On Source and Channel Codes for Multiple Inputs and Outputs: Does Multiple Description Meet Space Time? 1

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1 On Source nd Chnnel Codes for Muliple Inpus nd Oupus: oes Muliple escripion Mee Spce Time? Michelle Effros Rlf Koeer 3 Andre J. Goldsmih 4 Muriel Médrd 5 ep. of Elecricl Eng., 36-93, Cliforni Insiue of Technoy, Psden, CA 95, effros@clech.edu 3 Coordined Science Lborory, Univ. of Illinois Urbn-Chmpign, Urbn, IL 68, koeer@uiuc.edu 4 ep. of Elecricl Eng., Snford Universiy, Snford, CA 9435, ndre@snford.edu 5 Lb. for Informion nd ecision Sysems, Msschuses Insiue of Technoy, Cmbridge, MA 39, medrd@mi.edu Absrc We compre wo sregies for lossy source descripion cross pir of unrelible chnnels. In he firs sregy, we use brodcs chnnel code o chieve differen re for ech possible chnnel relizion, nd hen use muliresoluion source code o describe he source he resuling res. In he second sregy, we use chnnel coding sregy for wo independen chnnels coupled wih muliple descripion source code. In ech cse, we choose he coding prmeers o minimize he expeced end-o-end disorion in he source reconsrucion. We demonsre h in poin-o-poin communicion cross pir of non-ergodic chnnels, muliple descripion coding cn provide subsnil gins relive o muliresoluion nd brodcs coding. We hen invesige his comprison in simple MIMO chnnel. We demonsre he inferior performnce of spce ime coding wih muliresoluion source coding nd brodcs chnnel coding relive o muliple descripion codes nd ime shring chnnel coding sregy. These resuls indice h for non-ergodic chnnels, he rdiionl definiion of chnnel cpciy does no necessrily led o he bes chnnel code from he perspecive of end-o-end source disorion. I. Inroducion We consider he problem of communicion over chnnels subjec o non-ergodic link filures. An informion heoreic invesigion becomes ricky in his domin since even he definiion of chnnel cpciy for chnnels h cn fil compleely nd for ll ime is nonrivil. More precisely, consider nework comprising single rnsmier-receiver pir. Our chnnel model involves pir of prllel links from he rnsmier o he receiver, ech of which fils wih some nonzero probbiliy. Trdiionl definiions plce he cpciy of his chnnel since wih some nonzero probbiliy here re no working links beween he rnsmier nd he receiver. Alernive definiions, including he noions of ouge nd expeced cpciy inroduced in [], plce he cpciy vlue somewh higher by llowing chnnel coding sregies where we permi relible receip of subse of he rnsmied bis. This is ypicl for cpciy definiions bu gives incresed flexibiliy in non-ergodic communicion environmens. We pproch he problem hnd wih he gol of minimizing he expeced disorion chieved in rnsmiing coninuous source cross he given chnnel. More precisely, he source is memoryless Gussin wih men nd vrince This work ws suppored in pr by NSF Grn #3534. σ, nd disorion is mesured s he squred difference beween he observed source smple nd is reconsrucion he receiver. We consider wo communicion sregies. In one sregy, we re he chnnel s hree-receiver brodcs chnnel. The hree receivers here correspond o he decoder behviors employed when boh links succeed, when excly one link fils, nd when boh links fil. We use muliresoluion source code o give single source descripion h cn be decoded whever re he decoder receives. We choose he brodcs res o he hree receivers o lie on he ouer boundry of he brodcs chnnel s cpciy region nd o minimize he expeced disorion chieved in describing he given source cross he given brodcs chnnel. The expecion is here ken wih respec o he disribuion on he res ssocied wih he hree receivers. In he oher sregy, we use muliple descripion coding o send disinc descripions over he pir of chnnels. We choose he muliple descripion code h minimizes he expeced disorion wih respec o he given disribuion on chnnel filures. For simpliciy, we begin by compring hese sregies on pir of binry links. Ech link is eiher lossless cpciy ) or compleely bsen cpciy ) for ll ime, bu which of hese behviors will occur is unknown priori. Filures of he wo links occur s independen evens of probbiliy p. Since ny sregy for communicing cross he given pir of links my be viewed s muliple descripion coding sregy, n opiml muliresoluion-brodcs code cnno chieve lower end-o-end expeced disorion hn n opiml muliple descripion code. Comprison of hese sregies is sill of ineres, hough, becuse i lends insigh ino more compliced chnnel models where he oucome of such comprison becomes less cler. MIMO chnnels provide n ineresing second exmple for compring he sregies hnd. We here model he chnnel s complex chnnel from pir of rnsmi nenns o single receive nenn. Applicion of n Almoui spce ime code effecively yields single chnnel communicion re h is unknown o he rnsmier. Time shring yields pir of independen chnnels, ech of which my fil wih some nonzero probbiliy. While spce ime coding chieves higher chnnel res, i cnno exploi he dvnges of muliple descripion source code. In conrs, independen chnnel coding yields lower chnnel res bu is beer mched o muliple descripion source coding sregy. We herefore combine he muliresoluion-brodcs sregy wih he spce ime chnnel code nd combine muliple descripion The receiver for wo link filures lwys ges re zero.

2 coding wih he ime shring chnnel code. In compring he end-o-end disorion chieved by ech sregy, we find h for low vlues of he SNR muliple descripion source codes wih independen chnnel codes ouperform he spce ime chnnel code wih muliresoluion source coding. A low vlues of he SNR, he d re increse due o spce ime coding is insufficien o surpss he dvnge of muliple descripion coding observed in he previous chnnel model. In Secion II we derive simple bu surprisingly good) lower bound on he expeced disorion in he firs chnnel model nd hen derive he precise performnce of he muliresoluion-brodcs nd muliple descripion sregies on his chnnel. Secion III conins precise chrcerizion of he second chnnel model nd he corresponding expeced disorion resuls. Secion IV discusses he implicions of our resuls nd briefly describes few exensions of his work currenly under invesigion. A Lower Bound II. Trnsmission sregies The scenrio described in Secion I corresponds o he rnsmission of wo pckes hrough wo sepre links, ech of which my fil wih probbiliy p. We cn obin simple lower bound on he chievble performnce by ssuming h he rnsmier knows beforehnd if given chnnel will fil. Thus, if he rnsmier knows h boh chnnels will succeed, i cn rnsmi chnnel re r = on ech chnnel which my correspond o source coding re R = λr = λ. The prmeer λ dps he re which source symbols re produced o he chnnel re. In his cse normlized disorion of 4λ is chievble. If i is known he rnsmier h only one pcke will be vilble he receiver, normlized disorion level of λ is chievble. Thus n verge normlized disorion of σ = p) 4λ + p p) λ + p = p) λ + p) is chievble in he scenrio of n informed rnsmier. Clerly, we expec significnly decresed performnce for n uninformed rnsmier. Muliple escripion Coding The source coding problem of wo independenly vilble error-free chnnels hs been solved for Gussin sources by El Gml nd Cover nd Ozrow in [, 3]. We wn o compre heir soluion o n lernive pproch h uses muliresoluion code in conjuncion wih brodcs rnsmission scheme. Firs we rese he resul of [, 3] specilized o our seup. We rely on he clrificion of [4] in sing h resul. Le R nd R be he per symbol res of wo descripion code. Le nd be he per symbol disorions ssocied wih receiving only he firs descripion nd only he second descripion, respecively. Finlly, le be he per symbol disorion when boh descripions re received. Then he muliple descripion re-disorion region for Gussin source wih vrince σ is given by R σ where R = R σ, R + R R, σ σ / )δ δ δ δ d ) )) d < < d σ + σ > d, δ i = σ i, d = + σ, nd d = / + / /σ ). Bsed on he given chnnel model, we re priculrly ineresed in he cse of equl res R nd R, R = R = R = λr, nd equl disorions nd. Moreover, for our purposes i is more convenien o describe he disorion s funcion of he re. We begin by considering he boundry poins = d nd = d. When = d = σ, R + R = /) σ / ), giving /σ = R) nd /σ = /) + R) ). Since /σ R) for ll descripions of re no greer hn R nd he given poin minimizes subjec o he consrin = R), he poin ) σ, = R), ) σ + R) ) is he only poin of ineres in he region d. When = d = / /σ ) = σ /σ ), we ge R = /) σ / ), giving /σ = R nd /σ = R / R ). Since no smller vlue is chievble wih he given descripion re R, he poin σ, σ ) = R R, R ) is he only poin of ineres in he region d. In he mid-region, where d < < d, he poin of ineres is R = /) [σ / )σ ) /σ ) σ + ) )]. Leing = /σ nd b = /σ, we find [ ] b = + ) ) R) Thus he mid-region yields [ ]) ) σ, =, + ) ) R) σ for ll [ R), R / R )]. We summrize he bove equions in he following proposiion. Proposiion Le n i.i.d. Gussin source wih vrince σ be described by wo descripions boh of which hve re R. The disorions nd corresponding o observions of one or boh descripions. The chievble disorion region for fixed re R is described by: σ, σ ) σ 4R ) σ R ), + Here r = is he chnnel rnsmission re 4R )) 3)

3 isorion region for symmeric muliple descripion coding re R X Y H 3 Y L E / σ.6.5 R=.5 q q.4.3. R=.75 R= Figure : A single binry degrded brodcs chnnel / σ Figure : The expeced disorion region for n i.i.d. Gussin source wih double descripion coding nd idenicl descripion res. is he disorion chieved by observing only one descripion while is he disorion chievble by observing boh. for [ R), R / R )] Figure shows he chievble disorion region for vrious vlues of R. The Ersure Chnnel Model Sepre Source nd Chnnel Coding While he rnsmission sregy employing muliple descripions is, by definiion, opiml for he given pir of unrelible chnnels, i is ineresing o noe how sysem h uses brodcs coding performs. While muliple descripion coding sends wo descripions nd devises mechnisms for coping wih he complee loss of eiher descripion, brodcs coding gurnees relible rnsmission of fixed collecion of bis o ech receiver. This is ccomplished by reing he originl seup s he degrded brodcs ersure chnnel wih memory) depiced in Figure. The cpciy region wihou common informion) for he brodcs chnnel in Figure is given by r L = IU; Y L) = Hb)) r H = IX; Y H U) = Hb) for ll b [, /]. We here rely on he fc h he cpciy for n ersure chnnel wih he given mrginl disribuion is he sme wheher or no he chnnel hs memory. In fc, ime shring gives simple wy o chieve his cpciy. uring frcion of ime equling r H, we rnsmi informion o he firs sink; during he remining ime r H), we rnsmi informion o he second sink using n ersure correcing code opimized for ersure probbiliy /. Allowing for common informion gives wih = Hb) [, ]): r L = ) r L + r H = + ). The cpciy region for pir of independen brodcs chnnels simply doubles hese res. Since he Gussin source is successively refinble [5, 6, ] he disorions chieved in he firs nd second resoluions of muliresoluion code wih incremenl res R = λr L nd R = λr H he fcor of wo comes from he fc h he wo independen chnnels ogeher give rise o wice he cpciy) re σ λ ) nd σ λ+), giving expeced disorion over he choice of chnnels: = p + p p) λ ) + p) λ+). σ In order o opimize his expression, noice h [ ] = λ p) λ p λ p) λ). σ Seing his derivive equl o zero nd checking boundry poins gives {,, /4λ)) p)/p)}. Since [, ], he ler soluion is only pplicble for p [/ + 4λ+ ), /3]. Thus /σ = min{p + p p) λ + p) λ, p +p p)+ p) 4λ } for p [, /+ 4λ+ )] /3, ] nd /σ = min{p +p p) λ + p) λ, p + p p) + p) 4λ, p + p) λ+ p p)} for p [/ + 4λ+ ), /3], giving finlly p + p p) λ + p) λ p [, ] + 4λ+ σ = p + p) λ+ p p) p [, ] + 4λ+ 3 p + p p) + p) 4λ p [, ]. 3 Figure 3 compres he performnce of he muliresoluion nd muliple descripion sregies o he bound corresponding o n informed encoder. The muliple descripion code ouperforms he muliresoluion code for ll p, ). III. MIMO Coding The findings of he previous secion indice h he scenrio of non-ergodic chnnel behvior cn open he door o significn gins in end-o-end performnce. Nex, we pply his pproch o he re of MIMO chnnels. The gol of his secion is no o give self conined invesigion of source nd chnnel coding in his conex. Rher, we wn o illusre he problems h rise wih specific exmple. As in he simples seup for spce ime coding, consider nework wih wo rnsmi nenns nd single receive nenn. Le he wo complex chnnels rnsmier o he receiver nd rnsmier o he receiver) hve complex gins h nd h. We ssume very slowly fding chnnel modeled by he ssumpion h h nd h re ime invrin les wih respec o he mpliude of he h i). We lso ssume h he receiver knows he h i. While he rndom vribles h i

4 /σ.9 λ= λ= λ= λ=.. λ= p boh hve he sme squred complex gin h + h. I is ineresing o noe h hese wo chnnels re independen of ech oher, which is due o he fc h he Almoui scheme consiues n orhogonl design. From his observion i immediely follows h we obin in ime slos nd + wo independen chnnels, ech one wih gin h + h nd herefore cpciy C = + h + h )γ/). The erm γ/ rises becuse we hve o pump energy ino boh chnnels. We see h if boh h nd h hve he sme mpliude h, hen he resuls of boh sregies re idenicl, i.e. we obin cpciy of + h γ) per muli-inpu chnnel use. If h nd h re differen, hen due o concviy of he rihm funcion Sregy he ime-shring scheme) ins smller ergodic cpciy of Figure 3: The normlized expeced disorion /σ ) chieved in describing he Gussin source cross pir of unrelible links. The dshed nd solid lines show he performnce of he muliresoluion-brodcs coding nd muliple descripion coding, respecively. The dsh-do line gives lower bound on he opiml performnce. We perform he comprison for λ vlues rnging from.5 o. would ypiclly be disribued ccording o some coninuous disribuion, we wn o consider he somewh idelized cse where he mpliude of he h i ssume only wo vlues, nmely {, }. The phse of he complex numbers h i is uniformly disribued in [, π]. Ech rnsmi nenn rnsmis complex symbol x i) ime. The oupu y of he receiver nenn hus equls: y = h x ) + h x ) + n, where he n re independen, ideniclly disribued complex Gussin rndom vribles wih vrince N. We cn use he muli-nenn chnnel in wo differen wys. Employing Sregy, we se x ) o zero in even imeslos while x ) equls zero in odd imeslos. In oher words we use ech rnsmi nenn independenly of he oher one in even nd odd ime slos, effecively creing wo independen chnnels, one wih complex gin h nd he oher one wih complex gin h. Ech chnnel hs cpciy + h i γ) where γ is defined s he signl o noise rio. In Sregy, we use spce ime code. The nurl choice for he seup described bove is o choose he Almoui scheme [7] in order o rnsmi informion for his seup. We briefly describe his scheme for compleeness. The ide is o rnsmi complex number x ) = ) he firs nenn nd noher complex number x ) = ) he second nenn. In he nex ime slo we choose x ) on on + = ) nd x ) = ). Two new complex numbers ) +, ) + re chosen imesep +. Rewriing he received signl in mrix form, we ge: ) y = y + ) ) ) ) ) ) h + h n ). n + The resul of using he Almoui scheme for informion rnsmission is h we obin wo complex chnnels which hn Sregy + h γ) + + h γ)) + h + h )γ/), which seems o fvor spce ime coding over ime shring. We emphsize h he Almoui scheme crees independen, eqully good, nd relible chnnels. While his is good sregy from chnnel cpciy poin of view, i implies h here is no room lef o exploi he dvnges of muliple descripion coding. On he oher hnd, Sregy, while inferior from cpciy poin of view, opens he possibiliy of employing muliple descripion coding. In order o illusre he rde-offs involved, we nex compre he expeced disorion performnces under he ssumpion h he mpliudes h i re drwn independenly from he Bernoulli disribuion wih Pr h i = ) = p nd Pr h i = ) = p. Sregy : Time Shring & Muliple escripion Coding By Proposiion, re R = λ + γ) he muliple descripion code chieves normlized expeced disorion σ = p) + p p)b + p, where [ + γ) 4λ, + γ) λ / + γ) λ )] nd b = + + γ) 4λ. Numericl opimizion for s funcion of p, γ, nd λ yields he solid curves in Figures 4 nd 5. Sregy : Spce Time, Brodcs, & Muliresoluion Coding In his cse we hve pir of chnnels res + γ/) when h = nd h = or vice vers), nd pir of chnnels res + γ) when h = h =. Since he chnnel cpciy in operion is unknown o he sysem encoder, we employ brodcs code o simulneously rnsmi cross his pir of possible chnnels. We refer o Cover nd Thoms for he cpciy region of he Gussin brodcs chnnel. 3 3 The brodcs chnnel in quesion here is degrded

5 Muliple descripion solid lines) vs spce ime dshed lines) coding for end o end disorion Muliple descripion solid lines) vs spce ime dshed lines) coding for end o end disorion M /σ, MR /σ p= p=. p=. p=.3 p=.4 p=.5 p=.6 p=.7 p=.8 p=.9 p= γ M /σ, MR /σ 3 4 p= p=. p=. p=.3 p=.4 p=.5 5 p=.6 p=.7 p=.8 p=.9 p= γ Figure 4: Spce ime nd muliple descripion resuls for λ =.5. Figure 5: Spce ime nd muliple descripion resuls for λ =. The resuling cpciy region cross ech chnnel is chrcerized by:.6 Muliple descripion solid lines) vs spce ime dshed lines) coding for end o end disorion r L + r H + αγ) + + α)/α + /γ))) r L + α)/α + /γ))) for α [, ]. Employing muliresoluion source code incremenl res R = λr L nd R = λr H gives disorion σ R +R ) = σ + αγ) 4λ + α)/α + /γ)) 4λ when h = h = nd σ R = σ + α)/α + /γ)) 4λ when h, h ) {, ),, )} by he successive refinbiliy of he Gussin source. As resul, he end o end expeced normlized disorion of his pproch equls ) = p) + αγ) 4λ + α 4λ σ α + /γ ) +p p) + α 4λ + p. α + /γ Numericl opimizion over he choice of α for ech vlue of γ nd p yields he dshed curves in Figure 4 nd 5. The difference beween Sregy which uses muliresoluion source coding nd spce ime chnnel coding) nd Sregy which uses ime shring nd muliple descripion coding) for ech vlue of p ppers in Figures 6 nd 7. The resuls demonsre h he Almoui scheme is subopiml sregy if we re ineresed in end-o-end disorion. 4 IV. iscussion The resuls of he comprison described in Secion III re illusred in Figures 4-7. For fixed prmeer p we cn disinguish four regions in Figure 6 nd Figure 7. A very low SNRs boh schemes chieve similr normlized disorion σ very close o one since he source descripion res re jus oo smll. In priculr, he difference beween he wo schemes becomes very smll. As he SNR vlues increse we begin 4 This holds les for low vlues of γ. M /σ MR /σ p= p=. p=. p=.3 p=.4 p=.5 p=.6 p=.7 p=.8 p=.9 p= γ/[db] Figure 6: The difference beween muliple descripion nd spce ime resuls for λ =.5. M /σ MR /σ Muliple descripion solid lines) vs spce ime dshed lines) coding for end o end disorion p= p=. p=. p=.3 p=.4 p=.5 p=.6 p=.7 p=.8 p=.9 p= γ/[db] Figure 7: The difference beween muliple descripion nd spce ime resuls for λ =..

6 o observe non-vnishing descripion res nd he muliple descripion code begins o show beer performnce. In priculr, in his rnge of SNR vlues he spce ime code does no ye provide he subsnil gins h we would expec o see higher SNR vlues. As he SNR increses he incresed muul informion provided by he spce ime code begins o ouweigh he muliple descripion gins of Sregy. Evenully, for very high SNRs he chievble disorion becomes ouge deermined nd boh schemes converge o disorion = p σ. Their bsolue disorion difference vnishes gin. While he shown curves re specific for he Almoui scheme, we would expec similr behvior for generl spce ime codes. In priculr, he filure of spce ime codes in he low SNR regime cn be inerpreed in he conex of exising resuls on MIMO cpciy in he low SNR regime. Indeed, i is known h, in he low SNR regime, he cpciy of MIMO sysem scles s msnr, where m is he number of receive nenns [8, 9], for boh coheren nd non-coheren chnnel cses. In h cse, rnsmi nenns do no provide ny dded benefi from he poin of view of cpciy nd he MIMO sysem cs s single rnsmier, single receiver chnnel. The ypes of chnnel codes h chieve cpciy in h cse re rdiionl single sender, single receiver codes opering wih single descripion code. The role of he receive nenns is no o cquire dded refinemen in informion, bu merely o hrves s much energy s possible. Indeed, he inerference mong rnsmi nenns, which spce ime codes seek o mnge hrough orhogonliy, is negligible wih respec o he effec of he noise. In his rnge of SNR vlues muliple descripion coding ppers o be n inriguing lernive in order o cpilize on chnnel diversiy; muliple descripion codes should offer n effecive wy o hndle differen non-ergodic chnnel relizions. Noe h he scling wih SNR nd number of nenns is logeher differen from he high SNR cse, for which spce ime codes re designed. For sufficienly coheren chnnels, cpciy is roughly minm, n) SNR), where n is he number of rnsmi nenns [9]. The rihmic dependence on SNR, s well s he dependence on boh rnsmi nd receive nenns, suggess sysem which is limied by degrees of freedom rher hn energy []. In his regime, while he muliple descripion code is sill good sregy, he dvnges of spce ime coding wih respec o chievble res simply overwhelm he gins offered by he muliple descripion pproch. I is ineresing o noe h he incresed chnnel res re due o diversiy gin nd s such hey drw on he sme resource s he muliple descripion codes. In priculr, rding off he vilble diversiy wih respec o diversiy gin nd muliple descripion gin seems o be very inriguing problem. Combining his problem wih he design issues of muliple receive nd rnsmi nenn codes ppers o hold mny chllenges. As finl hough we would like o emphsize h we believe he pper ddresses much bigger problem hn muliple descripion coding over MIMO chnnels. In fc, MIMO is jus specil cse of exploiing non-ergodic diversiy h my be offered by prllel chnnels. This scenrio ppers for exmple lso in prllel chnnels hrough mcrodiversiy or sof hndoff, or prllel roues hrough wireless or wired nework. In ll of hese scenrios i ppers h he vilble diversiy my be eiher used o sbilize connecion, hus enbling higher rnsmission res, or i my be used for decresed end-o-end disorion by using muliple descripion code. References [] M. Effros nd A. J. Goldsmih. Cpciy definiions nd coding sregies for generl chnnels wih receiver side informion. In Proceedings of he IEEE Inernionl Symposium on Informion Theory, pge 39, Cmbridge, Msschuses, Augus 998. IEEE. [] A. A. El Gml nd T. M. Cover. Achievble res for muliple descripions. IEEE Trnscions on Informion Theory, IT- 86):85 857, November 98. [3] L. Ozrow. On source-coding problem wih wo chnnels nd hree receivers. Bell Sysem Technicl Journl, 59:446 47, ecember 98. [4] H. Feng nd M. Effros. On he chievble region for muliple descripion source codes on Gussin sources. In Proceedings of he IEEE Inernionl Symposium on Informion Theory, pge 95, Yokohm, Jpn, June 3. IEEE. [5] R. M. Gry nd A.. Wyner. Source coding for simple nework. Bell Sysems Technicl Journl, 539):68 7, November 974. [6] V. Koshelev. An evluion of he verge disorion for discree schemes of sequenil pproximion. Probl. Pered. Inform., 73): 33, 98. [7] S. M. Almoui, A simple rnsmier diversiy scheme for wireless communicions, IEEE J. Selec. Ares Commun., vol., pp , Ocober 998. [8]. Tse, P. Viswnh, Fundmenls of Wireless Communicions, preprin of book, hp:// dse/min.pdf [9] L. Zheng nd. N. C. Tse, Communicion on he Grssmn mnifold: A geomeric pproch o he noncoheren muliplenenn chnnel, IEEE Trnscions on Informion Theory, Vol. 48, pp , Februry. [] M. Médrd, C. Luo, Frequency-shif keying for ulrwidebnd - chieving res of he order of cpciy, 4h Annul Alleron Conference on Communicion, Conrol, nd Compuing,