Joint Message-Passing Symbol-Decoding of LDPC Coded Signals over Partial-Response Channels

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1 Joit Message-Passig Symbol-Decodig of LDPC Coded Sigals over Partial-Respose Chaels Rathakumar Radhakrisha ad ae Vasić Departmet of Electrical ad Computer Egieerig Uiversity of Arizoa, Tucso, AZ {ratha, Abstract We cosider the problem of joit detectio ad decodig of low-desity parity-check (LDPC) coded sigals over partial respose (PR) chaels. A method to graphically represet the costraits imposed by the chael ad the code o the chael output sequece is itroduced. This eables the desig of a detector ad decoder that estimates a posteriori probabilities of oiseless chael output symbols rather tha biary chael iputs. y ruig the sum-product algorithm (SPA) o this graph, a joit decoder is obtaied that is show to perform sigificatly better tha the turbo-equalizer. I. INTRODUCTION Low-desity parity-check (LDPC) codes, iveted by Gallager [] ad rediscovered by Mackay ad Neal [2] have bee show to be capacity achievig o memoryless chaels. It has also bee show to achieve ecellet error rate performace over chaels with memory, such as magetic storage [3] ad optical commuicatio chaels [4]. For reasos of badwidth efficiecy, these iter-symbol iterferece (ISI) chaels are equalized to a partial-respose (PR) target with relatively small memory compared to the uequalized chael. Ay LDPC decoder must cope with the cotrolled amout of ISI itroduced by the PR. For a ucoded system, the chael iput sequece is optimally detected i the presece of ISI by the Viterbi algorithm. For a LDPC coded system, the optimal maimum a posteriori (MAP) decodig is impractical, but good error rate performace ca be achieved by usig the turbo priciple [5]. I this techique, iformatio is iteratively passed back ad forth betwee a soft-iput softoutput (SISO) detector ad a SISO LDPC decoder. The ISI i the chael output is elimiated by a detector usig the chael iformatio ukow to the decoder, ad subsequetly, the output is decoded by a LDPC decoder usig the code structure iformatio ukow to the detector. Sice, Viterbi algorithm produces oly hard decisios, algorithms like softoutput Viterbi algorithm (SOVA) [6] ad ahl-cocke-jeliek- Raviv (CJR) algorithm [7] are used for SISO detectio ad the well-kow sum-product algorithm (SPA) [8] is used for SISO decodig. This iterative algorithm is kow as turboequalizer. Though sub-optimal, it is the best ad most widely used decoder kow today. The performace ca be improved beyod what is achieved by turbo-equalizer, if both chael ad the code iformatio are simultaeously used to make decisios o the chael iputs. This is referred to as joit detectio ad decodig or simply as joit decodig, ad is the focus of this paper. It has also bee the focus of some recet works i [9], [], [] ad [2]. A practical ad popular approach i this directio has bee to desig message-passig (MP) decodig algorithms that operate o a graph, which represets the costraits imposed by both the chael ad the LDPC code. I turbo-equalizer, the chael costraits are represeted by the chael trellis ad the code costraits are represeted by the bipartite graph of the code, kow as the Taer graph. Detectors like SOVA ad CJR operate serially o the trellis, while the LDPC decoder operates parallely o the Taer graph. While aimig to coceive a joit MP decodig algorithm operatig o a graph, it is logical to first cosider the problem of represetig the chael costraits as a graph ad the to desig a parallel MP detectio algorithm that operates o this graph. This problem has bee addressed by Kurkoski et. al. i [9], who itroduced parallel bit-based ad state-based MP algorithms for chael detectio. The bit-based MP algorithm is useful oly i chaels with uit memory legth, but has bee modified for use i chaels with loger memory by Colavolpe et. al. []. Eve though i [9], the state-based MP detector was combied with the LDPC MP decoder to obtai a joit MP decoder, it achieved at best the same performace as the turbo-equalizer. This is due to the fact that the joit MP decoder is simply a parallel schedule of the turbo-equalizer. The challege i joitly usig both the chael ad code iformatio arises from the fact that the chael imposes costraits o the chael output sequeces, whereas the code imposes costraits o the chael iput sequeces. The idea motivatig our approach is the observatio that by imposig costraits o the chael iput sequeces, the code also imposes certai costraits o the oiseless chael output sequeces. I this paper, we modify the LDPC MP decoder to produce iformatio o the oiseless chael output symbols rather tha o chael iputs. This eables us to combie it with a modified versio of the state-based MP detector to desig a joit decoder, that sigificatly surpasses the performace of the turbo-equalizer. The joit decoder estimates a posteriori probabilities (APPs) of chael output symbols, from which APPs of chael iputs are derived. Also, as it will be show, this algorithm ca be used irrespective of the chael memory legth, although as described i Sectio III, it may perform relatively better for chaels with small memory.

2 The rest of the paper is orgaized as follows. We describe a graphical model used to represet the chael ad describe a optimal MP detectio algorithm operatig o this graph i Sectio II. This model is eteded to iclude code costraits ad a joit MP decodig algorithm that operates o this combied graph is described i Sectio III. it error simulated performace for a LDPC code is show i Sectio IV ad fially, the paper is cocluded i Sectio V. II. MESSAGE-PASSING DETECTION ALGORITHM A. System model Fig. shows the system model cosidered, where the chael respose is represeted by a polyomial h(d) or the correspodig coefficiet vector h. A sequece u of k biary bits is ecoded by a LDPC code ito a codeword of biary bits. The codeword is trasmitted through the PR chael, whose o-biary output is corrupted by additive white Gaussia oise (AWGN). The oiseless chael output, y = h, is of legth + m, where deotes the covolutio operator ad m deotes the chael memory legth. The oisy chael output is give by r = y + z. We first cosider a ucoded system, where the optimal detector is the oe that estimates MAP probability of the bits p( i r), i =,,...,k, where i is the i th elemet of the vector ad i {,}. These quatities are efficietly determied by operatig the CJR algorithm o the chael trellis.. Chael graph The trellis of a chael represets the costraits imposed o the rage of oiseless chael output sequeces. A chael trellis ca be give as a factor graph [8] show i Fig. 2, where q,q,...,q + represets state (hidde) variables,,,..., represets chael iputs ad y,y,...,y represets oiseless chael outputs. The factor graph ca be divided ito sectios, where the i th sectio deoted by T i is defied by all valid triples {q i,y i,q i+ }. Therefore, each sectio acts as a local costrait of the chael. Cosequetly, a sequece of state ad chael output variables {q,q,...,q +,y,y,...,y } is valid if ad oly if it satisfies all local costraits T,T,...,T. Whe a global chael costrait is factored ito local chael costraits, umerous schedulig schemes for implemetig the CJR algorithm are possible. Typically, oe istace of a CJR algorithm is operated o oe sectio of the trellis at ay time istat ad is progressively moved to other sectios. This schedulig is kow as fully-serial [8]. O the other etreme, istaces of the CJR algorithm ca operate o each sectio of the trellis simultaeously, echagig iformatio through the state variables durig every iteratio. This schedulig is kow as fully-parallel, ad is referred to as the parallel state-based MP algorithm i [9]. Naturally, itermediate schedulig schemes are possible. For eample, the factor graph show i Fig. 2 ca be divided ito p sectios (p < ), ad durig every iteratio, p istaces of the CJR algorithm ca operate o each of these sectios, k u LDPC Ecoder h(d) +m y z +m r Detector (CJR) Joit decoder Turbo-Equalizer ˆ MP decoder Fig.. lock diagram of a PR system. Decoder is either turbo-equalizer (upper brach) or joit decoder (lower brach) ad oise is modeled as additive white Gaussia. i Ti q q q 2 q i q i+ Fig. 2. y y y i Geeralized factor graph represetatio of a PR chael trellis. echagig iformatio with the adjacet oes, but withi each sectio, the CJR ca operate serially. I this work, we cosider the fully-parallel MP schedulig. This ecessitates the eed for state variables to store ad relay iformatio betwee adjacet sectios after every iteratio. We modify the factor graph represetig the chael costraits to remove the eed for state variables ad correspodigly alter the detectio algorithm. The graph is ow simply represeted as a bipartite graph G show i Fig. 3, where circles correspod to the oiseless chael symbols y i, ad squares correspod to the local chael costraits. These sets of odes are referred to as symbol odes ad chael odes respectively. Every chael ode represets two sectios of the trellis. For eample, a chael ode s i acts as a local costrait ad represets all valid 5-tuples {q i,y i,q i+,y i+,q i+ }. Therefore, ulike the factor graph i Fig. 3, every sectio of the trellis is represeted by two chael odes i this graph. As will be described i the et sectio, iformatio pertaiig to state trasitio probabilities is echaged betwee symbol odes through the chael odes. Like the factor graph of the trellis, graph G is a geeric cycle-free represetatio of a PR chael, irrespective of its memory legth. However, the size of the set represeted by the chael odes icreases epoetially with icrease i memory legth. If memory legth is m, the size of this set is 2 (m+2). C. Message-passig symbol detector Now, we describe a MP detectio algorithm that operates o graph G ad produces APPs of output symbols, from which APPs of chael iputs are derived. If k ad y k are the chael iput ad oiseless output at time k, the ˆ y k = f ( k, k,..., k m ), () where the fuctio f() is determied by the chael respose h(d). Let A = {a,a,...,a 2 m } be the set of possible

3 y y y2 3 Symbol odes y y+m 2 y +m s s s 2 s 3 s +m-3 s +m-2 Chael odes Fig. 3. A graph that represets costraits imposed by the chael o the oiseless chael output sequeces. oiseless chael output symbols whe the curret chael iput k =, ad let = {b,b,...,b 2 m } be the set of possible oiseless chael output symbols whe the chael iput k =. The, APPs of the chael iput k are give as, p( k = r) = 2 m i= p(y i = a i r). (2) Though elemets of set A ad may ot be uique, we emphasize that they correspod to a uique state trasitio i the correspodig chael trellis. Sice the MP detectio algorithm is ot operated eplicitly o the trellis, we simply refer the quatities p(y i ) as symbol probabilities rather tha state trasitio probabilities. Let ϕ i deote the set of symbol pairs represeted by chael ode s i of the graph show i Fig. 3. Therefore, (y i,y i+ ) ϕ i, i. The costraits imposed o the output sequeces by the chael ca be viewed as a code, where each chael ode s i correspods to a code of legth 2 ad the set ϕ i correspods to its set of codewords. With the kowledge of ϕ i for all chael odes, the output sequece ca thus be decoded without the use of state variables. Sice the uderlyig chael graph G is a tree, APPs ca be estimated optimally usig the SPA. Now, the MP detectio algorithm is give as follows. MP Detectio Algorithm: ) Iitializatio: Sice, chael iputs are i.i.d, all state trasitios are iitially equally likely. Therefore, p(y i ) = /2 m+, i =,,..., + m. 2) Message from symbol odes to chael odes durig the t th iteratio: k s k = M s (t ) k y k k s k = M s (t ) k y k, k =,..., + m 2 (3) where, y k s k deotes the message set from symbol ode y k to chael ode s k durig the t th iteratio. Other terms are defied similarly. 3) Message from chael odes to symbol odes durig the t th iteratio: M s (t) k y k = p(y k r k,r k+,ϕ k ), y k p(r k y k )p(y k ) M s (t) k y k+ = p(y k+ r k,r k+,ϕ k ) p(r k+ y k+ )p(y k+ ), y k+. (4) Messages received from the symbol odes durig the curret iteratio serve as the a priori symbol probabilities for the chael ode operatio. 4) APPs of chael output symbols: After repeatig the above steps for a fied umber of iteratios, APPs of chael output symbols are calculated as, y k = s k y k s k y k p(r k y k ) p(y k ) (5) where, p(y k ) is the iitial a priori probability. 5) APPs of chael iput bits: The algorithm halts after calculatig chael iput APPs usig Eq. 2. Remark: p(y k r k,r k+,ϕ k ) ad p(y k+ r k,r k+,ϕ k ) i Eq. 4 ca be computed i a straightforward way, sice ϕ k 2 m+2, ad m is small. I the cotet of a trellis, this is same as operatig CJR oly o the two sectios of the trellis represeted by the chael ode s k. Whe the umber of iteratios equal the legth of the sequece, the APPs obtaied usig the above algorithm are same as that obtaied from the CJR algorithm. Usually, oly a small umber of iteratios are required to obtai performace close to optimal. However, as observed i [9], all scheduligs of the CJR algorithm, ecept the fully-serial schedulig, result i a error floor if eough iteratios are ot ru. Like other MP detectio algorithms proposed i the literature, this algorithm is more comple tha CJR, primarily due to the parallel schedulig of the algorithm, but it ca potetially reduce latecy time ad is suitable for high-speed applicatios. Its most importat advatage is that it ca be combied with a LDPC MP decoder to obtai a joit MP decoder. III. JOINT MESSAGE-PASSING SYMOL-DECODING ALGORITHM I this sectio, we eted the graphical model described earlier to iclude costraits imposed by the parity checks of the LDPC code. The tripartite graph show i Fig. 4 is obtaied by icludig the parity check odes to the chael graph of Fig. 3. Coectios betwee the parity check odes ad the symbol odes are defied i the same way as the coectios betwee the parity check odes ad the variable odes. Usig this combied graph, a joit decodig algorithm is developed that estimates the symbol APPs usig both the chael ad the code iformatio simultaeously. The joit decodig algorithm is outlied below. Joit MP Decodig Algorithm: ) Iitializatio: Symbol a priori probabilities p(y i ) = /2 m+. 2) Message from symbol odes to chael odes durig the t th iteratio: For every k, compute, y k s k = M (t ) s k y k y k s k = M (t ) s k y k j h jk = j h jk = M (t ) c j y k M (t ) c j y k. 3) Message from symbol odes to check odes durig the t th iteratio: If H = {h ij } is the parity check matri (6)

4 Check odes c c c 2 c 3 c q y y y 2 y3 y +m s s s 2 s 3 s +m-3 s +m-2 Chael odes Fig. 4. A graph that represets costraits imposed by the chael ad the parity checks of the LDPC code o the oiseless chael output sequeces. Parity checks imposes certai costraits o the chael output sequeces by imposig costraits o the chael iput sequeces. i j Fig. 5. Trellis diagram of a sigle parity check code of legth 3. The states idicate the parity of the icomig sequeces. k of the LDPC code, the for every i ad k such that h ik =, compute, k c i = M s (t ) k y k M s (t ) k y k M c (t ) j y k. j h jk =,j i (7) 4) Message from chael odes to symbol odes durig the t th iteratio: This is same as Eq. 4. 5) Message from check odes to symbol odes durig the t th iteratio: For every i ad k such that h ik =, compute, c i y k = p(y k r i,c i ) p(r k y k )p(y k ), y k (8) where, C i deotes the evet that the check c i is satisfied ad r i deotes the set of received samples at the locatios of the variable odes coected to check c i. 6) APPs of chael symbols: For every k, compute, y k = s k y k s k y k p(r k y k ) p(y k ) j h jk = c j y k where, p(y k ) is the iitial a priori probability. 7) APPs of chael iput bits: After every iteratio, the chael iput APPs are calculated usig Eq. 2. All the above steps are repeated util either the decoded sequece satisfies all chael ad code costraits or a preset maimum umber of iteratios is reached. (9) Remark: Observe that the check ode operatio of the LDPC MP decoder is modified to provide symbol iformatio for use i joit decodig. I the traditioal decoder, the check ode operatio is efficietly computed by the tah fuctio, whereas the ew check ode operatio is more comple. We describe et a efficiet method to compute this ew operatio. For ease of epositio, cosider a degree 3 check ode c l = i j k. The check ode c l implies that the three variable odes form a sigle parity check code, deoted by C X. This is compactly represeted by the code trellis show i Fig. 5. All paths begiig ad edig at state correspod to codewords of the code C X. Iput bit APPs coditioed o the evet C l ca be determied by operatig the CJR algorithm o this trellis, which simply turs out be the tah check ode operatio. The check ode c l is coected to symbol odes y i, y j ad y k i the combied graph. y imposig costraits o the variable odes i, j ad k, the check c l also imposes certai costraits o the correspodig output symbols. I other words, the check ode c l implies that the three symbol odes also form a code, which we deote by C Y. I order to obtai symbol APPs, a trellis for the code C Y, referred to as the epaded code trellis, is costructed by epadig the edges of the code trellis show i Fig. 5. The epaded code trellis is show i Fig. 6. A edge represetig a state trasitio i code trellis is ow replaced by 2 m edges represetig all possible oiseless chael outputs geerated durig the correspodig state trasitio. For eample, if i = is trasmitted, the correspodig oiseless chael output y i A. Further, the edge labels are elemets of set A or depedig o whether the correspodig edge label i code trellis is or. If the three symbol odes are idepedet, i.e. they have at least m other symbol odes betwee them, the all paths begiig ad edig at state of the epaded code trellis correspod to codewords of the code C Y. Therefore, the symbol APPs of y i, y j ad y k coditioed o the evet C l is determied by operatig the CJR algorithm o the epaded code trellis. If the symbol odes coected to a check ode are ot idepedet, the above method ca still be used for a approimate calculatio of Eq. 8. I priciple, the joit symbol decoder ca be applied for chaels with ay memory legth, although the umber of check odes that violates the idepedece property will be lower for chaels with small memory. The performace of the joit decoder would be severely affected if the LDPC code cotais may pairs of cosecutively occurrig variable odes coected to a check, as such codes would result i may four-cycles i the combied graph. I practice, the check ode degree is much higher tha 3, but the computatio of Eq. 8 usig the epaded trellis is still simple, sice the epaded code trellis will always have oly two states as log as the check ode represets a sigle parity check code. IV. SIMULATION RESULTS We illustrate the performace of the joit symbol-decodig algorithm by simulatig a LDPC coded PR system, where

5 yi y j A A A Fig. 6. Epaded code trellis of Fig. 5. The state trasitio edge labels are either elemets of set A or. ER A yk Turbo equalizer (ER after every iteratio) Joit decoder SNR(d) Fig. 7. it error rate compariso of (98,22) radom LDPC code o a PR4 chael whe decoded usig turbo-equalizer ad the joit decoder. the chael is give by the impulse respose [,, ] (PR4), ad the LDPC code is of legth 98 ad rate.89 [3]. The chael output sequeces are decoded by usig both turboequalizer ad the joit symbol-decoder. Whe turbo-equalizer is used, the umber of global iteratios is restricted to 5 ad the umber of iteral LDPC decoder iteratios is restricted to 3. These settigs give the best bit error rate (ER) performace. Icreasig the umber of global iteratios beyod 5 does ot improve the performace sigificatly. Whe the joit symboldecoder is used, the umber of iteratios is restricted to 6. The performace compariso is show i Fig. 7. At a sigalto-oise ratio (SNR) of 5.4 d the ER obtaied by the joit decoder is almost a order of magitude better tha the turboequalizer. Also, the figure suggests that the gai icreases with icreasig SNR. Amog the 22 parity checks of this code, parity checks cotai at least oe pair of variable odes (or symbol odes) that are ot idepedet. However, the joit decoder was applied to the code without ay modificatio, implyig that the computatio of Eq. 8 was approimate. I spite of this, the decoder was able to achieve sigificat gai over the turbo-equalizer. V. CONCLUSION The problem of joit detectio ad decodig of LDPC coded sigals over partial respose chaels is cosidered. I order to joitly use both the chael ad code iformatio, the LDPC decoder is modified to produce iformatio o chael output symbols rather tha o chael iputs. This is combied with a message-passig detector to develop a joit decoder that estimates chael iput APPs by first estimatig chael output symbol APPs. The performace of this decoder is show to sigificatly outperform that of the turbo-equalizer for a radom LDPC code of rate.89. VI. ACKNOWLEDGEMENT This work was supported by Seagate Techology ad the NSF uder grat ECCS REFERENCES [] R. G. Gallager, Low-desity parity-checks codes, IEEE Tras. Iform. Theory, vol. 8, o., pp. 2 28, Ja [2] D. J. C. Mackay ad R. M. Neal, Near Shao limit performace of low-desity parity-check codes, Electroics Lett., vol. 32, o. 8, pp , 996. [3] J. L. Fa, A. Friedma, E. Kurtas, ad S. W. McLaughli, Low desity parity check codes for magetic recordig, i Proc. 37th Aual Allerto Cof. o Commuicatio, Cotrol ad Computig, 999, pp [4]. Vasić,. Djordjevic, ad R. Kostuk, Low-desity parity check codes ad iterative decodig for log haul optical commuicatio systems, J. Lightwave Techol., vol. 2, o. 2, pp , Feb. 23. [5] C. errou, A. Glavieu, ad P. Thitimajshima, Near Shao limit error-correctig codig ad decodig: Turbo-codes, i Proc. IEEE It. Cof. o Commuicatios, vol. 2, Geeva, Switzerlad, May 993, pp [6] J. Hageauer ad P. Hoeher, A Viterbi algorithm with soft-decisio outputs ad its applicatios, i Proc. IEEE Global Telecomm. Cof. (GLOECOM 89), vol. 3, Dallas, Teas, Nov. 989, pp [7] L. R. ahl, J. Cocke, F. Jeliek, ad J. Raviv, Optimal decodig of liear codes for miimizig symbol error rate, IEEE Tras. Iform. Theory, vol. 2, o. 2, pp , Mar [8] F. R. Kschischag,. J. Frey, ad H.-A. Loeliger, Factor graphs ad the sum-product algorithm, IEEE Tras. Iform. Theory, vol. 47, o. 2, pp , Feb. 2. [9]. M. Kurkoski, P. H. Siegel, ad J. K. Wolf, Joit message-passig decodig of LDPC codes ad partial-respose chaels, IEEE Tras. Commu., vol. 48, o. 6, pp , Jue 22. [] G. Colavolpe ad G. Germi, O the applicatio of factor graphs ad the sum product algorithm to ISI chaels, IEEE Tras. Commu., vol. 53, o. 5, pp , May 25. [] P. Pakzad ad V. Aatharam, Kikuchi approimatio method for joit decodig of LDPC codes ad partial-respose chaels, IEEE Tras. Commu., vol. 54, o. 7, pp , July 26. [2] G. Colavolpe, O LDPC codes over chaels with memory, IEEE Tras. Wireless Commu., vol. 5, o. 7, pp , July 26. [3] D. Mackay. Ecyclopedia of sparse graph codes. [Olie]. Available: