Hybrid Digital-Analog Joint Source-Channel Coding for Broadcasting Correlated Gaussian Sources

Transcription

1 Hybrid Digitl-Alog Joit ource-chel Codig for Brodcstig Correlted Gussi ources Hmid Behroozi, Fdy Aljji d Tmás Lider Deprtmet of Mthemtics d ttistics, Quee s iversity, Kigsto, Otrio, Cd, K7L 3N6 Emil: {behroozi, fdy, lider}@mstqueesuc Abstrct We cosider the trsmissio of bivrite Gussi source =, cross power-limited two-user Gussi brodcst chel ser i i =, observes the trsmitted sigl corrupted by Gussi oise with power i d wts to estimte i We study hybrid digitl-log HDA joit source-chel codig schemes d lyze these schemes to obti chievble squred-error distortio regios Two cses re cosidered: source d chel bdwidths re equl, brodcstig with bdwidth compressio We dpt HDA schemes of Wilso et l [] d rbhkr et l [] to provide vrious chievble distortio regios for both cses sig umericl exmples, we demostrte tht for bdwidth compressio, three-lyered codig scheme cosistig of log, superpositio, d codig performs well compred to the other provided HDA schemes I the cse of mtched bdwidth, three-lyered codig scheme with log lyer d two lyers, ech cosistig of Wyer-Ziv coder followed by coder, performs best I INTRODCTION This pper cosiders brodcstig correlted Gussi sources d ims to chrcterize me squred-error ME distortio pirs tht re simulteously chievble t two receivers usig hybrid digitl-log HDA codig schemes It is kow tht the seprte desig of source d chel codig due to ho does ot i geerl led to the optiml performce theoreticlly ttible OTA i etworks O the other hd, for the poit-to-poit trsmissio of sigle Gussi source through dditive white Gussi oise AWGN chel it is well kow tht if the chel d source bdwidths re equl, simple ucoded trsmissio chieves OTA coded or log trsmissio i this cse d i the rest of this pper mes sclig the ecoder iput subject to the chel power costrit d trsmittig it without explicit chel codig I order to exploit the dvtges of both log trsmissio d digitl techiques, vrious HDA schemes hve bee itroduced i the literture, see eg, [], [3] [9] Brodcstig sigle memoryless Gussi source uder bdwidth mismtch usig HDA schemes is cosidered i [5], [8] Bross et l [] show tht there exists cotiuum of HDA schemes with optiml performce for the trsmissio of Gussi source over verge-power-limited Gussi chel with mtched bdwidth Ti d hmi [] geerlize this result to the mismtched bdwidth cse Brodcstig Gussi source with memory is lyzed i [9] This work ws supported i prt by ostdoctorl Fellowship from the Otrio Miistry of Reserch d Iovtio MRI d by the Nturl cieces d Egieerig Reserch Coucil NERC of Cd Bi vrite ource Ecoder V V Receiver Receiver Fig Brodcstig bivrite Gussi source over two-user powerlimited Gussi brodcst chel Our system model is illustrted i Fig We im to determie chievble distortio regios usig HDA schemes for two cses; the source bdwidth equls the chel bdwidth, brodcstig with bdwidth compressio To our kowledge, prt from [] i which Bross et l lyzed ucoded trsmissio for brodcstig correlted Gussi sources, o explicit distortio-regios hve bee estblished i the literture for brodcstig correlted Gussi sources We re lso ot wre of y prior work o HDA schemes for brodcstig correlted Gussis either whe the source d chel bdwidths re equl or whe there is bdwidth mismtch Note tht the source-chel seprtio theorem does ot hold i brodcstig correlted sources II ROBLEM TATEMENT We cosider brodcstig bivrite Gussi source cross two-user power-limited Gussi brodcst chel ser i i =, receives the trsmitted sigl corrupted by Gussi oise with power i d wts to estimte the ith compoet of the source We ssume > d cll user the wek user d user the strog user Let d be correlted Gussi rdom vribles d let { t, t} t= be sttiory Gussi memoryless vector source with mrgil distributio tht of, We ssume tht t d t hve zero me d vrice d, respectively, d correltio coefficiet ρ, We represet the first source smples by the dt sequeces = {,,, } d = {,,, }, respectively The system is show i Fig The source sequeces d re joitly ecoded to = ϕ,, where the ecoder fuctio is of the form ϕ : R R R The trsmitted sequece is verge-power limited to >, ie, E t= [ t ] ser i observes the trsmitted sigl t corrupted by Gussi oise V i t with power i so tht ech observtio time t =,,3, receiver i observes i t = t + V i t, i =,

2 where the V i t N,i re idepedetly distributed over i d t, d re idepedet of the t Bsed o its chel output i, user i provides estimte Ŝi = ψ i i, where ψ i : R R is decodig fuctio The qulity of the estimte is mesured by the verge ME distortio i = E[ i t Ŝit ] Let F t= deote ll ecoder d decoder fuctios ϕ,ψ,ψ defied s bove For prticulr codig scheme ϕ,ψ,ψ, the performce is determied by the chel power costrit d the icurred distortios d t the receivers For y give power costrit >, the distortio regio D is defied s the covex closure of the set of ll distortio pirs D,D for which,d,d is chievble, where power-distortio pir,d,d is chievble if for y δ >, there exists δ such tht for y δ there exists ϕ,ψ,ψ F with distortios i D i + δ i =, III DITORTION REGION WITH MATCHED BANDWIDTH A coded Trsmissio I [] for the bove problem chievble distortio regio is obtied bsed o lyzig the ucoded trsmissio i brodcstig bivrite Gussi source I this pproch, lier combitio of both compoets of bivrite Gussi source is trsmitted cross powerlimited Gussi brodcst chel The trsmitted sigl c be expressed s t = α i i t, 3 where α = Vr i it i= i=, i d Vr i i t = i= + + ρ The scle fctor α is chose such tht the chel power costrit is stisfied with equlity The received sigl t receiver i is the give by i t = t + V i t = α i i t + V i t 4 i= By evlutig the resultig ME distortio, the set of simulteously chievble distortio pirs t two users re s follows: D i = i α i i + j ρ i j + i, i,j =,, j i 5 It is show i [] tht the ucoded scheme is optiml below certi NR-threshold B Joit ource-chel Codig chemes I our schemes, we will closely follow the ottio d code costructios i [] Here we oly give high-level descriptio d lyses of the schemes without detiled proofs I prticulr, i my steps of the lysis we tret fiite-blocklegth codig schemes s idelized systems with symptoticlly lrge blocklegths Lyerig with Alog d Codig: This codig scheme hs three lyers d is similr to the scheme i [] for brodcstig sigle memoryless Gussi source The oly differece betwee the two schemes is tht we use Wyer-Ziv ecoder followed by ecoder i the secod lyer, while the secod lyer of the scheme i [] employs HDA coder which will be explied i ectio IV-A Block digrms of the ecoder d the decoder re show i Fig The first lyer is the log trsmissio lyer Here t = α i i t, where α = i= This lyer is met for both strog d Vr i it i= wek users Now fix d to stisfy = + + I the secod lyer, the first compoet of the source is first Wyer-Ziv coded t rte R = log + + usig estimte of t the receiver s side iformtio The Wyer-Ziv idex, m {,,, R }, is the ecoded usig s dirty pper codig tretig the log trsmissio lyer,, s iterferece Let be uxiliry rdom vrible give by = +α, where N, is idepedet of N, d the sclig fctor α is set to be + + We geerte legth iid Gussi codebook with I; codewords, where ech compoet of the codeword is Gussi with zero me d vrice + α, d ech codeword is the rdomly plced ito oe of R bis Let i be the idex of the bi cotiig For give m, we look for such tht i = m d d re joitly typicl The, we trsmit = α, where is met to be decoded by the wek user I the third lyer, which is met for the strog user, the secod compoet of the source,, is lso Wyer Ziv coded t rte R = log + usig the estimte of t the receiver s side iformtio The Wyer-Ziv idex, m {,,, R }, is the ecoded usig digitl codig tht trets both d s iterferece d uses power Let be uxiliry rdom vrible give by = + α +, where N,, d re idepedet from ech other d α = + Here we lso crete legth iid Gussi codebook with I; codewords, where ech compoet of the codeword is Gussi with zero me d vrice + α + d rdomly evely distribute them over R bis Let i be the idex of the bi cotiig For give m, we look for such tht i = m d,, re joitly typicl The, we trsmit = α + As show i Fig, we merge ll three lyers d trsmit = + + A chievble distortio-regio c be obtied by vryig, d subject to = + + For give, d, the chievble distortio pirs c be computed s follows At the receiver Fig b, estimte of the first compoet of the source,, is first obtied from the log lyer This estimte cts s side iformtio tht c be used i refiig the estimte of for the wek user usig the R decoded Wyer-Ziv bits obtied by the decoder of the secod lyer ice R equls the cpcity of the chel with kow iterferece t the ecoder oly, I ; I ; = log + the distortio i estimtig t the wek user is give by the Wyer-Ziv distortio-rte fuctio, D R, where D = E[ E[ ] ] is the idelized MME from +,

3 α 5 Alog Trsmissio Alog, uperpositio d Codig Alog d Codig Bi vrite ource Wyer Ziv m Ecoder Wyer Ziv m Ecoder Ecoder Ecoder Ecoder log D 5 5 V V MME Estimtor Decoder Decoder Decoder b Decoder Wyer Ziv Decoder Wyer Ziv Decoder MME Estimtor Fig Brodcstig bivrite source, by doptig the lyerig scheme with log d codig lyers i [] the received o the overll distortio see t the wek, user c be expressed s D = D + + where D = α + ρ The, estimte of c be determied from the first d the secod lyers This estimte cts s side iformtio for estimtig for the strog user from the R decoded Wyer-Ziv bits Here, gi, R equls the cpcity of the chel with kow iterferece, d, t the ecoder oly, ie, R = I ; I ;, = log+ Thus, the distortio i estimtig t the strog user is give by the Wyer-Ziv distortio-rte fuctio, D R, where D is the MME from the received d the decoded o the overll distortio for the strog user, is give by D = D + where D = Γ T Υ Γ, d Υ = Γ = α + ρ α α, + ρ α + α + α 7 Lyerig with Alog, uperpositio d Codig: This scheme lso hs three codig lyers: log, superpositio, d codig I the secod lyer, we hve two merged strems, similr to the cse of brodcstig sigle memoryless source over brodcst chel [4], [3] The first compoet of the source is brodcsted to two users The first source ecoder is optiml Wyer-Ziv ecoder with rte R = log + λ λ + + +, d the secod source ecoder is optiml Wyer-Ziv ecoder for the residul error of the first ecoder with rte R R = log + λ + + The, we ecode the Wyer-Ziv bits with cpcity-chievig chel codes d trsmit with log D Fig 3 Distortio regios i brodcstig bivrite source with the correltio coefficiet ρ = powers λ d λ, respectively ice we require rte of oe chel use per source symbol, d the Gussi source is successively refible, by combiig the Wyer-Ziv rte-distortio fuctio with the pir of chievble rtes for brodcst chel R,R, the correspodig chievble distortio pirs re [4]: D R d D R, where D is give i 6 The codig scheme i the third lyer is similr to tht i the previous scheme The fil distortio i estimtig t the wek user is D = D D R = + λ λ At the strog user, first estimte of the first compoet of the source c be obtied withi distortio D = D R D = R + λ + + = + D λ + + The we obti estimte of from the bove estimte of with the followig distortio: D = ρ D 9 This estimte of cts s side iformtio i refiig the estimte of for the strog user usig the decoded Wyer-Ziv bits The overll distortio for the strog user i estimtig is thus give by D = D + 3 Numericl Exmple: We trsmit smples of bivrite [ Gussi ] source with the covrice mtrix Λ = i uses of power-limited brodcst chel to two users with observtio oise vrices = 5dB d = db, respectively The two-user brodcst chel hs the power costrit = db The boudries of the distortio regios for the schemes preseted i this sectio re show i Fig 3 We observe tht the lyerig with log trsmissio d codig outperforms ll other JCC schemes, icludig log trsmissio IV DITORTION REGION WITH BANDWIDTH COMREION We ext cosider the problem of brodcstig bivrite Gussi source with : bdwidth compressio We wt to trsmit k = smples of bivrite Gussi source k, k i uses of power-limited brodcst chel to

4 Bi vrite ource k= k k,, k, k, Hybrid Digitl Alog HDA Ecoder Wyer Ziv Ecoder α Ecoder Fig 4 Brodcstig bivrite source k, k with bdwidth compressio usig three-lyered codig provided i [] two users The two-user brodcst chel hs the power costrit We split both compoets of the bivrite Gussi source ito two equl legth prts, ie, we split smples of ech source vector i ito two vectors of legth : i, d i, A Lyerig with Alog, HDA d Codig This scheme is itroduced i [] for brodcstig memoryless Gussi source with bdwidth compressio; see Fig 4 I the first log trsmissio lyer, lier combitio of the first smples of the bivrite Gussi source compoets re scled such tht the power of the trsmitted sigl i this lyer becomes Here t = α i i, t where α = i= + + ρ I the secod d the third lyers, we work o the remiig smples of the source compoets, ie,, d,, respectively I the secod lyer, we pply the HDA codig, preseted i [], to, i order to produce with power Here, the source is ot explicitly qutized d it ppers i log form i the trsmitted sigl [] Let be uxiliry rdom vrible give by = + α + K,, where N,, N,, d, re idepedet of ech other, α = + +, d K = + + As i [], we geerte rdom iid codebook with R codewords, where ech compoet of ech codeword is Gussi with zero me d vrice + α + K d R = log +α +K For give, d, we fid such tht,,, is joitly typicl d trsmit = α K, I the third lyer, smples of the secod compoet of the source,, re Wyer Ziv coded t rte R = log + usig the estimte of, t the receiver s side iformtio The Wyer-Ziv idex is the ecoded usig codig tht trets both d s iterferece d uses power = The code costructio s well s the ecodig d decodig procedures re logous to the oes described i ectio III-B Therefore, we trsmit = α + We merge ll three lyers d trsmit = + + At the decoder, we look for tht is joitly typicl with The wek user estimtes k =,, by MME estimtio from the received sigl d the decoded Thus, the overll distortio see t the wek user is []: D = k D + k D = D + D, where D j j =,, the MME distortio i estimtig,j from d, is give by where d Γ = Υ HDA = D j = Γ T jυ HDA Γ j, α + ρ α α, Γ + ρ = K, α + α + α + K The, estimte of k is obtied from the first d the secod lyers This estimte cts s side iformtio for estimtig for the strog user usig the decoded Wyer- Ziv bits The strog user estimtes the secod compoet of the source k =,, from, the decoded d Hece the overll distortio for the strog user is give by D = D + D, where D j j =,, the distortio i estimtig,j, is determied vi the Wyer- Ziv distortio-rte fuctio: where Γ = d D j = Γ T jυ HDA Γ j + j, Υ HDA = α + ρ α α,γ + ρ =, K ρ α + α + α + K B Lyerig with Alog d Codig Here, we lso use three codig lyers d they re the sme s the oes i ectio IV-A, except for the secod lyer I the secod lyer, the smples of the secod hlf of the first compoet of the source,,, re qutized t rte R = log + + The qutiztio idex is the ecoded usig codig tht trets s iterferece d uses power Therefore, we trsmit = α, where α = + + We merge ll three lyers d trsmit = + + At the receiver, the wek user estimtes =,, by MME estimtio from the received sigl d the decoded Thus the overll distortio see t the wek user is give by D = Γ T Υ Γ +, where Γ is give i d + Υ = α + α + α The strog user estimtes the secod compoet of the

5 source =,, withi the overll distortio Γ T Υ Γ D = + ρ D + 4 where Γ is give i, Υ is provided i 7 d D = C Lyerig with Alog, uperpositio d Codig Alogously to the previous codig schemes, this scheme is three-lyered with its lyers ideticl to the oes preseted i ectio IV-A, except for the secod lyer I the secod lyer, s i ectio III-B, we use two merged strems The secod prt of the first compoet of the source,,, is brodcsted to two users The first source ecoder is optiml source ecoder with rte R log+ λ = λ + + +, d the secod source ecoder is optiml ecoder for the residul error of the first ecoder with rte R R = λ log+ + + The, we ecode the qutiztio bits with cpcity-chievig chel codes d trsmit the resultig strems uder powers λ d λ, respectively The wek user forms MME estimte of with the followig distortio: D = α + ρ λ λ λ At the strog user, first estimte of, c be obtied withi distortio D = + + λ + + λ λ This estimte cts s side iformtio for obtiig the estimte of, usig the decoded Wyer-Ziv bits The resultig distortio for the strog user is thus give by D = Γ T Υ Γ + ρ D + 6 Filly, ote tht if we set ρ = d =, the the results of [], [9], which curretly pper to be the best kow results for brodcstig Gussi source with bdwidth compressio, re obtied D Numericl Results We trsmit k = smples of bivrite Gussi ρ source, k k with the covrice mtrix Λ = ρ i uses of power-limited brodcst chel to two users with observtio oise vrices = 5dB d = db, respectively The distortio regios for the schemes preseted i this sectio re show i Fig 5 for two differet correltio coefficiets, ρ = d ρ = 8 We observe tht the lyerig with log, superpositio log D 5 5 Alog d Codig Alog, HDA d Codig Alog, uperpositio d Codig ρ=8 ρ= log D Fig 5 Distortio regios of differet HDA codig schemes ystem prmeters re = db, = 5dB d = db d codig of ectio IV-C outperforms ll other schemes i both cses Whe the source compoets re highly correlted, lyerig with log, HDA, d codig scheme performs better th the lyerig with log d codig scheme; however, the two two schemes perform similrly for smll vlues of the correltio coefficiet REFERENCE [] M Wilso, K Nry, d G Cire, Joit source chel codig with side iformtio usig hybrid digitl log codes, i roc IEEE If Theory d Applictios ITA Workshop, L Joll, CA, J 7, pp [] V M rbhkr, R uri, d K Rmchdr, Colored Gussi source-chel brodcst for heterogeeous log/digitl receivers, IEEE Trs If Theory, vol 54, o 4, pp 87 84, Apr 8 [3] hmi, Verdu, d R Zmir, ystemtic lossy source/chel codig, IEEE Trs If Theory, vol 44, o, pp , Mr 998 [4] B Che d G W Worell, Alog error-correctig codes bsed o chotic dymicl systems, IEEE Trs Commu, vol 46, o 7, pp 88 89, Jul 998 [5] Mittl d N hmdo, Hybrid digitl-log HDA joit sourcechel codes for brodcstig d robust commuictios, IEEE Trs If Theory, vol 48, o 5, pp 8, My [6] esi, G Cire, d G Vivier, Lossy trsmissio over slowfdig AWGN chels: compriso of progressive, superpositio d hybrid pproches, i roc IEEE IIT, Adelide, Austrli, ep 5 [7] M koglud, N hmdo, d F Aljji, Hybrid digitl-log source-chel codig for bdwidth compressio/expsio, IEEE Trs If Theory, vol 5, o 8, pp , Aug 6 [8] Z Rezic, M Feder, d R Zmir, Distortio bouds for brodcstig with bdwidth expsio, IEEE Trs If Theory, vol 5, o 8, pp , Aug 6 [9] V M rbhkr, R uri, d K Rmchdr, Hybrid logdigitl strtegies for source-chel brodcst, i roc 43rd Allerto Cof Commu, Cotr, Comput, Allerto, IL, ep 5 [] Bross, A Lpidoth, d Tiguely, uperimposed coded d ucoded trsmissios of Gussi source over the Gussi chel, i roc IEEE IIT, ettle, WA, Jul 6, pp [] C Ti d hmi, A uified codig scheme for hybrid trsmissio of Gussi source over Gussi chel, i roc IEEE IIT, Toroto, ON, Jul 8 [] Bross, A Lpidoth, d Tiguely, Brodcstig correlted Gussis, i roc IEEE IIT, Toroto, ON, Jul 8 [3] M C Gstpr, eprtio theorems d prtil orderigs for sesor etwork problems, I ligrm, Vektesh Ed, Networked esig Iformtio d Cotrol, priger, 8