Finite Length Weight Enumerator Analysis of Braided Convolutional Codes

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1 Fnte Length Weght Enumerator Analyss of Braded Convolutonal Codes Saeedeh Moloud, Mchael Lentmaer, and Alexandre Graell Amat Department of Electrcal and Informaton Technology, Lund Unversty, Lund, Sweden Department of Sgnals and Systems, Chalmers Unversty of Technology, Gothenburg, Sweden Abstract Braded convolutonal codes (BCCs are a class of spatally coupled turbo-lke codes (SC-TCs wth excellent belef propagaton (BP thresholds. In ths paper we analyze the performance of BCCs n the fnte block-length regme. We derve the average weght enumerator functon (WEF and compute the unon bound on the performance for the uncoupled BCC ensemble. Our results suggest that the unon bound s affected by poor dstance propertes of a small fracton of codes. By computng the unon bound for the expurgated ensemble, we show that the floor mproves substantally and very low error rates can be acheved for moderate permutaton szes. Based on the WEF, we also obtan a bound on the mnmum dstance whch ndcates that t grows lnearly wth the permutaton sze. Fnally, we show that the estmated floor for the uncoupled BCC ensemble s also vald for the coupled ensemble by provng that the mnmum dstance of the coupled ensemble s lower bounded by the mnmum dstance of the uncoupled ensemble. I. ITRODUCTIO Low-densty party-check (LDPC convolutonal codes [1], also known as spatally coupled LDPC (SC-LDPC codes [2], have attracted a lot of attenton because they exhbt a threshold saturaton phenomenon: the belef propagaton (BP decoder can acheve the threshold of the optmal maxmuma-posteror (MAP decoder. Spatal couplng s a general concept that s not lmted to LDPC codes. Spatally coupled turbo-lke codes (SC-TCs are proposed n [3], where some block-wse spatally coupled ensembles of parallel concatenated codes (SC-PCCs and serally concatenated codes (SC- SCCs are ntroduced. Braded convolutonal codes (BCCs [4] are another class of SC-TCs. The orgnal BCC ensemble has an nherent spatally coupled structure wth couplng memory m = 1. Two extensons of BCCs to hgher couplng memory, referred to as Type-I and Type-II BCCs, are proposed n [5]. The asymptotc behavor of BCCs, SC-PCCs and SC-SCCs s analyzed n [6] where the exact densty evoluton (DE equatons are derved for the bnary erasure channel (BEC. Usng DE, the thresholds of the BP decoder are computed for both uncoupled and coupled ensembles and compared wth the correspondng MAP thresholds. The obtaned numercal results demonstrate that threshold saturaton occurs for all consdered SC-TC ensembles f the couplng memory s large enough. Moreover, the occurrence of threshold saturaton s proved analytcally for SC-TCs over the BEC n [6], [7]. Ths work was supported n part by the Swedsh Research Councl (VR under grant # Whle the uncoupled BCC ensemble suffers from a poor BP threshold, the BP threshold of the coupled ensemble mproves sgnfcantly even for couplng memory m = 1. Comparng the BP thresholds of the SC-TCs n [6] ndcates that for a gven couplng memory, the Type-II BCC ensemble has the best BP threshold almost for all code rates. Motvated by the good asymptotc performance and the excellent BP thresholds of BCCs, our am n ths paper s analyzng the performance of BCCs n the fnte block-length regme by means of the ensemble weght enumerator. As a frst step, we derve the fnte block-length ensemble weght enumerator functon (WEF of the uncoupled ensemble by consderng unform random permutatons. Then we compute the unon bound for uncoupled BCCs. The unexpectedly hgh error floor predcted by the bound suggests that the bound s affected by the bad performance of codes wth poor dstance propertes. We therefore compute the unon bound on the performance of the expurgated ensemble by excludng the codes wth poor dstance propertes. The expurgated bound demonstrates very low error floors for moderate permutaton szes. We also obtan a bound on the mnmum dstance of the BCC ensemble whch reveals that the mnmum dstance grows lnearly wth the permutaton sze. Fnally, we prove that the codeword weghts of the coupled ensemble are lower bounded by those of the uncoupled ensemble. Thus, the mnmum dstance of the coupled BCC ensemble s larger than the mnmum dstance of the uncoupled BCC ensemble. From ths, we conclude that the estmated error floor of the uncoupled ensemble s also vald for the coupled ensemble. II. COMPACT GRAPH REPRESETATIO OF TURBO-LIKE CODES In ths secton, we descrbe three ensembles of turbo-lke codes, namely PCCs, uncoupled BCCs, and coupled BCCs, usng the compact graph representaton ntroduced n [6]. A. Parallel Concatenated Codes Fg. 1(a shows the compact graph representaton of a PCC ensemble wth rate R = 3 = 1 3, where s the permutaton sze. These codes are bult of two rate-1/2 recursve systematc convolutonal encoders, referred to as the upper and lower component encoder. The correspondng trellses are denoted by T U and T L, respectvely. In the graph, factor nodes, represented

2 v U v U T U T U T U t 1 T U t T U t+1 T U t T U t 1 T U t+1 u T L u T L ut u t u t 1 u t+1 ut+1 ut 1 T L t 1 T L t T L t 1 T L t T L t+1 T L t+1 v L v L v L t 1 v L v L t t+1 v L t 1 v L v L t t+1 (a (b (a (b Fg. 1. (a Compact graph representaton of (a PCCs (b BCCs. by squares, correspond to trellses. All nformaton and party sequences are shown by black crcles, called varable nodes. The nformaton sequence, u, s connected to factor node T U to produce the upper party sequence v U. Smlarly, a reordered copy of u s connected to T L to produce v U. In order to emphasze that a reordered copy of u s used n T L, the permutaton s depcted by a lne that crosses the edge whch connects u to T L. B. Braded Convolutonal Codes 1 Uncoupled BCCs: The orgnal BCCs are nherently a class of SC-TCs [3], [4], [5]. An uncoupled BCC ensemble can be obtaned by talbtng a BCC ensemble wth couplng length L = 1. The compact graph representaton of ths ensemble s shown n Fg. 1(b. The BCCs of rate R = 1 3 are bult of two rate-2/3 recursve systematc convolutonal encoders. The correspondng trellses are denoted by T U and T L, and referred to as the upper and lower trellses, respectvely. The nformaton sequence u and a reordered verson of the lower party sequence v L are connected to T U to produce the upper party sequence v U. Lkewse, a reordered verson of u and a reordered verson of v U are connected to T L to produce v L. 2 Coupled BCCs, Type-I: Fg. 2(a shows the compact graph representaton of the orgnal BCC ensemble, whch can be classfed as Type-I BCC ensemble [5] wth couplng memory m = 1. As depcted n Fg. 2(a, at tme t, the nformaton sequence u t and a reordered verson of the lower party sequence at tme t 1, vt 1, L are connected to T U t to produce the current upper party sequence vt U. Lkewse, a reordered verson of u t and vt 1 U are connected to TL t to produce vt L. At tme t, the nputs of the encoders only come from tme t and t 1, hence the couplng memory s m = 1. 3 Coupled BCCs, Type-II: Fg. 2(b shows the compact graph representaton of Type-II BCCs wth couplng memory m = 1. As depcted n the fgure, n addton to the couplng of the party sequences, the nformaton sequence s also coupled. At tme t, the nformaton sequence u t s dvded nto two sequences u t,0 and u t,1. Lkewse, a reordered copy of the nformaton sequence, ũ t, s dvded nto two sequences u t,0 and u t,1. At tme t, the frst nputs of the upper Fg. 2. Compact graph representaton of coupled BCCs wth couplng memory m = 1 (a Type-I (b Type-II. and lower encoders are reordered versons of the sequences (u t,0, u t 1,1 and (ũ t,0, ũ t 1,1, respectvely. III. IPUT-PARITY WEIGHT EUMERATOR A. Input-Party Weght Enumerator for Convolutonal Codes Consder a rate-2/3 recursve systematc convolutonal encoder. The nput-party weght enumerator functon (IP-WEF, A(I 1, I 2, P, can be wrtten as A(I 1, I 2, P = A 1, 2,pI 1 I 2 P p, 1 2 p where A 1, 2,p s the number of codewords wth weghts 1, 2, and p for the frst nput, the second nput, and the party sequence, respectvely. To compute the IP-WEF, we can defne a transton matrx between trells sectons denoted by M. Ths matrx s a square matrx whose element n the rth row and the cth column [M] r,c corresponds to the trells branch whch starts from the rth state and ends up at the cth state. More precsely, [M] r,c s a monomal I 1 1 I2 2 P p, where 1, 2, and p can be zero or one dependng on the branch weghts. For a rate-2/3 convolutonal encoder wth generator matrx G = ( 1 0 1/ /7 n octal notaton, the matrx M s M(I 1, I 2, P =, (1 1 I 2 P I 1 I 2 I 1 P I 1 I 1 I 2 P I 2 P I 2 P 1 I 1 P I 1 I 2 I 1 I 2 P I 1 P I 2. (2 Assume termnaton of the encoder after trells sectons. The IP-WEF can be obtaned by computng M. The element [M ] 1,1 of the resultng matrx s the correspondng IP-WEF. The WEF of the encoder s defned as A(W = A w W w = A(I 1, I 2, P I1=I 2=P =W, w=1 where A w s the number of codewords of weght w.

3 In a smlar way, we can obtan the matrx M for a rate-1/2 convolutonal encoder. The IP-WEF of the encoder s [M ] 1,1 and s gven by A(I, P = A,p I P p, p where A,p s the number of codewords of nput weght and party weght p. BER Vs. FER Smulaton BER of PCC Smulaton FER of PCC Smulaton FER of BCC Smulaton BER of BCC Bound on BER of BCC Bound on FER of BCC Bound on FER BER of PCC B. Parallel Concatenated Codes For the PCC ensemble n Fg. 1(a, the IP-WEFs of the upper and lower encoders are defned by A T U(I, P and A T L(I, P, respectvely. The IP-WEF of the overall encoder, A PCC (I, P, depends on the permutaton that s used, but we can compute the average IP-WEF for the ensemble. The coeffcents of the IP-WEF of the PCC ensemble [8] can be wrtten as Ā PCC,p = p 1 A T U C. Braded Convolutonal Codes,p1 A T L,p p 1 (. (3 For the uncoupled BCC ensemble depcted n Fg. 1(b, the IP-WEFs of the upper and lower encoders are denoted by A T U(I 1, I 2, P and A T L(I 1, I 2, P, respectvely. To derve the average WEF, we have to average over all possble combnatons of permutatons. The coeffcents of the IP-WEF for the uncoupled BCC ensemble can be wrtten as Ā BCC,p = p 1 A T U,p1,p p 1 A T L,p p 1,p 1 ( ( p 1 ( p p 1. (4 Remark: It s possble to nterpret the BCCs n Fg. 1(b as protograph-based generalzed LDPC codes wth trells constrants. As a consequence, the IP-WEF of the ensemble can also be computed by the method presented n [9], [10]. IV. PERFORMACE BOUDS FOR BRAIDED COVOLUTIOAL CODES A. Bounds on the Error Probablty Consder the PCC and BCC ensembles n Fg. 1 wth permutaton sze. For transmsson over an addtve whte Gaussan nose (AWG channel, the bt error rate (BER of the code s upper bounded by P b 2 =1 p=1 Ā,pQ 2( + pr, (5 and the frame error rate (FER s upper bounded by 2 P F Ā,p Q 2( + pr, (6 =1 w=1 where Q(. s the Q-functon and s the sgnal-to-nose rato. The truncated unon bounds on the BER and FER of the PCC ensemble n Fg. 1(a are shown n Fg. 3. We have consdered dentcal component encoders wth generator / Fg. 3. Smulaton results and bound on performance of the PCC and BCC ensembles. BER Vs. FER FER of BCC, Unformly random permutaton BER of BCC, Unformly random permutaton FER of BCC, Fxed permutaton BER of BCC, Fxed permutaton / Fg. 4. Smulaton results for BCC wth unformly random permutatons and fxed permutaton. matrx G = (1, 5/7 n octal notaton and permutaton sze = 512. We also plot the bounds for the uncoupled BCC ensemble wth dentcal component encoders wth generator matrx gven n (1. The bounds are truncated at a value greater than the correspondng Glbert-Varshamov bound. Smulaton results for the PCC and the uncoupled BCC ensemble are also provded n Fg. 3. To smulate the average performance, we have randomly selected new permutatons for each smulated block. The smulaton results are n agreement wth the bounds for both ensembles. It s nterestng to see that the error floor for the BCC ensemble s qute hgh and the slope of the floor s even worse than that of the PCC ensemble. Fg. 4 shows smulaton results for BCCs wth randomly selected but fxed permutatons. Accordng to the fgure, for the BCC wth fxed permutatons, the performance mproves and no error floor s observed. For example, at = 2.5dB, the FER mproves from to Comparng the smulaton results for permutatons selected unformly at random and fxed permutatons, suggests that the bad performance of the BCC ensemble s caused by a fracton of codes wth poor dstance propertes. In the next subsecton, we demonstrate that the performance of BCCs mproves sgnfcantly f we use expurgaton.

4 Mnmum dstance α=0 α=0.5 α= Permutaton sze Fg. 5. Bound on the mnmum dstance for the BCC ensemble. B. Bound on the Mnmum Dstance and Expurgated Unon Bound Usng the average WEF, we can derve a bound on the mnmum dstance. We assume that all codes n the ensemble are selected wth equal probablty. Therefore, the total number of codewords of weght w over all codes n the ensemble s c Āw, where c s the number of possble codes. As an example, c s equal to (! 3 for the BCC ensemble. Assume that ˆd 1 Ā w < 1 α, (7 w=1 for some nteger value ˆd > 1 and a gven α, 0 α < 1. Then a fracton α of the codes cannot contan codewords of weght w < ˆd. If we exclude the remanng fracton 1 α of codes wth poor dstance propertes, the mnmum dstance of the remanng codes s lower bounded by d mn ˆd. The best bound can be obtaned by computng the largest ˆd that satsfes the condton n (7. Consderng Āw for dfferent permutaton szes, ths bound s shown n Fg. 5 for α = 0, α = 0.5, and α = Accordng to the fgure, the mnmum dstance of the BCC ensemble grows lnearly wth the permutaton sze. The bound correspondng to α = 0.95, whch s obtaned by excludng only 5% of the codes, s very close to the exstence bound for α = 0. Ths means that only a small fracton of the permutatons leads to poor dstance propertes. Excludng the codes wth d mn < ˆd, the BER of the expurgated ensemble s upper bounded by P b 1 α k =1 (n k p=1 +p ˆd Ā,pQ 2( + pr. (8 For the BCC ensemble, the expurgated bounds on the BER are shown n Fg. 6 for α = 0.5 and permutaton szes = 128, 256, and 512. The error floors estmated by the expurgated bounds are much steeper than those gven by the unexpurgated bounds. The expurgated bounds on the BER are also shown n Bounds on BER Expurgated, BCC, =512 Expurgated, BCC, =256 Expurgated, BCC, =128 BCC, =512 BCC, =256 BCC, =128 Expurgated, PCC, =128 Expurgated, PCC, =256 Expurgated, PCC, = / Fg. 6. Expurgated unon bound on performance of PCC and BCC. Fg. 6 for the PCC ensemble. These bounds demonstrate that expurgaton does not mprove the performance of the PCC ensemble sgnfcantly. V. SPATIALLY COUPLED BRAIDED COVOLUTIOAL CODES The performance of BCCs n the waterfall regon can be sgnfcantly mproved by spatal couplng. To demonstrate t, we provde smulaton results for the uncoupled BCCs and Type-II BCCs for = 1000 and For coupled BCCs, we consder couplng length L = 100 and a sldng wndow decoder wth wndow sze W = 5 [11]. For all cases, the permutatons are selected randomly but fxed. Smulaton results are shown n Fg. 7. Accordng to the fgure, for a gven permutaton sze, Type-II BCCs perform better than uncoupled BCCs. As an example, for = 5000, the performance mproves almost 1.5 db. We also compare the uncoupled and coupled BCCs wth equal decodng latency. In ths case, we consder = 5000 and = 1000 for the uncoupled and coupled BCCs, respectvely. Consderng equal decodng latency, the performance of the coupled Type-II BCCs s stll sgnfcantly better than that of the uncoupled BCCs. The coupled BCCs have good performance n the waterfall regon and ther error floor s so low that t cannot be observed. It s possble to generalze equaton (4 for the coupled BCC ensembles n Fg. 2 but the computatonal complexty s sgnfcantly ncreased. In the followng theorem, we establsh a connecton between the WEF of the uncoupled BCC ensemble and that of the coupled ensemble. More specfcally, we show that the weghts of codewords cannot decrease by spatal couplng. A smlar property s shown for LDPC codes n [12], [13], [14]. Theorem 1: Consder an uncoupled BCC C wth permutatons Π, Π U and Π L. Ths code can be obtaned by means of talbtng an orgnal (coupled BCC C wth tme-nvarant permutatons Π t = Π, Π U t = Π U and Π L t = Π L. Let v = (v 1,..., v t,..., v L, v t = (u t, vt U, vt U, be an arbtrary code sequence of C. Then there exsts a codeword ṽ C that satfes w H (ṽ w H (v,

5 BER Type II BCC, =5000 Type II BCC, =1000 Uncoupled BCC, =1000 Uncoupled BCC, = / Fg. 7. Smulaton results for uncoupled and coupled BCCs, fxed permutaton, = 1000 and = e., the couplng does ether preserve or ncrease the Hammng weght of vald code sequences. Proof: A vald code sequence of C has to satsfy the local constrants ( ut vt 1 L Π U t vt U H T U = 0 (9 ( ut Π t vt 1 U ΠL t vt L H T L = 0 (10 for all t = 1,..., L, where H U and H L are some party-check matrces that represent the contrants mposed by the trellses of the upper and lower component codes, respectvely. Snce these constrants are lnear and tme-nvarant, t follows that any superposton of vectors v t = (u t,, from dfferent tme nstants t {1,..., L} wll also satsfy (9 and (10. In partcular, f we let then ũ = u t, ṽ L = vt L, ṽ U = (ũ ṽl Π U ṽ U H T U = 0 (11 (ũ Π ṽu Π L ṽ L H T L = 0. (12 Here we have mplctly made use of the fact that v t = 0 for t < 1 and t > L. But now t follows from (11 and (12 that ṽ = (ũ, ṽ U, ṽ L C,.e., we obtan a codeword of the uncoupled code. If all non-zero symbols wthn v t occur at dfferent postons for t = 1,..., L, then w H (ṽ = w H (v. If, on the other hand, the support of non-zero symbols overlaps, the weght of ṽ s reduced accordngly and w H (ṽ < w H (v. Corollary 1: The mnmum dstance of the coupled BCC C s larger than or equal to the mnmum dstance of the uncoupled BCC C: d mn (C d mn ( C. From Corollary 1, we can conclude that the estmated floor for the uncoupled BCC ensemble s also vald for the coupled BCC ensemble. VI. COCLUSIO The fnte block length analyss of BCCs performed n ths paper, together wth the DE analyss n [6] show that BCCs are a very promsng class of codes. They provde both close-to-capacty thresholds and very low error floors even for moderate block lengths. We would lke to remark that the bounds on the error floor n ths paper assume a maxmum lkelhood decoder. In practce, the error floor of the BP decoder may be determned by absorbng sets. Ths can be observed, for example, for some ensembles of SC-LDPC codes [15]. Therefore, t would be nterestng to analyze the absorbng sets of BCCs. REFERECES [1] A. Jménez Feltström and K.Sh. Zgangrov, Perodc tme-varyng convolutonal codes wth low-densty party-check matrces, IEEE Trans. Inf. Theory, vol. 45, no. 5, pp , Sep [2] S. Kudekar, T.J. Rchardson, and R.L. Urbanke, Threshold saturaton va spatal couplng: Why convolutonal LDPC ensembles perform so well over the BEC, IEEE Trans. Inf. Theory, vol. 57, no. 2, pp , Feb [3] S. Moloud, M. Lentmaer, and A. Graell Amat, Spatally coupled turbo codes, n Proc. 8th Int. Symp. on Turbo Codes and Iteratve Inform. Process. (ISTC, Bremen, Germany, [4] W. Zhang, M. Lentmaer, K.Sh. Zgangrov, and D.J. Costello, Jr., Braded convolutonal codes: a new class of turbo-lke codes, IEEE Trans. Inf. Theory, vol. 56, no. 1, pp , Jan [5] M. Lentmaer, S. Moloud, and A. Graell Amat, Braded convolutonal codes - a class of spatally coupled turbo-lke codes, n Proc. Int. Conference on Sgnal Process. and Commun. (SPCOM, Bangalore, Inda, [6] S. Moloud, M. Lentmaer, and A. Graell Amat, Spatally coupled turbo-lked codes, submtted to IEEE Trans. Inf. Theory, avalable onlne at TC Thresholds.pdf, Dec [7] S. Moloud, M. Lentmaer, and A. Graell Amat, Threshold saturaton for spatally coupled turbo-lke codes over the bnary erasure channel, n Inf. Theory Workshop (ITW, Jeju, South Korea, Oct [8] S. Benedetto and G. Montors, Unvelng turbo codes: some results on parallel concatenated codng schemes, IEEE Trans. Inf. Theory, vol. 42, no. 2, pp , Mar [9] S. Abu-Surra, W.E. Ryan, and D. Dvsalar, Ensemble enumerators for protograph-based generalzed LDPC codes, IEEE Global Telecommuncatons Conference (GLOBECOM, pp , ov [10] S. Abu-Surra, D. Dvsalar, and W. E. Ryan, Enumerators for protograph-based ensembles of LDPC and generalzed LDPC codes, IEEE Trans. on Inf. Theory, vol. 57, no. 2, pp , Feb [11] M. Zhu, D. G. M. Mtchell, M. Lentmaer, D. J. Costello, and B. Ba, Wndow decodng of braded convolutonal codes, n Inf.Theory Workshop (ITW, Jeju, South Korea, Oct [12] D. Truhachev, K. S. Zgangrov, and D. J. Costello, Dstance bounds for perodcally tme-varyng and tal-btng LDPC convolutonal codes, IEEE Trans. on Inf. Theory, vol. 56, no. 9, pp , Sep [13] D. G. M. Mtchell, A. E. Pusane, and D. J. Costello, Mnmum dstance and trappng set analyss of protograph-based LDPC convolutonal codes, IEEE Trans. on Inf. Theory, vol. 59, no. 1, pp , Jan [14] R. Smarandache, A. E. Pusane, P. O. Vontobel, and D. J. Costello, Pseudocodeword performance analyss for ldpc convolutonal codes, IEEE Trans. on Inf. Theory, vol. 55, no. 6, pp , June [15] D. G. M. Mtchell, L. Dolecek, and D. J. Costello, Absorbng set characterzaton of array-based spatally coupled LDPC codes, n IEEE Int. Sym. on Inf. Theory (ISIT, Honolulu, HI, USA, June 2014.