Threshold Saturation of Spatially-Coupled Codes on Intersymbol-Interference Channels

Transcription

1 hreshold Saturaton of Spatally-Coupled Codes on Intersymbol-Interference Channels Phong S Nguyen, Arvnd Yedla, Henry D Pfster, and Krshna R Narayanan Department of Electrcal and Computer Engneerng, exas A&M Unversty {psn, yarvnd, hpfster, krn}@tamuedu Abstract Recently, t has been observed that termnated lowdensty-party-check (LDPC) convolutonal codes (or spatallycoupled codes) appear to approach the capacty unversally across the class of bnary memoryless channels hs s facltated by the threshold saturaton effect whereby the belef-propagaton (BP) threshold of the spatally-coupled ensemble s boosted to the maxmum a-posteror (MAP) threshold of the underlyng consttuent ensemble In ths paper, we consder spatally-coupled codes over ntersymbol-nterference (ISI) channels under jont teratve decodng where we emprcally show that threshold saturaton also occurs hs can be observed by frst dentfyng the GEXI curve that naturally obeys the general area theorem From ths curve, the correspondng MAP and the BP threshold estmates are then numercally obtaned Gven the fact that regular LDPC codes can acheve the symmetrc nformaton rate (SIR) under MAP decodng, we conjecture that spatally-coupled codes wth jont teratve decodng can unversally approach the SIR of ISI channels I INRODUCION LDPC convolutonal codes, whch were ntroduced n [] and shown to have excellent BP thresholds n [2], have recently been observed to unversally approach the capacty of varous memoryless channels he fundamental mechansm behnd ths s explaned well n [3], where t s proven analytcally for the BEC that the BP threshold of a partcular spatally-coupled ensemble converges to the maxmum a-posteror (MAP) threshold of the underlyng ensemble A smlar result was also observed ndependently n [4] and stated as a conjecture Such a phenomenon s now called threshold saturaton va spatal couplng and has also been emprcally observed for general bnary-nput symmetrc-output memoryless (BMS) channels [5] In fact, threshold saturaton seems to be qute general and has now been observed n a wde range of problems In the realm of channels wth memory and partcularly ntersymbol nterference (ISI) channels, the capacty may not be achevable va equprobable sgnalng For lnear codes, a popular practce s to compare nstead wth the symmetrc nformaton rate (SIR), whch s also known as C ud [6], because ths rate s achevable by random lnear codes wth maxmum-lkelhood (ML) decodng For LDPC codes over ISI channels, a jont teratve BP decoder that operates on a large graph representng both the channel and the code constrants [6], [7] can perform qute well Progress has also been made on the desgn of SIR-approachng rregular LDPC hs materal s based upon work supported by the Natonal Scence Foundaton under Grant No he work of P Nguyen was also supported n part by a Vetnam Educaton Foundaton fellowshp Any opnons, fndngs, conclusons, or recommendatons expressed n ths materal are those of the authors and do not necessarly reflect the vews of the Natonal Scence Foundaton codes for some specfc ISI channels [8], [9], [0], [], [2] However, channel parameters must be known at the transmtter for such desgns and therefore unversalty appears dffcult to acheve Snce spatally-coupled codes and the threshold saturaton effect have now shown benefts n many communcaton problems, t s qute natural to consder them as a potental canddate to unversally approach the SIR of ISI channels wth low decodng complexty In fact, the combnaton of spatallycoupled codes and ISI channels was recently consdered by Kudekar and Kasa [3] for the smple dcode erasure channel (DEC) from [2], [4] hey provded a numercal evdence that the jont BP threshold of the spatally coupled codes can approach the SIR over the DEC However, the EXIlke curves they consdered were not equpped wth an area theorem and therefore could not be drectly connected wth the MAP threshold of the underlyng ensemble hus, the threshold saturaton effect was only ndrectly observed In ths paper, we frst focus on the case of general ISI channels where, by dervng the approprate GEXI curve and assocated area theorem, the MAP threshold upper bound can be computed and threshold saturaton can be seen As a consequence, t s possble for spatally-coupled codes to closely approach the SIR of ISI channels under jont teratve BP decodng because regular LDPC codes can acheve the SIR under MAP decodng [5] Also, we revst the transmsson of the spatally-coupled codes over the DEC as a specal case For ths channel, wth the MAP threshold estmated from the (G)EXI curve, threshold saturaton can also be observed to occur exactly Furthermore, the smplcty of the DEC allows us to provde a rgorous analyss of the upper bound on the MAP threshold A ISI Channels and the SIR II BACKGROUND Let the nput alphabet X be fnte, {X } Z be the dscretetme nput sequence (e, X X ) and {Y } Z be the dscretetme output sequence wth Y R Many ISI channels of nterest can be modeled by Y = ν t=0 a t X t + N, where the channel memory s ν, {a t } ν t=0 s the set of tap coeffcents and {N } Z s a sequence of ndependent nose random varables One can also wrte the above as Y = Z + N where Z = ν t=0 a t X t s the ISI channel output wthout nose In ths paper, we restrct ourselves to the class of bnary-nput ISI channels Often, the tap coeffcents are represented through a transform doman polynomal a(d) = ν t=0 a t D t he subject of Secton IV s the dcode erasure channel (DEC), whch s bascally a st-order dfferentator whose

2 d {f} a {x} DE c {L(y)} b {y} random permutaton channel outputs trells nodes bt nodes check nodes Fgure Gallager-anner-Wberg graph of the jont BP decoder for ISI channels he notatons a, b, c, d denote the average denstes of the messages traversng along the graph used n densty evoluton (DE) he quanttes nsde the brackets are erasure rates used n DE for the DEC case output s erased wth probablty ɛ and transmtted perfectly wth probablty ɛ Meanwhle, Secton III consders more general ISI channels among whch the most common s lnear ISI channels wth addtve whte Gaussan nose (AWGN) Unfortunately, no closed-form solutons for the SIR are known n ths case Instead, the numercal method descrbed n [6], [7] s typcally used to gve tght estmates of the SIR B LDPC Ensembles and the Jont BP Decoder he standard rregular LDPC ensemble s charactered by ts degree dstrbuton (dd), whch represents the fracton of nodes (or edges) of partcular degrees [8] From the edge perspectve, the dd par conssts of two polynomals λ(x) = λ x and ρ(x) = ρ x he LDPC ensemble can also be vewed from the node perspectve where ts dd par L(x) = L x and R(x) = R x he desgn rate of an LDPC ensemble s gven by r = L ()/R () When LDPC codes are transmtted over the ISI channel, one can construct a large graph by jonng the code graph and the channel graph together as depcted n Fg Workng on ths jont graph, a jont teratve decoder typcally passes the nformaton back and forth between the channel detector and the LDPC decoder hs technque s termed as turbo equalaton and was frst consdered by Doullard et al n the context of turbo codes [9] For analyss, we also requre the addton of a random scramblng vector to symmetre the effectve channel [20] hs s very smlar to usng a random coset of the LDPC code to allow analyss of the decoder usng the all-ero codeword assumpton; ths technque was also used n [6] where they proved a concentraton theorem and derved the densty evoluton (DE) equatons for ISI channels C Spatally-Coupled Ensembles he class of spatally-coupled ensembles can be defned qute broadly In ths paper, we manly consder two basc varants (see detals n [3]) as dscussed below ) he (l, r, L) ensemble: he (l, r, L) spatally-coupled ensemble (wth l odd so that ˆl = l N) can be constructed 2 from the underlyng (l, r)-regular LDPC ensemble At each poston from [, L] one has M bt nodes and l r M check nodes just lke n the (l, r)-regular case However, each bt node at poston s connected to one check node at each poston from ˆl to +ˆl In dong ths, one also needs to add l r M extra check nodes at each of ˆl extra postons on each sde For example, a jont code/channel graph for the (3, 6, L) ensemble and the ISI channels s shown n Fg 2 2) he (l, r, L, w) ensemble: he (l, r, L, w) can be obtaned wth the ntroducton of a smoothng parameter w One stll places M varable nodes at each poston n [, L] but places l M check nodes at each poston n [, L+w ] Each r bt node at poston s connected unformly and ndependently to a total of l check nodes at postons from the range [, + w ] III GENERAL ISI CHANNELS he man part of ths paper focuses on ISI channels wth general nose models he MAP upper bound for general bnary memoryless symmetrc channels was presented by Méasson et al and conjectured to be tght [2] For general ISI channels, we apply a smlar technque to gve an estmate of the MAP threshold of the underlyng ensemble by frst constructng the BP-GEXI curve that follows an area theorem he BP thresholds of the coupled ensembles are then computed va DE and the threshold saturaton effect s observed In addton, smulatons on the performance of the jont BP decoder for coupled codes of fnte length are conducted to valdate these thresholds A GEXI Curves for the ISI channels When the channel nput X n (X, X 2,, X n ) s chosen unformly at random from a sutable bnary lnear code, the ISI output wthout nose Z at some ndex s a dscrete random varable charactered by ts probablty mass functon p Z () for all n the alphabet Z For example, n the case of a dcode channel, Z = {0, +2, 2} and p Z (0) = 2, p Z (+2) = p Z ( 2) = 4 he channel from Z to Y s a Z -ary nput memoryless channel charactered by ts transton probablty densty p Y Z (y ) Wthout specfyng the ndex, we denote h H(Z Y ) and get h=h(z) p(y ) p()p(y ) log 2 { p( )p(y ) } dy Instead of lookng at a partcular channel, we assume that the channel from Z to Y s from a smooth famly {M(h )} h of Z -ary nput memoryless channels charactered by condtonal entropy h A further assumpton s made that all ndvdual channel famles are parametered n a smooth way by a common parameter 2 ɛ, e, h = H(Z Y )(ɛ) Wth the conventon that y y n y, defne φ (y ) {P Z Y ( y ) Z} and the random vector Φ φ (Y ) Each value of φ s a vector of length Z n the ( Z )- dmensonal probablty smplex he ndex of the vector assocated wth Z s denoted by [] One can see that Φ s a suffcent statstc for estmatng Z gven Y, e, Z Φ (Y ) Y forms a Markov chan 3 he code s proper [8, p 4] and ts dual code contans no codewords nvolvng only 0 s and a run of (ν + ) s 2 For the AWGN case, a convenent choce for ɛ s ɛ = 2σ 2 3 One way to see ths s to wrte P Y Z (y ) = P Z Y ( y ) Φ e [ P Y (y ) = ] P Z ( ) P Z ( ) P Y (y ), where e s the standard bass column vector wth a n the ndex [], and [] apply the result from [8, p 29]

3 L on on L Po s Po s t t trells node bt node check node M Fgure 2 he jont graph for the (l, r, L) ensemble over the ISI channels Illustrated n ths fgure s the case where l = 3 and r = 6 In the setup we consder, the bt transmsson s row by row where the order of transmsson wthn each row s mpled by the (green) arrows he (red) stars are to connect consecutve rows Defnton : Suppose the ntal state n the trells s S0 Let Xn chosen accordng to pxn (xn ) be the nput sequence, Zn be the ISI output sequence wthout nose and Yn be the fnal channel output sequence, e, Y s the result of transmttng Z over the smooth famly {M(h )}h of memoryless channels hen the th GEXI functon s H(Xn Yn (h,, hn ), S0 ) () G (h,, hn ) = h and the average GEXI functon s defned by G(h,, hn ) = n G (h,, hn ) For the case where all channel famles n = are the same, e, h = h, we have dh(xn Yn (h), S0 ) G(h) = n dh Lemma : Assume that all the channel famles are the same4, e, h = h hen, the th GEXI functon s gven by G (h) = p() v a, (v)κ, (v)dv where a, s the dstrbuton of the vector Φ gven Z =, v s a vector of length Z n the ( Z )-dmensonal probablty smplex and the GEXI kernel (for and ) s5 κ, (v) = v[ ] p(y ) p(y ) log2 { v[] p(y ) } dy p() p(y ) log2 { p( )p(y ) } dy p()p(y ) Proof: Suppose the ntal state s S0, we start by wrtng H(Xn Yn, S0 ) = H(Zn Yn, S0 ) = H(Z Yn, S0 ) + H(Z Yn, Z, S0 ) (2) For smplcty of notaton, we drop S0 n all the expressons although the dependency on S0 s always mpled From () and (2), t s clear that G (h) = H(Z Yn (h,, hn ) h =h h We also have H(Z Yn ) = H(Z Y, Φ (Y )) = p( )p(φ )p(y ) φ y p( φ )p(y ) log2 dy dφ p( φ )p(y ) (3) 4 Note that for the case of dfferent channel famles, one can stll compute the th GEXI functon as a functon of the common parameter 5 p(y ) s dependent on h and hence s dependent on where (3) follows from the Bayes theorem and the fact that p(, φ, y ) = p(, φ )p(y φ, ) = p( )p(φ )p(y ) (4) Note that (4) s true snce Y and Φ (Y ) are ndependent gven Z, e, Y Z Φ (Y ) akng dervatve and usng p( φ ) = p( y ), we get d p(y ) dh p( y )p(y ) log2 dy dφ p( y )p(y ) G (h) = p( ) p(φ ) φ y = p() a, (v)κ, (v)dv where κ, (v) = = y y v v[ ] p(y ) d p(y ) log2 { } dy dh v[] p(y ) v[ ] p(y ) h p(y ) log2 { } dy / v[] p(y ) Fnally, by seeng that h H(Z Y ( )) = p( )p(y ) = p() p(y ) log2 { } dy p()p(y ) y we obtan the result Remark : For σ = 0 n the AWGN case (or = 0 n erasure nose), h = 0 and a, s delta at v = e[] where e[] s the standard bass vector At ths extreme, G(0) = 0 snce κ, (v) = 0 At the other extreme σ (or at = for erasure nose), h = H(Z) (eg, 5 for the dcode channel) and G(h) = snce n ths case a, s delta at v[ ] = p( ) ) BP-GEXI curve (wth AWGN): In ths secton, we are partcularly nterested n computng the BP-GEXI functon for ISI channels wth AWGN In ths case, let ΦBP,` denote the extrnsc estmate of Z at the `th round of jont BP decodng If ΦBP,` s used nstead of Φ n the above formulas then one has the BP-GEXI (at the `th round) GBP,` n a smlar manner to [2] and the overall BP-GEXI GBP (h) = lm` GBP,` (h) Also, notce that the two extremes n Remark stll apply when the BP decoder s used nstead of the MAP decoder

4 (y e )2 2σ 2 2πσ 2 Next, AWGN mples that p(y ) = and then ɛ p(y ) = ((y ) 2 σ 2 )p(y ) herefore, the correspondng th BP-GEXI s G BP,l (h) = A B where A = B = p() v a BP,l, p() (v) log 2 { p(y ) { (y ) 2 σ 2 } v [] e ( )(2y ) 2σ v 2 } dy dv, [] p(y ) { (y ) 2 σ 2 } log 2 { p( ) p() e ( )(2y ) 2σ 2 } dy In the lmt of l, one can run the DE for ISI channels [6] to obtan the DE-FP and compute the quanttes A and B at ths FP Wth some abuse of notaton, let a (l), b (l), c (l) and d (l) denote the average densty of the bt-to-check, check-tobt, bt-to-trells and trells-to-bt messages, respectvely (see Fg ), at teraton l wth ntal values (at l = 0) beng 0, the delta functon at 0 Also, let n denote the densty of channel nose he DE update equaton for jont BP decodng of a general bnary-nput ISI channels s a (l) = d (l ) λ(b (l ) ), b (l) = ρ(a (l) ), c (l) = L(b (l) ), d (l) = Γ(c (l), n), where for a densty x, λ(x) = λ x ( ), ρ(x) = ρ x ( ) and L(x) = L x he operators and are the standard densty transformatons used n [8, p 8] he map Γ(, ) s not easy to compute n closed form for general trellses and often one needs to resort to the Monte Carlo methods (e, runnng the wndowed BCJR algorthm wth wndow parameter W on a long enough trells - see detals n [6]) to gve the estmates A smlar method was used to upper bound the MAP threshold for turbo codes over BMS channels [22] he denomnator B can be computed ether by numercal ntegraton or by Monte Carlo methods Meanwhle, the numerator A nvolves n the quantty v [] = p (Z = l ) where l denotes the computaton tree of depth l, rooted at ndex, whch ncludes all channel and code constrants assocated wth l teratons of decodng hs computaton tree l excludes the tree root y and s mpled by the decodng schedule n the DE equaton he quantty v [], due to complcatons from the trells, s not easy to obtan n closed form However, one can readly compute v [] as an extra output of the BCJR algorthm (whch s already used n DE) usng v [] s,s Z = α (s ) γ (s, s ) β (s ) where γ (s, s ) s probablty of the nput x that corresponds to the transton from state s (at tme ndex ) to state s at (tme ndex ) gven the computaton tree l Here, α ( ) and β ( ) are the standard forward and backward state probabltes n the BCJR algorthm Note that the scalng constant can be chosen so that v [] = G BP (h) h BP (3, 6) h MAP (3, 6) Area = r h Fgure 3 BP-GEXI curve for a (3, 6)-regular LDPC code over an AWGN dcode channel he upper bound h MAP s obtaned by settng the area under the BP-GEXI curve (the shaded regon) equal to the code rate B Upper Bound for the MAP hreshold he above-mentoned GEXI curve naturally obeys the area theorem H(Z) h G(h)dh = MAP H(Z) 0 G(h)dh = r herefore, one can apply the dscussed boundng technque, e, by fndng the largest value h MAP such that the area under the BP-GEXI curve equals the code rate, H(Z) G BP (h)dh = r, h MAP to obtan the MAP upper bound h MAP h MAP For example, the BP-GEXI curve for the (3, 6)-regular LDPC code over an AWGN dcode channel wth a(d) = ( D)/ 2 followng the analyss n Secton III-A s shown n Fg 3 In ths case, h BP (3, 6) 085 (the correspondng threshold measured n db s σ BP (3, 6) 703 ± 000 db) whle h MAP (3, 6) 0920 (or σ MAP (3, 6) 0959 ± 000 db) C Spatally-Coupled Codes on the ISI Channels Consder the (l, r, L) spatally-coupled ensemble For the ISI channels, the DE equaton for ths ensemble can be obtaned from the protograph chan n a smlar manner to the case of memoryless channels dscussed n [2] For each, j [ ˆl, L + ˆl], let a (l) j (and b(l) j ) denote the average densty of the messages from bt nodes at poston to check nodes at poston j (and the other way around) 6 Wth all the ntal message denstes (at l = 0) beng 0, the DE update equaton (for all [, L]) s a (l) j = d(l ) { b (l ) j }, j [ ˆl, + ˆl], j [ ˆl,+ˆl] j b (l) j = [j ˆl,j+ˆl] c (l) = b (l) j, j [ ˆl,+ˆl] d (l) = Γ(c (l), n) a (l) j, j [ ˆl, + ˆl], where j {j,,j t} x j and {,, t} x denote the operatons x j x j2 x jt and x x 2 x t, respectvely D Smulaton Results We start wth the (l, r, L) crcular ensemble obtaned by consderng all the postons > L of the protograph chan 6 For [, L], set a (l) j = +, the delta functon at +

5 Bt Error Rate σ SIR σ BP (5, 0, 44) σ BP (3, 6, 22) (3, 6, 22), M = 5000 (target) (3, 6, 22), M = 5000 (overall) (3, 6, 22), M = 502 (target) (5, 0, 44), M = 5000 (target) σ BP (3, 6) E b /N 0 (db) Fgure 4 BER and BP thresholds for the (3, 6)-regular, (3, 6, 22) and (5, 0, 44) spatally-coupled ensembles over the AWGN dcode channel to be the same as postons L (smlar to [5]) he order of bt transmssons s left to rght n each length-l row and then start wth the next row (n a total of M rows, see Fg 2) he I max(ν, l ) frst bts n each row are known hese bts wll break the crcular ensemble nto the (l, r, L I) ensemble and also serve as the plot bts to fx the trells state Due to ths fxng, one only needs to run the BCJR ndependently n each row and n a parallel manner [0], [] In our experments, we conduct smulatons over the AWGN dcode channel wth a(d) = ( D)/ 2 and memory ν = Frst, we use the DE n Sec III-C to compute the BP thresholds of the spatally-coupled codng scheme he results n Fg 4 reveal that σ BP (3, 6, 22) s roughly 0959 ± 000 db and approxmately the same as σ BP (3, 6, 44) whose rate loss s smaller Notce that ths s also roughly σ MAP (3, 6) - the MAP threshold estmate of the underlyng (3, 6)-regular ensemble, obtaned by the boundng technque, and s a sgnfcant mprovement over σ BP (3, 6) 703±000 db hs suggests that threshold saturaton occurs for regular ensembles Snce MAP decodng of regular ensembles can acheve the SIR [5], f threshold saturaton occurs, one can unversally approach the SIR of general ISI channels usng coupled codes wth jont teratve decodng o support ths, one can also see that for the (5, 0, 44) ensemble of the same rate as the (3, 6, 22) one, the threshold σ BP (5, 0, 44) 0834±000 db gets very close to the sgnal-to-nose rato (SNR) correspondng to the SIR (σ SIR 0823 ± 000 db usng the numercal method n [6], [7]) Although only smulatons wth the dcode channel are shown, the overall method s readly applcable to channels wth hgher memory Also shown n Fg 4 s the bt error rate (BER) versus SNR plot for the ensembles derved from the (l, r, L) crcular ensembles of fnte M = 502 and M = 5000 For each smulaton, we use l outer = 20 channel updates and between two such channel updates, we run l nner = 5 BP teratons on the code part alone he curves labeled target s the BER for the bts at poston I + (rght after the known bts) n the coupled chan whle the curve labeled overall s the overall BER for all the postons [I +, L] together One mght expect that the overall BER wll get closer to the target BER for large enough M and large enough number of teratons usng able I HRESHOLD ESIMAES OF (l, r)-regular ENSEMBLES OVER HE DEC AND DICODE AWGN CHANNEL FOR AWGN NOISE, HE HRESHOLDS ARE MEASURED IN db (l, r)- DEC Dcode AWGN regular ɛ BP ɛ MAP ɛ SIR σ BP σ MAP σ SIR (3, 6) (5, 0) an nducton argument From Fg 4, one can also observe that the overall BER for (3, 6, 22) and M = 5000 keeps gettng closer to the target BER as SNR slghtly ncreases hose BER curves are sgnfcantly mproved wth respect to ɛ BP (3, 6) - the BP threshold for the underlyng (3, 6)-regular ensemble IV ISI CHANNELS WIH ERASURE NOISE: HE DEC In ths secton, we brefly dscuss threshold saturaton on the DEC For ths channel, the GEXI curves mentoned above becomes dentcal (after scalng) to the EXI curves derved n [23] From these curves, one can also obtan a numercally tght upper bound on the MAP threshold of the underlyng ensemble and observe the threshold saturaton effect A Upper Bound on the MAP hreshold For the DEC and regular LDPC ensembles, the MAP upper bound was frst consdered n [23] In a recent report [24], the authors further provde a closed-form soluton for the BP- EXI and extended BP (EBP) EXI curves that can quckly gves an upper bound ɛ MAP on the MAP threshold by settng the area under the EBP curve (the shaded area n Fg 5) equal to the code rate he tghtness of the boundng technque s strongly suggested by a countng argument and for the (l, r)-regular ensemble, ths upper bound can be shown to quckly approach the erasure rate assocated wth the SIR when ncreasng l, r such that the code rate r s fxed (see arguments n [24] and facts n able I) B Spatally-Coupled Codes for the DEC Consder the (l, r, L, w) spatally-coupled ensemble We also follow the DE equaton dscussed n [3] to compute the BP thresholds of the coupled ensembles he man dfference s that we use the correct EBP-EXI curves wth ther operatonal meanng nstead of the EXI-lke curves used n [3] Let x (l) denote the expected erasure rate at teraton l from bt nodes at poston to check nodes where for [, L], one sets x (l) = 0 o compute both the stable and unstable FPs of DE, one can use the fxed entropy DE procedure outlned n [2, Sec VIII] where the normaled entropy of a constellaton x (l) = (x (l),, x(l) L ), whch s defned as χ(x (l) ) = L L = x (l), s kept constant at every teraton by varyng the channel parameter Wth each FP x obtaned, one obtans the EBP-EXI value of the spatally-coupled ensemble as L L = h EBP (x ) where h EBP ( ) s the EBP-EXI functon defned n [24] he threshold saturaton effect of couplng can be ncely seen by plottng the EBP-EXI curves for the uncoupled and coupled codes For example, Fg 5 shows the EBP curves for the (3, 6, L, 5) ensembles wth varous L along wth the

6 h EBP (ɛ) ɛ BP (3, 6) EBPcurve (3, 6) ɛ MAP (3, 6) L = 33 L = 7 L = 9 L = ɛ Fgure 5 EBP-EXI curves for (3, 6, L, 5) wth L = 2 ˆL + where ˆL = 2, 4, 8, 6, 32, 64, 28, 246 over the DEC As L grows larger, the rate loss becomes neglgble and the curves keep movng left, but they saturate at the MAP threshold of the underlyng regular ensemble EBP curve of the underlyng (3, 6)-regular ensemble From the EBP curves, one can determne ɛ BP (3, 6) and ɛ MAP (3, 6) he BP thresholds of spatally-coupled ensembles for small L due to rate-loss can have larger values, eg, ɛ BP (3, 6, 7, 6) > ɛ MAP (3, 6) However, for a wde range of L, e, L = 33, 65, 29, 257, 53, we observe that ɛ BP (3, 6, L, 5) whch s essentally ɛ MAP (3, 6) whle the rate loss gradually becomes nsgnfcant In [3], Kudekar and Kasa provded a smlar plot based on the EXIlke functon borrowed from the EXI functon of the BEC; though the pcture s smlar, t s not assocated wth an area theorem for the DEC Instead, we use a proper EXI functon h EBP and obtan the MAP threshold estmate ɛ MAP V CONCLUDING REMARKS In ths paper, we consder bnary communcaton over the ISI channels and numercally show that the threshold saturaton effect occurs on both the DEC and dcode channel wth AWGN o do ths, we construct the (G)EXI curves that satsfy the area theorem and obtan an upper bound on the threshold of the MAP decoder he upper bound s conjectured to be tght and, for the DEC, a numercal evdence can be shown to strongly support ths conjecture he observed threshold saturaton effect s valuable because by changng the underlyng regular LDPC ensemble combned wth the results of [5], t s shown that the jont BP decodng of spatally-coupled codes can unversally approach the SIR of the ISI channels Also, the convolutonal structure of the codes allows one to consder a wndowed decoder smlar to the one dscussed n [25], [26] All of these propertes suggest that spatally-coupled codes may be compettve n practce for systems wth ISI REFERENCES [] J Felstrom 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