Equalization on Graphs: Linear Programming and Message Passing

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1 Equalzaon on Graphs: Lnear Programmng and Message Passng Mohammad H. Taghav and Paul H. Segel Cener for Magnec Recordng Research Unversy of Calforna, San Dego La Jolla, CA , USA Emal: (maghav, Absrac We propose an approxmaon of maxmumlkelhood deecon n ISI channels based on lnear programmng or message passng. We conver he deecon problem no a bnary decodng problem, whch can be easly combned wh LDPC decodng. We show ha, for a ceran class of channels and n he absence of codng, he proposed echnque provdes he exac ML soluon whou an exponenal complexy n he sze of channel memory, whle for some oher channels, hs mehod has a non-dmnshng probably of falure as SNR ncreases. Some analyss s provded for he error evens of he proposed echnque under lnear programmng. I. INTRODUCTION Inersymbol nerference (ISI) s a characersc of many daa communcaons and sorage channels. Sysems operang on hese channels employ error-correcng codes n conjuncon wh some ISI reducon echnque, whch, n magnec recordng sysems, s ofen a convenonal Verb deecor. I s known ha some gan wll be obaned f he equalzaon and decodng blocks are combned a he recever by exchangng sof nformaon beween hem. A possble approach o achevng hs gan s o use sof-oupu equalzaon mehods such as he BCJR algorhm [1] or he sof-oupu Verb algorhm (SOVA) [2] along wh erave decoders. However, boh BCJR and SOVA suffer from exponenal complexy n he lengh of he channel memory. Kurkosk e al. [3] proposed a b-based and a sae-based graph represenaon of he ISI channel ha can be combned wh he Tanner graph of a low-densy pary-check (LDPC) code for jon message-passng (MP) decodng. They showed ha he b-based mehod suffers from a sgnfcan performance degradaon due o he abundance of 4-cycles, bu he sae-based mehod has a performance and overall complexy smlar o BCJR, whle benefng from a parallel srucure and reduced delay. Lnear programmng (LP) has been recenly appled by Feldman e al. [4] o he decodng of LDPC codes, as an alernave o message-passng echnques. In hs mehod, he bnary pary-check consrans of he code are relaxed o a se of lnear consrans n he real doman, hus urnng he neger problem no an LP problem. Whle LP decodng performs closely o message-passng algorhms such as he sum-produc algorhm (SPA) and he mn-sum algorhm (MSA), s much easer o analyze for fne code lenghs. Fg. 1. Bnary-npu ISI channel. Movaed by he success of LP decodng, n hs work we sudy he problem of ML deecon n he presence of ISI, whch can be wren as an neger quadrac program (IQP). We conver hs problem no a bnary decodng problem, whch can be used for message-passng decodng, or, afer relaxng he bnary consrans, LP decodng. Furhermore, decodng an underlyng LDPC code can be ncorporaed no hs problem smply by ncludng he pary checks of he code. By a geomerc analyss we show ha, n he absence of codng, f he mpulse response of he ISI channel sasfes ceran condons, he proposed LP relaxaon s guaraneed o produce he ML soluon a all SNR values. Ths means ha here are ISI channels, whch we call LP-proper channels, for whch uncoded ML deecon can be acheved wh a complexy polynomal n he channel memory sze. On he oher end of he specrum, some channels are LP-mproper,.e. he LP mehod resuls n a nonnegral soluon wh a probably bounded away from zero, even n he absence of nose. Furhermore, we observe some nermedae asympocally LP-proper channels where he performance approaches ha of ML deecon a hgh SNR. When message passng s used nsead of LP, we observe a smlar behavor. Moreover, when LDPC decodng s ncorporaed n he deecor, LPproper channels acheve very good performance, whle some oher channels canno go below a ceran word error rae. The res of hs paper s organzed as follows. In Secon II, we descrbe he channel, and nroduce he LP relaxaon of ML deecon. The performance analyss s presened n Secon III. Smulaon resuls are gven n Secon IV, and Secon V concludes he paper. II. RELAXATION OF THE EQUALIZATION PROBLEM A. Channel Model We consder a paral-response (PR) channel wh bpolar (BPSK) npus, as descrbed n Fg. 1, and use he followng noaon for he ransmed symbols /07/$25.00 c 2007 IEEE 2551

2 Noaon 1: The bpolar verson of a bnary symbol, b {0, 1}, s denoed by b { 1, 1}, and s gven by b = 1 2b. (1) The paral-response channel ransfer polynomal s h(d) = µ =0 h D, where µ s he channel memory sze. Thus, he oupu sequence of he PR channel n Fg. 1 before addng he whe Gaussan nose can be wren as µ y = h x. (2) =0 B. Maxmum-lkelhood (ML) Deecon Havng he vecor of receved samples r = [r 1 r 2 r n ] T, he ML deecor solves he opmzaon problem Mnmze r y 2 Subjec o x C, (3) where C {0, 1} n s he codebook and 2 denoes he L 2 -norm. By expandng he square of he objecve funcon, he problem becomes equvalen o mnmzng (r y ) 2 = ( ) 2 r 2 2r h x + h x = [ r 2 2r h x + h 2 x2 + h h j x x j ], (4) j where, for smplcy, we have dropped he lms of he summaons. Equvalenly, we can wre he problem n a general marx form Mnmze q T x xt P x, Subjec o x C, (5) where n hs problem q = h r +, and P = H T H, wh H defned as he n n Toeplz marx h H = h µ h h µ h 0 0. (6) h µ h 0 Here we have assumed ha µ zeros are padded a he begnnng and he end of he ransmed sequence, so ha he rells dagram correspondng o he ISI channel sars and ends a he zero sae. If he sgnals are uncoded,.e. C = {0, 1} n, and q and P are chosen arbrarly, (5) wll represen he general form of an neger quadrac programmng (IQP) problem, whch s, n general, NP-hard. In he specfc case of a PR channel, where we have he Toeplz srucure of (6), he problem can be solved by he Verb algorhm wh a complexy lnear n n, bu exponenal n µ. However, hs model can also be used o descrbe oher problems such as deecon n MIMO or wo-dmensonal ISI channels. Also, when he source symbols have a non-bnary alphabe wh a regular lace srucure such as he QAM and PAM alphabes, he problem can be reduced o he bnary problem of (5) by nroducng some new varables. C. Problem Relaxaon A common approach for solvng he IQP problem s o frs conver o an neger LP problem by nroducng a new varable for each quadrac erm, and hen relax he negraly condon; e.g. see [5]. Whle hs relaxed problem does no necessarly have an neger soluon, can be used along wh branch-and-cu echnques o solve neger problems of reasonable sze. A more recen mehod s based on dualzng he IQP problem wce o oban a convex relaxaon n he form of a sem-defne program (SDP) [6][7]. In hs work, we use he lnear relaxaon due o he lower complexy of solvng LPs compared o SDPs. Unlke n [5], where he auxlary varables are each defned as he produc of wo 0-1 varables, we defne hem as he produc of ±1 varables, whch, as we wll see, ranslaes no he modulo-2 addon of wo bs when we move o he 0-1 doman. Ths relaxaon s more suable for our purpose, snce modulo- 2 addve consrans are smlar o pary-check consrans; hus, message-passng decoders desgned for lnear codes can be appled whou any modfcaon. However, can be shown ha hs relaxaon gves he exac same soluon as n [5]. To lnearze (4), we defne z,j = x x j, j = 1,...,µ, = j + 1,...,n. (7) In he bnary doman, hs wll be equvalen o z,j = x x j, (8) where sands for modulo-2 addon. Hence, he rgh-hand sde of (4) s a lnear combnaon of {x } and {z,j }, plus a consan, gven ha x 2 = 1 s a consan. Wh some smplfcaons, he IQP n (5) can be rewren as Mnmze x,z q x + Subjec o x C, λ,j z,j, z,j = x x j, j = 1,..., µ, where, n he equalzaon problem, j = j + 1,...,n, (9) mn(µ j,n ) λ,j = P, j = h h +j. (10) =0 In hs opmzaon problem, we call {x } he nformaon bs, and {z,j } he sae bs. I can be seen from (10) ha λ,j s ndependen of, excep for ndces near he wo ends of he block;.e. 1 µ and n µ + 1 n. In pracce, hs edge effec can be negleced due o he zero 2552

3 paddng a he ransmer. For clary, we somemes drop he frs subscrp n λ,j, when he analyss s specfc o he PR deecon problem. The combned equalzaon and decodng problem (9) has he form of a sngle decodng problem, whch can be represened by a low-densy Tanner graph. Fg. 2 shows an example of he combnaon of a PR channel of memory sze 2 wh an LDPC code. We call he upper and lower layers of hs Tanner graph he code layer and he PR layer (or he PR graph), respecvely. The PR layer of he graph consss of µn check nodes c,j of degree 3, each conneced o wo nformaon b nodes x, x j, and one dsnc sae b node, z,j. Also, he PR layer can conan cycles of lengh 6 and hgher. If a coeffcen, λ,j, s zero, s correspondng sae b node, z,j, and he check node s conneced o can be elmnaed from he graph, as hey have no effec on he decodng process. I follows from (10) ha he coeffcens of he sae bs n he objecve funcon, {λ,j }, are only a funcon of he PR channel mpulse response, whle he coeffcens of he nformaon bs are he resuls of mached flerng he nosy receved sgnal by he channel mpulse response, and herefore dependen on he nose realzaon. Once he varable coeffcens n he objecve funcon are deermned, LP decodng can be appled o solve a lnear relaxaon of decodng on hs Tanner graph. We call hs mehod LP deecon. In he relaxaon of [4], he bnary pary-check consran correspondng o each check node c s relaxed as follows. Le N c be he ndex se of neghbors of check node c,.e. he varable nodes s drecly conneced o n he Tanner graph. Then, we nclude he followng consrans x x V 1, V N c s.. V s odd. (11) V N c\v In addon, he negraly consrans x {0, 1} are relaxed o box consrans 0 x 1. Ths relaxaon has he ML cerfcae propery,.e. f he soluon of he relaxed LP s negral, wll also be he soluon of (9). The coeffcens n he lnear objecve funcon, afer some normalzaon, can also be reaed as log-lkelhood raos (LLR) of he correspondng bs, whch can be used for erave message-passng decodng. In hs work, we have used he Mn-Sum Algorhm (MSA), snce, smlar o LP decodng, s no affeced by he unform normalzaon of he varable coeffcens n (9). III. PERFORMANCE ANALYSIS In hs secon, we sudy he performance of LP deecon n he absence of codng,.e. solvng (5) wh C = {0, 1} n. I s known ha f he off-dagonal elemens of P are all nonposve;.e. λ,j 0, j 0,, he 0-1 problem s solvable n polynomal me by reducng o he MIN- CUT problem; e.g. see [8]. As an example, Sankaran and Ephremdes [9] argued usng hs fac ha when he spreadng sequences n a synchronous CDMA sysem have nonposve cross correlaons, opmal muluser deecon can be done n polynomal me. In hs secon, we derve a slghly weaker Fg. 2. PR channel and LDPC code represened by a Tanner graph. condon han he nonnegavy of λ,j, as he necessary and suffcen condon for he success of he LP relaxaon o resul n an neger soluon for any value of q n (9). Ths analyss also sheds some lgh on he queson of how he algorhm behaves when hs condon s no sasfed. For a check node n he Tanner graph connecng nformaon b nodes x and x j and sae b node z,j, he consrans (11) can be summarzed as z,j max[x x j, x j x ] z,j mn[x + x j, 2 x x j ], (12) whch can be furher smplfed as x x j z,j 1 x + x j 1. (13) Snce here s exacly one such par of upper and lower bounds for each sae b, n he soluon vecor, z,j wll be equal o eher he lower or upper bound, dependng on he sgn of s coeffcen n he lnear objecve funcon, λ,j. Hence, havng he coeffcens, he cos of z,j n he objecve funcon can be wren as { λ,j x x j f λ,j 0, λ,j z,j = (14) λ,j λ,j x + x j 1 f λ,j < 0, where he frs erm n he second lne s consan and does no affec he soluon. Consequenly, by subsung (14) n he objecve funcon, he LP problem wll be projeced no he orgnal n-dmensonal space, gvng he equvalen mnmzaon problem Mnmze f(x) = q x + λ,j x x j,j:λ,j>0 + λ,j x + x j 1,,j:λ,j<0 Subjec o 0 x 1, = 1,...,n, (15) whch has a convex and pecewse-lnear objecve funcon. Each absolue value erm n hs expresson corresponds o a check node n he PR layer of he Tanner graph represenaon. A. LP-Proper Channels: Guaraneed ML Performance For a class of channels, whch we call LP-proper channels, he proposed LP relaxaon of uncoded ML deecon always gves he ML soluon. The followng heorem provdes a creron for recognzng LP-proper channels. 2553

4 Theorem 1: The LP relaxaon of he neger opmzaon problem (9), n he absence of codng, s exac for every ransmed sequence and every nose confguraon f and only f he followng condon s sasfed for {λ,j }: Weak Nonnegavy Condon (WNC): Every check node c,j, conneced o varable nodes x and x j, whch les on a cycle n he PR Tanner graph corresponds o a nonnegave coeffcen;.e. λ,j 0. Proof: We frs prove ha WNC s suffcen for guaraneed convergence of LP o he ML sequence, and hen show ha f hs condon s no sasfed, here are cases where he LP algorhm fals. Some of he deals n he proof are omed, and he reader s referred o a fuure longer paper [10]. In he proof, we make use of he followng defnon. Defnon 1: Consder a pecewse-lnear funcon f : R n R. We call a a breakpon of f f he dervave of f(a+sv) wh respec o s changes a s = 0, for any nonzero vecor v R n. 1) Suffcency: I s suffcen o show ha under WNC, he soluon of (15) s always a one of he verces of he un cube, [0, 1] n. In he absence of he box consrans, he mnmum of he objecve funcon has o occur eher a nfny or a a breakpon of hs pecewse-lnear funcon. Each breakpon, a, s deermned by makng n of he absolue value erms acve,.e. seng her argumens equal o zero. When he feasble regon s resrced o he un cube [0, 1] n, he opmum can also occur a he boundares of hs regon. We can assume ha he opmum les on a number, k, of hyperplanes correspondng o he box consrans, makng exacly k varables equal o eher 0 or 1. In order o deermne he remanng n k fraconal varables, a leas n k oher equaons are needed, resulng from makng a number of absolue value erms acve. Each of hese equaons wll have one of he wo forms x = x j or x + x +j = 1, dependng on wheher λ,j > 0 or λ,j < 0, respecvely. In each of hese equaons, eher boh, or neher of s varables can be neger. Snce he former case does no provde an equaon n erms of he fraconal varables, we can assume ha all hese equaons only nvolve fraconal varables. Now he queson becomes under wha condon such equaons can have a unque and nonnegral soluon. We can llusrae hs sysem of equaons by a dependence graph, where he verces correspond o he n k fraconal varable nodes, and beween verces x s and x s here s a posve edge f λ s, > 0 and a negave edge f λ s, < 0. An example of a dependence graph sasfyng WNC s shown n Fg. 3. In general, hs graph conans L clusers, C 1, C 2,...,C L, of cycles, and also J addonal edges whch do no le on any cycle. If WNC s sasfed, each cluser wll conan only posve edges. Ths means ha f we separae he equaons correspondng only o verces whn a cluser C of cycles, hese equaons wll all have he form x = x j, where verces and j belong o C. Consequenly, all verces n C have he same value, β. The values of {β } should be deermned by he edges (equaons) beween he clusers. Fg. 3. The dependence graph of he sysem of lnear equaons wh one cluser of cycles. Sold lnes represen posve edges and dashed lnes represen negave edges. If we modfy he dependence graph by concenrang each cluser o a verex of value β, he graph wll no conan any cycle. Therefore, he sysem of equaons for deermnng he varables s under-deermned, and none of he nodes n hs graph wll have a unque soluon. Hence, he only case ha wll have a unque soluon for all he varables s k = n, whch means ha all of he varables {x } are negral. 2) Necessy: (Oulne) We prove he necessy by a couner-example. Consder a case where he realzaon of he nose sequence s such ha he receved sequence s zero. Snce hs makes he {q } equal o zero, we wll be lef wh he posve-weghed sum of a number of absolue value erms. The objecve funcon wll become zero f and only f all hese erms are zero, whch s sasfed f x = [ 1 2,..., 1 2 ]T. By usng smlar deas o hose n he suffcency proof, we can show ha f WNC s no sasfed, hs s he unque vecor ha makes all he erms equal o zero, and herefore, hs fraconal vecor wll be he soluon of LP deecon. Corollary 1: The soluons of he LP relaxaon of uncoded ML equalzaon are n he space {0, 1 2, 1}n. (Proof omed) In a PR channel, due o he perodc srucure of he Tanner graph, WNC mples ha eher he graph s acyclc, or all he coeffcens {λ,j } are nonnegave. An neresng applcaon of hs resul s n 2-D channels, for whch here s no feasble exenson of he Verb algorhm. In a 2-D channel, he receved sgnal r,s a coordnae (, s) n erms of he ransmed symbol array { x,s } has he form µ ν r,s = h,j x,s j + n,s. (16) =1 j=1 Hence, he sae varable defned as z (,s),(k,l) = x,s x k,s l wll have he coeffcen µ ν γ k,l = h,j h +l,j+l. (17) =1 j=1 Theorem 1 guaranees ha ML deecon can be acheved by lnear relaxaon f γ k,l 0, k, l > 0. An example of a 2-D channel sasfyng hs condon s gven by he marx [ ] 1 1 [h,j ] =. (18) 1 1 B. Hgh SNR Analyss: asympocally LP-Proper Channels Here we oulne a mehod o characerze he even ha LP deecon fals o fnd he ML soluon a hgh SNR. Ths 2554

5 analyss s movaed by he observaon ha for some channels no sasfyng WNC, hs even a hgh SNR s domnaed by he falure of he ML deecor o fnd he ransmed sequence. We call hese channels asympocally LP-proper, snce her hgh-snr performance s smlar o ha of ML. Gven he soluon ˆx of LP deecon, le he fraconal se, F {1,...,n}, be he se of ndces of elemens of ˆx ha have fraconal values n he soluon. We know from Corollary 1 ha hese fraconal values are all equal o 1 2. A reasonable assumpon suppored by our smulaons a hgh SNR s ha f he ML soluon, x, s correc, he neger elemens of ˆx are correc, as well. For he objecve funcon f of (15) we have BER CH1, LP CH1, MSA CH2, LP CH2, MSA CH2, ML CH3, LP CH3, MSA f(ˆx) < f(x). (19) By expandng f, hs nequaly can be wren n erms of {λ,j }, x, and q. Moreover, each q s a funcon of he channel coeffcens, he ransmed sequence x, and he addve nose. Hence, (19) wll provde a condon n erms of x and he nose. In he absence of nose, and for a gven fraconal se, F, we can se up an neger opmzaon problem o fnd he ransmed sequence ha mnmzes f(ˆx) f(x). We have suded he specal case where F = {1,..., n}. In hs case, f(ˆx) f(x) = δ, where δ, whch we call he all- 1 2 dsance of he channel, becomes ndependen of x for large block lenghs. Ths dsance can be used o esmae he hgh-snr probably of falure. In parcular, f for a channel δ 0, LP deecon fals wh a non-dmnshng probably. We refer o hese channels as LP-mproper channels. IV. SIMULATION RESULTS We have smulaed graph-based deecon usng LP decodng and MSA for hree PR channels of memory sze 3: 1) CH1: h(d) = 1 D 0.5D 2 0.5D 3 (sasfes WNC), 2) CH2: h(d) = 1 + D D 2 + D 3, 3) CH3: h(d) = 1 + D D 2 D 3 (EPR4 channel). Uncoded b error raes (BER) of deecon on hese channels usng LP and MSA are shown Fg. 4. Snce CH1 sasfes WNC, LP wll be equvalen o ML on hs channel. For CH2, we have also provded he BER of ML. Excep a very low SNR where we see a small dfference, he performance of LP and ML are nearly equal, whch means ha CH2 s an asympocally LP-proper channel. For boh CH1 and CH2, MSA converges n a mos 3 eraons and has a BER very close o ha of LP. On he oher hand, for CH3, we observe ha he BERs of LP and MSA are almos consan. Hence, CH3 s an LP-mproper channel. The resuls for deecon wh LDPC codng are omed due o space lmaons [10]. However, for he cases ha we have smulaed, all he LPproper and asympocally LP-proper channels also showed a good performance n he presence of codng. V. CONCLUSION We nroduced a new graph represenaon of ML deecon n ISI channels, whch can be used for combned equalzaon and decodng usng LP relaxaon or erave message-passng mehods. By a geomerc sudy of he problem, we derved Normalzed SNR, db Fg. 4. BER for CH1-CH3. SNR s defned as he ransmed sgnal power o he receved nose varance. necessary and suffcen condons for he LP relaxaon of he channel equalzaon problem o gve he ML soluon for all ransmed sequences and all nose confguraons. In addon, for ceran oher channels, he performance of LP approaches ha of ML a hgh SNRs. For a hrd class of channels, LP deecon has a probably of falure bounded away from zero even n he absence of nose. In a sep o characerze hese channels, we showed how a condon can be derved for he falure of LP deecon, whch can be used o esmae he asympoc behavor of hs deecon mehod. Smulaon resuls show ha message-passng decodng echnques have a smlar performance o ha of LP decodng for mos channels. ACKNOWLEDGMENT Ths work s suppored n par by NSF Gran CCF REFERENCES [1] L. R. Bahl, J. Cocke, F. Jelnek, and J. Ravv, Opmal decodng of lnear codes for mnmzng symbol error rae, IEEE Trans. Inform. Theory, vol. IT-20, pp , Mar [2] J. Hagenauer and P. Hoecher, A Verb algorhm wh sof-decson oupus and s applcaons, n Proc. IEEE GLOBECOM, Dallas, TX, Nov. 1989, vol. 3, pp [3] B. M. Kurkosk, P. H. Segel, and J. K. Wolf, Jon message-passng decodng of LDPC codes and paral-response channels, IEEE Trans. Inform. Theory, vol. 48, no. 6, pp , Jun [4] J. Feldman, M. J. Wanwrgh, and D. Karger, Usng lnear programmng o decode bnary lnear codes, IEEE Trans. Inform. Theory, vol. 51, no. 3, pp , Mar [5] L. J. Waers, Reducon of neger polynomal programmng problems o zero-one lnear programmng problems, Operaons Research, vol. 15, no. 6, pp , Nov.-Dec [6] C. Lemaréchal and F. Ousry, Semdefne relaxaons and Lagrangan dualy wh applcaon o combnaoral opmzaon, Rappor de Recherche no. 3710, INRIA, Jun [7] M.X. Goemans and D.P. Wllamson, Improved approxmaon algorhms for maxmum cu and sasfably problems usng semdefne programmng, J. ACM, vol. 42, no. 6, pp , Nov [8] J. M. W. Rhys, A selecon problem of shared fxed coss and nework flows, Managemen Sc, vol. 17, no. 3, pp , Nov [9] C. Sankaran and A. Ephremdes, Solvng a class of opmum muluser deecon problems wh polynomal complexy, IEEE Trans. Inform. Theory, vol. 44, no. 5, pp , Sep [10] M. H. Taghav and P. H. Segel, Graph-based decodng n he presence of ISI, In preparaon. 2555

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