Hybrid Digitl-Anlog Coding for Interference Brodcst Chnnels Ahmd Abou Sleh, Fdy Aljji, nd Wi-Yip Chn Queen s University, ingston, O, 7L 36 Emil: hmd.bou.sleh@queensu.c, fdy@mst.queensu.c, chn@queensu.c Abstrct We consider the trnsmission of bivrite Gussin sources (V, V over the two-user Gussin brodcst chnnel in the presence of interference tht is correlted to the source nd known to the trnsmitter. Ech user i is interested in estimting V i. We study hybrid digitl-nlog (HDA schemes nd nlyze the chievble (squre-error distortion region under mtched nd expnsion bndwidth regimes. These schemes require proper combintions of power splitting, bndwidth splitting, rte splitting, nd HDA Cost coding. An outer bound on the distortion region is lso derived by ssuming knowledge of V t the second user nd full/prtil knowledge of the interference t both users. umericl results show tht the HDA schemes outperform tndem nd liner schemes nd perform close to the derived bound for certin system settings. I. ITRODUCTIO The trditionl pproch for nlog source trnsmission over noisy chnnels is to use seprte source nd chnnel coders. This pproch is well known to be optiml for point-to-point communictions. For multi-terminl systems, tndem coding is no longer optiml; joint source-chnnel coding (JSCC scheme my be required to chieve optimlity. One simple scenrio where tndem scheme is suboptiml concerns the brodcst of Gussin sources over Gussin chnnels []. For single Gussin source sent over Gussin brodcst chnnel with mtched source-chnnel bndwidth, the distortion region is known, nd cn be relized by liner scheme []. For mismtched source-chnnel bndwidth, the best known coding schemes re bsed on JSCC with hybrid signlling [] [5]. One extension to this problem is the brodcsting of two correlted sources to two users, ech of which is interested in recovering one of the two sources; in [6], it ws proven tht the liner scheme is optiml when the system s signl-to-noise rtio is below certin threshold under mtched bndwidth. In [7], hybrid digitl-nlog (HDA scheme is proposed for the sme mtched bndwidth system nd is shown to be optiml whenever the liner scheme of [6] is not; hence providing complete chrcteriztion of the distortion region. Under mismtched bndwidth, vrious HDA schemes re proposed in [8], consisting of different combintions of severl known schemes using either superposition or dirty pper coding. Recently, in [9], tndem scheme bsed on successive coding is studied nd shown to outperform the HDA schemes of [8]. In [0], the uthors investigte the trnsmission of Gussin source over correlted interference chnnel nd propose hybrid lyered scheme for point-to-point communictions. This work ws supported in prt by SERC of Cnd. In this work, we consider the trnsmission of two correlted sources over brodcst interference chnnels, where the interference is ssumed to be correlted with the sources. This brodcst system with correlted source-interference cn model situtions where two nerby nodes re trnsmitting correlted informtion simultneously. One node sends directly its signl; the other, however, hs knowledge bout its neighbour signl nd trets it s correlted interference. We propose nd nlyze HDA schemes for this system bsed on Wyner- Ziv [], Cost [] nd HDA Cost coding [3]. The rest of the pper is orgnized s follows. Section II presents the problem formultion. Section III introduces n outer bound on the system s distortion region nd some reference schemes. In Section IV, inner bounds on the distortion region under mtched nd expnsion bndwidth re studied by proposing HDA schemes. umericl results re included in Section V. Finlly, conclusions re drwn in Section VI. II. PROBLEM FORMULATIO We consider the trnsmission (Fig. of pir of correlted Gussin sources (V,V over two-user Gussin brodcst chnnel in the presence of Gussin interference S known to the trnsmitter. User i (i =, receives the trnsmitted signl corrupted by dditive white Gussin noise W i nd interference S with vrinces Wi nd S, respectively. Ech user i ims to estimte Vi =(V i (,V i (,...,V i (, where ech smple V i (j,j =,...,,is drwn from n independent nd identiclly distributed (i.i.d. Gussin. In this work, we ssume tht (V (i,v (i,s(i, i =,...,, re correlted vi the following covrince mtrix 3 V V V V S VV S = 4 5 ( V V V V S V S V S where V nd V re the vrinces of V nd V, respectively,, nd re the correltion coefficients between V nd V, S nd V nd S nd V, respectively. The covrince mtrix in ( being positive definite restricts the possible vlues of, nd. As shown in Fig., the source pir vector (, S is trnsformed into n dimensionl chnnel input X R vi (, mpping from (R R R! R. The received vector t user i is Yi = X S Wi, where ddition is component-wise, X = (,,S, S is the i.i.d. Gussin interference (S (0, S known to the trnsmitter, nd ech smple in the dditive noise Wi is drwn from n i.i.d. Gussin distribution (W i (0, Wi
α(. X S W Y Y γ (. γ (. ˆ ˆ W Fig.. System model structure. independently from both sources nd interference. The system opertes under n verge power constrint P given by E[ (,,S ]/ pple P ( where E[( ] denotes the expecttion opertor. The reconstructed signl is given by ˆV i = i (Yi, where the decoder functions i(. re mppings from R! R. The system s rte is given by = chnnel use/source symbol nd the reconstruction qulity t ech user is the men squre error (MSE D i = E[ Vi ˆV i ]/ for i =,. We ssume tht W > W, nd hence user is the wek user nd user is the strong one. For given power constrint P nd rte the distortion region is defined s the closure of ll distortion pirs ( D, D for which (P, D, D is chievble, where powerdistortion triple is chievble if for ny > 0, there exist sufficiently lrge integers nd with / =, encoding nd decoding functions (,, stisfying (, such tht D i < D i, i =,. In this work, we re interested in nlyzing the distortion region of this system under mtched ( = nd expnsion bndwidth modes ( >. ote tht for >, Vi nd the first interference smples S in S =[S,S ] re correlted vi the covrince mtrix in (, while Vi nd S re independent. III. OUTER BOUD AD REFERECE SCHEMES A. Outer Bound In [4] nd [8], n outer bound on the distortion region for sending correlted sources over the brodcst chnnel without interference ws obtined for =nd 6=, respectively, by ssuming knowledge of the source t the strong user. In [0], [5], severl bounds re derived for point-to-point communictions under correlted interference. In this section, we derive n outer bound on the distortion region for the interference brodcst chnnel for. Since S(i nd V (i re correlted for i =,...,, we hve S(i =S I (is D (i, with S D (i = S V V (i nd S I (0, ( S.To derive n outer bound, we ssume knowledge of t the strong user (this is resonble ssumption for smll correltion coefficients; this bound, however, might not be tight for high correltion vlues nd ( S,S t both users, where S = SI SD. The knowledge of the liner combintion S is motivted by [5]. The outer bound on the distortion region cn be expressed s follows Vr(V S( P W D (MSE(Y ; S(P W,D Vr(V V,S P W where [0, ], Vr(V V,S= V Vr(V S = V ( (3, nd MSE(Y ; S is the distortion from estimting Y bsed on S using liner minimum MSE estimtor (LMMSE. This distortion is function of,, E[XS I ] nd E[XS D ]. By Cuchy-Schwrtz, we hve E[XS I ] pple p p E[X ]E[SI ] nd E[XS D] pple E[X ]E[SD ]. For given nd, the mximum vlue of MSE(Y ; S hs to be used in (3. ote tht we need to mximize D over the prmeters nd. Proof: For : system with, we hve log V pple I( ; D ˆV pple I( ; Y,, S,S = I( ;, S,S I( ; Y, S,S where the first nd the second terms re nd h( h(,s = log V Vr(V V,S, h(y,s h(y,,s = log e( P W log e( W = log ( P W, (4 respectively. ote tht we used log e( W pple h(y,s pple log e(p W ; hence there exists n [0 ] such tht h(y,s = log e( P W. To get bound on estimting, we cn write the following log V pple I( ; D ˆV pple I( ; Y, S,S = I( ; S,S I( ; Y S,S (5 with the first nd the second terms stisfy nd h( h( S,S = log V Vr(V S, h(y S,S h(y,s pple h(y S h(y S h(y pple log e(mse(y ; S log e(p W,,S log e( P W respectively, where h(y,s log e( P W due to the entropy power inequlity nd since Y = Y Z with Z (0, W W. Moreover, we used h(y S pple h(y lmse( S, where lmse ( S is the LMMSE estimtor of Y bsed on S. ote tht most inequlities follow from rte-distortion theory, the dt processing inequlity, the nonnegtivity of mutul informtion, conditioning reduces differentil entropy nd the fct tht the Gussin distribution mximizes differentil entropy. Remrk: The bound in (3 reduces to the one in [4] when there is no interference. This cn be seen by setting = = 0 nd = =0. eglecting the strong user (i.e., reducing the brodcst problem to point-to-point communictions, the bound on V in (3 reduces to the bounds derived in [0], [5] for point-to-point communictions over the interference chnnel under equl bndwidth. This cn be seen by setting =0nd =for different vlues of nd.
B. Liner Scheme In this section, we ssume tht the encoder trnsforms the dimensionl sources (, into n dimensionl chnnel input X using liner trnsformtion ccording to X = A B CS (6 where A, B re mtrices nd C is mtrix. At the receiver side, we use liner decoder tht minimizes the MSE distortion. The estimted source is ˆV i = F i G i Yi, where F i is the correltion mtrix between Vi nd Yi, nd G i is the covrince mtrix of Yi, for i =,. C. Tndem Digitl Scheme This strtegy is bsed on successive coding where the sources re encoded jointly t both the common nd the refinement lyers. Using [9], the chievble source coding rte (R,R for ny distortion (D,D is given by R ( = log D ( ( pple R ( = log (7 D h where, min (,, p i ( D /, [x] = mx (x, 0, nd = D. For Gussin interference brodcst chnnel, the rte (R,R cn be chieved if nd only if there exists 0 pple pple such tht R pple log nd R pple log ( P W P ( P W. By plugging these rtes into (7, we get the chievble distortions for D nd D. The bove rtes cn be chieved vi Cost coding. ote tht this is the best tndem scheme for uncorrelted interference in terms of chievble distortion region. IV. HDA CODIG SCHEMES A. HDA Scheme for Mtched Bndwidth As shown from the encoder structure in Fig., this scheme hs four lyers tht re merged to output X. The first lyer, which uses n verge power of P u, outputs Xu = p u( V V 3 S, liner combintion of the sources nd the interference, where,, 3 [, ], nd u = P u /( V V 3 S V V 3 V S 3 V S is gin fctor relted to power constrint P u. The second lyer, which outputs X with power P, uses HDA Cost coding on the liner combintion X 0 = p Xu, where = P /P u is gin fctor relted to power constrint P. This lyer is ment for both users nd trets Xu nd S s known interference. The uxiliry rndom vrible of the HDA Cost encoder is U = X (S Xu pple X 0, where X (0,P, P = P P u, pple P W = (P P u W D, nd D is defined in (8 below. The HDA Cost encoder forms codebook U with codeword length nd R codewords (R is defined lter. Every codeword is generted following the rndom vrible U. The codebook is reveled to both the encoder nd decoder. The encoder serches for U U such tht (X 0, (S Xu,U re jointly typicl. The third lyer encodes the source using coding t rte R = log ( P P P u P P. The index W m is then encoded using Cost coding tht trets S, Xu, nd X s interference nd uses n verge power of P ; the output of this lyer is denoted by X. Similrly, in the fourth lyer the source is first encoded using t rte R 0 = P log (, where P = P P u P P, W followed by Cost coder tht trets S s well s the outputs of the first three lyers s interference nd outputs X. S β β β 3 u Encoder Encoder X HDA Cost Encoder Cost Encoder Cost Encoder X u X X X Fig.. HDA Scheme encoder structure for rte = =. X At the wek user, from the noisy received signl Y, LMMSE estimtor is used to get n estimte of X 0 denoted 0 by ˆX. The distortion D = E[(X 0 ˆX 0 ], used in the prmeter pple, is given by E[X D = P Y 0 ] p ( Pu E[X E[Y ] = P S] 0 P S W E[SX u ]. (8 With our choice of prmeters nd pple, the HDA Cost decoder cn estimte U with low probbility of error (for sufficiently lrge by serching for U such tht (U,Y 0, ˆX is jointly typicl. ote tht the HDA codeword U cn be decoded t both users. This cn be proved by noting tht the HDA Cost rte R stisfies I(U ;(S X u,x 0 pple R pple I(U ; Y i, ˆX, 0 for i =,. The HDA Cost decoder then forms LMMSE estimte of, denoted by ˆV, bsed on Y nd the decoded codeword U. The resulting distortion is given by D = V T (9 where is the covrince mtrix of [U Y ], nd is the correltion vector between V nd [U Y ]. ote tht fter some mnipultions, the distortion in (9 cn be written s D = V D x 0 v x (0 0 P where vx 0 is the correltion coefficient between X0 nd V, D nd D x 0 = P /(P P u P. A better estimte of V W is obtined from the third lyer by using the decoded Wyner- Ziv codeword T nd the previous estimte r ˆV. ote tht T = wz H, where wz = P W P P W, nd H (0,D/( P P. The overll distortion in W reproducing is then given by D D =. ( P P W
The bove distortion is found by equting log D to R. The strong user, tht is ble to decode ll codewords used by the wek user, estimtes the source by first finding liner MMSE estimte of, denoted by ˆV, bsed on the HDA Cost codeword U, the codeword T, nd Y. The distortion in reproducing is D = V T ( where is the covrince mtrix of [U Y T ], nd is the correltion vector between V nd [U Y T ].A better estimte ˆV is then found using the decoded Wyner- Ziv codeword T nd ˆV. The resulting overll distortion in estimting is given by equting log D D to R 0 s follows D = D. (3 P W The inner bound for HDA Scheme is given by ( nd (3. B. HDA Scheme for Bndwidth Expnsion D This scheme comprises two lyers tht re conctented to output the trnsmitted signl s shown in Fig. 3. The first lyer, which outputs X, consists of the HDA Scheme encoder for =(composed of four sublyers s described in the previous section. The second lyer is composed of two sublyers. The first sublyer encodes using t rte R = log ( P 0 P P 0 followed by Cost coder W with n verge power P. 0 ote tht the Cost coder trets S s interference nd outputs X tht is ment for both users. The second sublyer encodes using Wyner- Ziv t rte R 0 = log ( P 0 followed by Cost coder W with n verge power P 0 = P P 0 tht trets X nd S s interference nd outputs X. The output of the second lyer is then given by X = X X. ote tht X is the conctention of X nd X Encoder3 Encoder4 HDA Scheme Encoder Cost Encoder3 Cost Encoder4 X X Fig. 3. HDA Scheme encoder structure for rte = >.. X X X At the wek user, LMMSE decoder bsed on the decoded HDA Cost codeword U nd the first received smples Y is used to get n estimte of denoted by. The distortion in estimting cn be expressed in similr wy s given in (0. A better estimte, V 0, is then chieved using the decoder (of the HDA scheme. The resulting distortion is then D 0 D =. Using the lst P P W smples of the received signl Y, better refinement of cn be obtined using the decoder 3. The overll distortion in reconstructing is then D = D 0 P 0 P 0 W. (4 At the strong user, using the received signl Y, the HDA Cost codeword U, the decoded codeword T nd T3 of the encoder (of HDA scheme nd 3, we cn obtin n estimte of using LMMSE estimtor. The resulting distortion is D = V T, where is the covrince mtrix of [U Y T T 3 ], nd is the correltion vector between V nd [U Y T T 3 ]. ote tht T 3 = wz3 V H 3, where wz3 = s P 0 W P 0P 0 W nd H 3 (0,D. A refinement of this estimte cn be obtined using the decoder (of HDA scheme nd 4. The resulting distortion in estimting is then D = P W D P 0 W. (5 The inner bound for HDA Scheme is given by (4 nd (5. Remrk: The presented lyered schemes use purely nlog lyer tht consists of liner combintion of the sources nd the interference. The use of this lyer is to benefit from the interference when the source-interference correltion is high. V. UMERICAL RESULTS In this section, we ssume tht the source pirs, with vrince V = V =, re brodcsted to two users with interference vrince S =0dB, nd observtion noise vrince W =0dB nd W = 5 db, respectively. The system s verge power is set to P =. To evlute the performnce, we plot the inner nd outer bounds derived in the previous sections for =nd. Fig. 4 focuses on the 0 log 0 (D 0 ρ =0.8, ρ = ρ =0 λ = ρ =0., ρ = ρ =0 λ = HDA Scheme Liner scheme Tndem scheme.5.5.5 0.5 0 0 log 0 (D Fig. 4. Distortion regions for HDA Scheme for =. uncorrelted source-interference cse ( = = 0 under mtched bndwidth ( =. For low correltion between the source pirs ( =0., the proposed scheme gives some improvement over the tndem scheme nd considerbly performs very close to the outer bound (derived in Section III; for high source correltion levels ( =0.8, however, the HDA scheme outperforms the tndem system but hs lrger gp with respect to the outer bound. ote tht for uncorrelted sourceinterference, the liner scheme gives poor performnce. Fig. 5 shows the performnce of the HDA scheme for =0.5,
= =0.nd =. We cn notice tht the purely nlog scheme outperforms slightly the tndem scheme without being ble to pproch the HDA scheme. This cn be explined from the fct tht we operte t high noise levels, nd since the liner scheme cn benefit from the source-interference correltion. For the tndem scheme, which uses Cost coding, the trnsmitted signl is designed to be orthogonl to the interference; hence it cnnot exploit the source-interference correltion nd no performnce improvement cn be detected. ote tht for moderte to low noise levels, the liner scheme does not outperform the tndem scheme for low sourceinterference correltions. Moreover, from other simultions, we noticed tht for =, =0.8, nd = =0.5, the liner scheme gives the best performnce (our HDA scheme reduces to liner scheme in this cse. This cn be explined by noting tht in [6], the uthors proved tht under some conditions on the noise power nd source correltion (which re in ccordnce with the conditions for the lst simultion, the liner scheme is optiml for brodcsting bivrite Gussins under no interference. As result, under similr conditions, the liner scheme is expected to give good performnce for our problem when the source-interference correltion gets high s in [0] for point-to-point system. Fig. 6 0 log 0 (D Liner scheme Tndem digitl scheme HDA Scheme.5.5.5 0.5 0 0 log 0 (D Fig. 5. Distortion regions for HDA Scheme for =. shows tht the HDA scheme outperforms the tndem scheme under bndwidth expnsion ( =. ote tht for =0. nd = =0, it is hrd to notice (from Fig. 6 the gin of the HDA scheme over the tndem system on the plotted scle; the outer bound for this cse is not shown, since both schemes perform very closely to it. Moreover, the tndem scheme cnnot benefit from the source-interference correltion nd its performnce depends solely on in Fig. 6. VI. SUMMARY AD COCLUSIOS In this pper, we consider the trnsmission of pir of correlted Gussin sources over the two-user Gussin brodcst chnnel in the presence of interference tht is correlted to the source. We propose lyered HDA schemes under mtched nd expnsion bndwidth scenrios bsed on nd HDA Cost coding nd nlyze their inner bounds. An outer bound 0 log 0 (D 9 0 ρ =0.8 ρ = ρ =0.5 λ = ρ =0.8 ρ = ρ =0 λ = HDA Scheme ρ =0. ρ = ρ =0 λ = Tndem scheme 0 0 log 0 (D Fig. 6. Distortion regions for HDA Scheme for =. on the system s distortion region is lso derived. umericl results indicte tht the HDA schemes outperform the best tndem scheme nd perform close to the derived outer bound under some system settings. REFERECES [] M. Gstpr, B. Rimoldi, nd M. Vetterli, To code, or not to code: lossy source-chnnel communiction revisited, IEEE Trns. Inform. Theory, vol. 49, no. 5, pp. 47 58, My 003. [] U. Mittl nd. Phmdo, Hybrid digitl nlog (HDA joint source chnnel codes for brodcsting nd robust communictions, IEEE Trns. Inform. Theory, vol. 48, pp. 08 0, My 00. [3] M. Skoglund,. Phmdo, nd F. Aljji, Hybrid digitl-nlog sourcechnnel coding for bndwidth compression/expnsion, IEEE Trns. Inform. Theory, vol. 5, no. 8, pp. 3757 3763, Aug 006. [4] Z. Reznic, M. Feder, nd R. Zmir, Distortion bounds for brodcsting with bndwidth expnsion, IEEE Trns. Inform. Theory, vol. 5, no. 8, pp. 3778 3788, Aug 006. [5] V. M. Prbhkrn, R. Puri, nd. Rmchndrn, Hybrid nlogdigitl strtegies for source-chnnel brodcst, in Proc. 43rd Allerton Conf. Communiction, Control nd Computing, Allerton, IL, Sep 005. [6] A. Lpidoth nd S. Tinguely, Brodcsting correlted Gussins, IEEE Trns. Inform. Theory, vol. 56, no. 7, pp. 3057 3068, July 00. [7] C. Tin, S.. Diggvi, nd S. Shmi, The chievble distortion region of sending bivrite Gussin source on the Gussin brodcst chnnel, IEEE Trns. Inform. Theory, vol. 57, no. 0, pp. 649 647, Oct 0. [8] H. Behroozi, F. Aljji, nd T. Linder, On the performnce of hybrid digitl-nlog coding for brodcsting correlted Gussin sources, IEEE Trns. Communictions, vol. 59, no., pp. 3335 334, Dec 0. [9] Y. Go nd E. Tuncel, Seprte source-chnnel coding for brodcsting correlted Gussins, in Proc. IEEE Int. Symp. on Inform. Theory, Sint Petersburg, Russi, July 0. [0] Y.-C. Hung nd. R. rynn, Joint source-chnnel coding with correlted interference, IEEE Trns. Commun., vol. 60, no. 5, pp. 35 37, My 0. [] A. D. Wyner nd J. Ziv, The rte-distortion function for source coding with side informtion t the decoder, IEEE Trns. Inform. Theory, vol., pp. 0, Jn 976. [] M. Cost, Writing on dirty pper, IEEE Trns. Inform. Theory, vol. 9, no. 3, pp. 439 44, My 983. [3] M. P. Wilson,. R. rynn, nd G. Cire, Joint source-chnnel coding with side informtion using hybrid digitl nlog codes, IEEE Trns. Inform. Theory, vol. 56, no. 0, pp. 49 4940, Oct 00. [4] R. Soundrrjn nd S. Vishwnth, Hybrid coding for Gussin brodcst chnnels with Gussin sources, in Proc. IEEE Int. Symp. on Inform. Theory, Seoul, ore, July 009. [5] Y.-. Chi, R. Soundrrjn, nd T. Weissmn, Estimtion with helper who knows the interference, in Proc. IEEE Int. Symp. on Inform. Theory, Cmbridge, MA, July 0.