 1;Y 1 S ) = h(y 1 S ) h(y 1 S 1,S ) = logπe( ) h(z 1) (3) = logπe( ) logπe = ( logπe 1 α ) 1P (4) The equality i (3) results from the followig argumet Sice logπe h(y1 S ) logπe(p ), there exists a α 1 [0,1] such that h(y 1 S ) = logπe( ) (5) Therefore, from () ad (4), we get D 1 σ (1 ρ ) For source S, usig DPI we observe that The rate distortio fuctio for S Also, I(S ;Y ) = h(y ) h(y S ) I(S ;Ŝ ) I(S ;Y ) (6) implies that σ log I(S D ;Ŝ ) (7) logπe(p N ) h(y S ) (8) logπe(p N ) logπe( N ) (9) = ( log 1 (1 α ) 1)P, (10) α 1 P N sice i (8), a Gaussia radom variable maximizes etropy for a give variace ad (9) is true due to the followig discussio Note that due to the physically degraded ature of the broadcast chael, Y may be writte as Y = Y 1 W where W has variace N Thus usig (5) ad etropy power iequality, we get h(y S ) = h(y 1 W S ) logπe( N ) = logπe( N ) Combiig (6), (7) ad (10), we obtai D

3 Note that the outer boud o the coditioal distortio regio is obtaied as a fuctio ofα 1 We ow state a corollary for the case of idepedet sources Corollary 1: The distortio regio for trasmissio of idepedet Gaussia sources over a degraded broadcast chael, D(,0,P,,N ), cosists of all pairs (D 1,D ) such that ad D where α 1 [0,1] Proof: We oly preset the proof for D 1 sice the result for D is the same as i the theorem above Note that I(S 1;Ŝ 1 S ) = h(s 1 S ) h(s 1 S,Ŝ 1) = h(s 1 ) h(s 1 S,Ŝ 1 ) (11) h(s 1) h(s 1 Ŝ 1) (1) logπeσ (13) D 1 where (11) uses the idepedece of S1 ad S, (1) is true because coditioig reduces etropy ad (13) follows from the rate distortio fuctio for Gaussia sources Now combiig the above with (1) ad (4), we get IV ACHIEVABLE DISTORTION REGION I this sectio, we preset achievable distortio regios for trasmittig idepedet ad correlated Gaussia sources We briefly discuss aspects of the codig scheme that are commo for both the idepedet ad correlated cases The hybrid schemes proposed i the followig subsectios are based o lattices Let Λ be a lattice of dimesio Let the quatized value ofx R,Q(x) = argmi r Λ x r The fudametal Vorooi regio of Λ is defied as V 0 = x R : Q(x) = 0 Also, we deote xmodλ = x Q(x) We choose Λ to be a good lattice for both source ad chael codig [10] ad R require it to have a secod momet costrait (Λ) = x dx V 0 = P dx RV where P will be specified later Note that the 0 trasmitter has a average power costrait P A Idepedet Gaussia Sources We ow compare a hybrid codig strategy with ucoded trasmissio for commuicatig idepedet Gaussia sources A hybrid codig scheme is basically a superpositio of coded ad ucoded trasmissio Let the distortio regio achieved by the hybrid scheme be D h (,0,P,,N ) = (D 1,D ) : D 1 ad D are achieved by the hybrid scheme Theorem : D h (,0,P,,N ) = D(,0,P,,N ) Proof: Cosider a hybrid codig scheme i which the coded portio is give by X 1 = [S 1 βs U ]modλ, where β ad are costats which will be specified later ad U is the dither which is kow apriori to both the trasmitter ad receivers ad is uiformly distributed i V 0 We sed S ucoded after scalig it appropriately to meet the power costrait I the followig, α 1 represets the power allocatio i the outer boud discussio The chael iput is a superpositio of coded ad ucoded trasmissio ad is expressed as X = αx 1 S, where α satisfies α P = α 1 P (14) ad (1 α1 )P = (15) Note thatx1 ads are idepedet of each other o accout of additio of the uiform dither before the modulo operatio We also observe that the scheme is similar to the dirty paper codig strategy i [8] where S resembles the iterferece kow at the trasmitter The output at the receivers is give by Yk = X Zk (16) for k = 1, At Receiver 1, we perform the followig series of operatios: R 1 =[δy 1 U ]modλ (17) =[(δα 1)X 1 S 1 (β δ)s δz 1 ]modλ (18) =[S 1 (δ β)s (δα 1)X 1 δz 1]modΛ (19) =[S1 W 1 ]modλ (0) where W1 = (δβ)s (δα 1)X 1 δz 1 is the effective oise term idepedet of S1 Choosig β = δ ad δ = αp α P, we reduce the variace of the effective oise to P α P Sice Λ is a good chael lattice, we require P to satisfy P α P P, (1) for correct decodig with high probability as [11][1] This leads to R 1 = [S 1 W 1 ]modλ = S 1 W 1 Allowig P to satisfy (1) with equality ad usig (14), we obtai P = α1 P ad α = α 1 P P Thus W 1 has variace P α P = σ Ŝ1 = R1, achieves a distortio i S 1 of D 1 = σ 4 = The estimator

4 O the other had, Receiver observes Y = αx 1 S Z = S W, wherew = αx1 Z is treated as the effective oise which is idepedet of S We ow costruct the liear estimator Ŝ = σ P N Y Usig (15), this estimator results i a distortio D = Thus the superpositio scheme described above achieves the optimal distortios for sources S 1 ad S We ow show that ucoded trasmissio is sub-optimal for idepedet sources through the followig theorem Let the distortio regio achieved by ucoded trasmissio be D u (,0,P,,N ) Theorem 3: D u is equal to the set of all distortio pairs (D 1,D ) such that 1 where α 1 [0,1] Further, ad D (1 α 1)P D u (,0,P,,N ) D(,0,P,,N ) Proof: The ucoded trasmissio strategy is to sed a liear combiatio of S1 ad S The power allocated for sedig S1 ad S are ad (1 α 1 )P respectively Therefore we trasmit, X α1 P = S 1 (1 α1 )P S Receiver 1 obtais a miimum mea squared error (MMSE) estimate of S1 give Y1 as Ŝ 1 = P Y1 This leads to a distortio i S 1, D 1 = 1 (1 α 1)P The estimate ofs is give byŝ = (1 α1)p PN Y resultig i D = We observe that while ucoded trasmissio scheme achieves the optimal distortio i S, the distortio i S 1 is higher tha the optimal distortio Thus D u (,0,P,,N ) D(,0,P,,N ) B Correlated Gaussia Sources Next, we exted our hybrid codig scheme to the problem of correlated sources, with the goal of achievig a better distortio regio tha source chael separatio We cosider two source chael separatio schemes i this sectio, Scheme A ad Scheme B Scheme A treats the messages obtaied by compressig S 1 ad S as idepedet ad commuicates them reliably over the broadcast chael Scheme B explores the idea of usig Wyer Ziv codig for commuicatig S 1 Let the distortio regio achieved by the hybrid scheme be D h (,ρ,p,,n ) ad the distortio regio achieved by Scheme A ad Scheme B be D A (,ρ,p,,n ) ad D B (,ρ,p,,n ) Theorem 4: If α 1 < 1 ad P < 1 α1, the α 1 D A (,ρ,p,,n ) D h (,ρ,p,,n ) For ay P > 0 ad 0 α 1 1, D B (,ρ,p,,n ) D h (,ρ,p,,n ) Proof: We begi by otig that S1 may be represeted as S1 = ρs V with V idepedet of S ad Gaussia distributed with mea zero ad variace (1 ρ ) The mai idea of the hybrid codig scheme is to use the scheme proposed i the previous subsectio to sedv ads, which are idepedet Thus, the trasmitter forms the coded portio of the chael iput similar to the idepedet case usig the lattice Λ as X 1 = [V βs U ]modλ As before, S is set ucoded ad superposed o the coded portio after a appropriate scalig to satisfy the power costrait Thus the chael iput is give by X = αx 1 S, where α ad satisfy (14) ad (15) respectively The chael output at Receiver 1 is expressed as Y 1 = αx 1 S Z 1 () The receiver ca ow perform the same sequece of operatios as i Equatios (17)-(0) to obtai R 11 = [V W 11]modΛ where W11 = (δ β)s (δα 1)X 1 δz 1 represets the effective oise idepedet of V By choosig P = (1 ρ ) α1pn1 α, α = 1P P, δ = αp ad β = δ, we get R11 = V W11, (3) where the variace of W 11 is give by σ (1 ρ) Observe that we ca rewrite () as a oisy observatio of S i the form R1 = S W 1 (4)

5 where W1 = (αx 1 Z 1 ) variace (α1pn1)σ (1 α 1)P is idepedet of S ad has Now, usig the oisy observatios of V ad S i (3) ad (4) respectively, we costruct a liear estimator of S1 give R 11 ad R 1 Before fidig the estimator, we observe that W11 ad W1 are ucorrelated for the choice of costats α ad δ stated above For each time istat i, E[W 11i W 1i ] = E[((δα 1)X 1i δz 1i ) (αx 1i Z 1i ) ] = 1 ) ((δα 1)αP δ = 0 sice δ = αp α P Therefore R 11 ad R 1 are ucorrelated as well ad the liear estimator of S 1 is give by Ŝ1 = 1 R 1 N1 11 ρ N1 (1 α 1)P R 1 The distortio resultig i S 1 is calculated to be D 1 = σ (1 ρ ) ρ Receiver obtais the estimate of S i the same fashio as the idepedet case by treatig the coded portio of the received sigal as oise to obtai a distortio Thus D h = D = (D 1,D ) : σ (1 ρ ) D ρ We ow compare the distortio regio achieved by the hybrid codig scheme with two possible source chael separatio schemes, Scheme A ad Scheme B I Scheme A, S1 ad S are compressed to obtai messages that ca be reliably trasmitted over the broadcast chael The distortio regio achieved by this scheme is give by the set D A = (D 1,D ) :,D The distortio i S 1 icurred by the hybrid scheme is smaller tha the distortio that is achieved by the above source chael separatio scheme if (1 ρ ) P < 1 α 1 α 1 ρ < Thus D A D h for α 1 < 1 ad P < 1 α1 α 1 I Scheme B, we use the represetatio S1 = ρs V, stated earlier i this sectio The trasmitter compresses V ad S to obtai messages that ca be reliably commuicated to Receivers 1 ad respectively Due to the degraded ature of the broadcast chael, Receiver 1 ca mimic Receiver to obtai a estimate of S Now Receiver 1, combies the estimates of S ad V to costruct a estimate of S1 The distortio regio thus achieved is give by D B = (D 1,D ) : σ (1 ρ ) D ρ, Hece, D B D h for P N > 0 ad 0 α 1 1 V CONCLUSIONS We preset a hybrid codig scheme for source chael commuicatio of correlated Gaussia sources over broadcast chaels, that resembles dirty paper codig We show that the scheme is optimal i terms of achievig the smallest distortio for commuicatig idepedet sources Further, we prove that for a o-trivial set of SNR, the scheme achieves a lower distortio tha source chael separatio As a ext step, we pla to compare ucoded, hybrid coded ad separately coded trasmissio schemes to determie regimes where each outperforms the others REFERENCES [1] T M Cover, Broadcast chaels, IEEE Tras Iform Theory, vol 18, o 1, pp 14, Ja 197 [] P Bergmas, A simple coverse for broadcast chaels with additive white gaussia oise, IEEE Tras Iform Theory, vol 0, o, pp 79 80, March 1974 [3] R G Gallager, Iformatio Theory ad Reliable Commuicatio Wiley, 1968 [4] K Marto, A codig theorem for the discrete memoryless broadcast chael, IEEE Tras Iform Theory, vol 5, o 3, pp , May 1979 [5] T S Ha ad M H M Costa, Broadcast chaels with arbitrarily correlated sources, IEEE Tras Iform Theory, vol 33, o 5, pp , Sep 1987 [6] S Bross, A Lapidoth, ad S Tiguely, Broadcastig Correlated Gaussias, i Proc IEEE It Symp Ifo Theory, Toroto, Caada 008 [7] D Guduz, J Nayak, ad E Tucel, Wyer-ziv codig over broadcast chaels usig hybrid digital/aalog trasmissio, i Proc IEEE It Symp Ifo Theory, Toroto, Caada 008 [8] U Erez, S S (Shitz), ad R Zamir, Capacity ad lattice strategies for cacellig kow iterferece, i Proceedigs of It Symp Iform Theory ad its Applicatios, Nov 000, pp [9] T M Cover ad J A Thomas, Elemets of Iformatio Theory Wiley, 1991 [10] U Erez, S Litsy, ad R Zamir, Lattices which are good for (almost) everythig, IEEE Tras If Theory, vol 51, pp , Oct 005 [11] U Erez ad R Zamir, Achievig 1 log(1 SNR) o the AWGN chael with lattice ecodig ad decodig, IEEE Tras Iform Theory, vol 50, pp , Oct 004 [1] Y Kochma ad R Zamir, Joit Wyer-Ziv/Dirty-Paper Codig by Modulo-Lattice Modulatio, IEEE Tras If Theory, 008, submitted for publicatio Preprit available at zamir/publicatioshtml

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