A Sub-Optimal Log-Domain Decoding Algorithm for Non-Binary LDPC Codes

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1 Procdings of th 9th WSEAS Intrnational Confrnc on APPLICATIONS of COMPUTER ENGINEERING A Sub-Optimal Log-Domain Dcoding Algorithm for Non-Binary LDPC Cods CHIRAG DADLANI and RANJAN BOSE Dpartmnt of Elctrical Enginring Indian Institut of Tchnology, Dlhi INDIA Phon: , rbos@.iitd.ac.in Abstract: - Low-Dnsity Parity-Chck (LDPC cods hav gaind significant popularity rcntly and hav bn rcommndd for various wirlss standards of th futur. Non-binary LDPC cods provid all th wll-known advantags of binary LDPC cods but allow for much shortr block lngths. Th sum-product dcoding algorithm for non-binary LDPC cods is too compl for fficint implmntation. Hnc, w hav rprsntd th sum-product algorithm in a novl logdomain form in trms of oprators dfind arlir in litratur. This form gratly simplifis th ky stp in th chck-nod procssing and allows us to rplac a ma* opration in th dcodr by a ma opration. This lads to a sub-optimal dcoding algorithm which w propos in this papr. Ky-Words: - Low Dnsity Parity Chck (LDPC Cods, Dcoding Algorithm 1 Introduction Low-Dnsity Parity-Chck Cods Low-Dnsity Parity- Chck (LDPC Cods ar linar block cods which ar dfind in trms of a vry spars parity-chck matri [1-6]. LDPC cods wr discovrd by R. G. Gallagr [2], in 1962 who dfind thm as A Low-Dnsity Parity- Chck Cod is a cod spcifid by a parity chck matri which has th following proprtis: ach column contains a small fid numbr 3 of 1s and ach row contains a small fid numbr k > of 1s. Rgular LDPC cods ar thos in which th numbr of symbols participating in ach parity chck as wll as th numbr of parity chcks a symbol participats in is constant. Cods which do not satisfy this proprty ar known as irrgular cods [6]. Ths two valus ar rspctivly known as th chck nod dgr (d c and variabl nod dgr (d v. LDPC cods hav bn shown to achiv arbitrarily low probability of rror for transmission rats blow but vry clos to th channl capacity as th block-lngth gos to infinity [5]. LDPC cods hav bn shown to b ffctiv rror-corrcting cods whn dcodd using soft dcoding. This papr is organizd as follows. W bgin with an introduction to EXIT charts and thir proprtis. W thn prsnt a novl rprsntation of th sum-product dcoding algorithm in th log domain. This rprsntation lads us to a sub-optimal dcoding algorithm. Finally, w prov th convrgnc bhaviour of this algorithm using EXIT charts. 2 EXIT Charts An important tool usd to track th convrgnc bhaviour of itrativ dcoding algorithms for LDPC cod is EXIT (Etrinsic Information Transfr Function charts [4]. Etrinsic Information Transfr Function (EXIT Charts track th flow of mutual information in th dcodr through th dcoding itrations. EXIT charts provid us an ffctiv tool to study th bhaviour of dcodrs by tracking ust on scalar paramtr. EXIT charts hav bn usd in this papr to study and compar th convrgnc proprtis of dcoding algorithms. Hnc, w hav assumd that all mssags which ar fd to th componnt dcodrs (th variabl nod dcodr and th chck nod dcodr ar Gaussian distributd and indpndnt[3]. Without loss of gnrality, all simulations hav bn carrid out for rgular LDPC cods with variabl dgr d v 3 and chck dgr d c 6 and hnc th rat of th cod is givn by dv R 1 d (1 c ISSN: ISBN:

2 Procdings of th 9th WSEAS Intrnational Confrnc on APPLICATIONS of COMPUTER ENGINEERING 3 Th Sum-Product Dcoding Algorithm Th sum-product algorithm for dcoding non-binary LDPC cods has bn studid widly in litratur. Th variabl nod procssing stp is simpl and computationally light to implmnt. Th chck nod dcoding stp on th othr hand, rquirs a sris of lngthy computations. Th lft-bound itration of th sum-product algorithm in th probability domain, as dscribd in litratur, can b rprsntd in a compact form as follows [1]: + g g d 1 1 ( ( d n g g n n 1 r [ ( ] whr r ( n is a right-bound mssag vctor (input mssag to th lft-bound stp and convolution oprator oprats on d -1 incoming mssags whr d is th chck nod dgr. Th + g oprator usd abov is dfind as follows [1]:,,..., g 1 + g ( q 1 + g Similarly, th g oprator is dfind as follows:,,..., 0 g 2. g ( q 1. g It should b notd that th ky stp in this procssing is th convolution oprator. Hnc, w focus on writing th convolution opration in a simplr form. 4 Log-domain GF(q Convolution Lt us considr two probability vctors p ( p0, p1, p2, p3 and q ( q0, q1, q2, q3 dfind ovr GF(4. Th corrsponding GF(4 vctor lmnts can b writtn as follows: p0 p0 w0 log( 0 w1 log( p0 p1 p0 p0 w2 log( w3 log( p2 p3 q0 q0 v0 log( 0 v1 log( q0 q1 q0 q0 v 2 log( log( q q (5 2 3 (2 (3 (4 By dfinition, th rsult of th convolution opration on p and q probability vctors is givn by: q (,,, q (0, Th corrsponding vctor of this convolution is givn by:, p0q0+ p1q1 + p2q2+ p3q3 (21, log( log( p0q3+ p1q2 + p2q1+ p3q0 log( p0q0+ p1q1 + p2q2+ p3q3 (22 Also, it must b notd that for any i and, q (0 p q p q ( wi+ v 0 0 ( w0+ v1 ( w1+ v0 ( w2+ ( w log(, ( w0+ v0 ( w1+ v1 ( w2+ ( w ( w0+ ( w1+ ( w2+ v0 ( w3+ v1 + + log(, ( w0+ v0 ( w1+ v1 ( w2+ ( w ( w0+ ( w1+ ( w2+ v1 ( w3+ v0 + + log(, ( w0+ v0 ( w1+ v1 ( w2+ ( w i (6 (7 (8 Thrfor, th vctor of th convolution of th two vctors can b writtn as: W dfin th ma oprator as follows [6]: (9 ISSN: ISBN:

3 Procdings of th 9th WSEAS Intrnational Confrnc on APPLICATIONS of COMPUTER ENGINEERING ma ( ( log 1 (10 Using this ma* oprator w rdfin th convolution oprator as: q (0, ma [ ( w + v, ( w + v, ( w + v, ( w + v ] J ma [ ( w + v, ( w + v, ( w + v, ( w + v ], ma [ ( w + v, ( w + v, ( w + v, ( w + v ] ma [ ( w + v, ( w + v, ( w + v, ( w + v ], ma [ ( w + v, ( w + v, ( w + v, ( w + v ] ma [ ( w + v, ( w + v, ( w + v, ( w + v ] (11 Hnc, th convolution oprator can now b dfind for th gnral GF(q cas as follows: (1 (2 ( a k a + (1 (2 [ ] ma* k GF( q a GF( q (12 ma ( a, b ma( a, b + ln[1 + p( a b ] (13 From this altrnat dfinition it can b sn that th ma* oprator can b split as th sum of th ma oprator and an additional trm. Hnc, if it can b shown that th contribution of th additional trm is ngligibl, w will b abl to optimiz th sum-product dcoding algorithm by approimating th ma* oprator with th ma oprator in th chck nod procssing stag. It must b notd that this is possibl only whn th dcoding is don in th domain and is du to th rprsntation of th GF(q convolution oprator as a sris of ma* oprations. Th charactristics of th ma* function can b studid by plotting a graph of th ln[1 + p( ] function in Fig.1 for varying valus of. Hr rprsnts th diffrnc btwn th inputs of th ma* function. Hnc, sinc th contribution of th scond trm in th ma* oprator is ngligibl for largr valus, w hav nglctd it and significantly rducd th computational burdn of th chck nod procssing stp. It can b sn that th convolution oprator in GF(q now contains a sris of ma* oprations whn cutd in th domain as compard to a sris of multiplication oprations whn cutd in th probability domain. Thrfor, by oprating in th domain w can rduc th computational complity of th convolution oprator. Our optimization of th sumproduct algorithm is basd upon this particular structur of th convolution oprator. 5 Sub-optimal Dcoding Algorithm Th sub-optimal dcoding algorithm for non-binary LDPC cods is basd upon am approimation similar to th on mad in th sum-min dcoding algorithm for binary LDPC cods [6-9]. Th algorithm works in th log-liklihood ratio domain. Th ky sub-optimal stp in th algorithm is th rplacmnt of a ma function by a ma function. Th ma oprator which was dfind and usd in th prvious chaptr for rprsnting th convolution oprator in th domain can b altrnatly dfind as follows [6]: Figur 1: Proprtis of th ma* function 6 EXIT Chart Simulations EXIT chart simulations for th sub-optimal algorithm ar prsntd in Fig. 3 and Fig. 4 compars th chck nod dcodr curvs of th sub-optimal algorithm and th sum-product algorithm. Th chck nod procssing is th sam in both algorithms and hnc, only a chang in th chck nod curv was obsrvd. In cas of th EXIT chart for sum-product dcoding, th dcoding tractory has also bn plottd along with th EXIT chart. Both dcodrs hav bn implmntd in MATLAB and dcoding has bn tstd for block- ISSN: ISBN:

4 Procdings of th 9th WSEAS Intrnational Confrnc on APPLICATIONS of COMPUTER ENGINEERING lngths of up to 50,000 symbols. Th dcodrs convrg in a finit numbr of stps. Figur 2: EXIT chart for sum-product dcoding of non-binary LDPC cods Figur 4: Comparison of chck nod curvs for sub-optimal and sum-product dcoding 7 Conclusions Th chck nod curvs for both dcodrs ar compard in Fig. 4. Also, w s that thr is a loss in thrshold SNR of approimatly 0.5 db for th sub-optimal algorithm as sn from Fig. 5. Th thrshold of th sumproduct dcoding lis at 1.8 db whras that of th suboptimal algorithm is at 2.3 db. Hnc, th sub-optimal algorithm rquirs an additional 0.5 db of input SNR for th dcodr to convrg. Figur 3: EXIT chart for sub-optimal dcoding algorithm Figur 5: Comparison of numbr of itrations ndd by both algorithms ISSN: ISBN:

5 Procdings of th 9th WSEAS Intrnational Confrnc on APPLICATIONS of COMPUTER ENGINEERING Howvr, a spdup of 7% (for 100 symbol long cods and 9% (for 1000 symbol long cods has bn obsrvd in th MATLAB implmntation of th sub-optimal dcodr. Whn th dcodr is implmntd on a DSP, significant prformanc improvmnts will b sn bcaus th sub-optimal dcodr now consists ntirly of additions, subtractions, comparisons and log and division functions hav bn liminatd. Rfrncs: [1] Bnnatan and D. Burshtin, Dsign and analysis of nonbinary LDPC cods for arbitrary discrtmmorylss channls, IEEE Transactions on Information Thory, vol. 52, no. 2, pp , Fbruary 2006 [2] R. G. Gallagr, Low Dnsity Parity Chck Cods. Cambridg, MA MIT Prss, 1963 [3] E. Sharon, A. Ashikhmin, and S. Litsyn, EXIT functions for th Gaussian channl, in Proc. 40th Annu. Allrton Conf. Communication, Control, and Computrs, Allrton, IL, Oct. 2003, pp [4] S. tn Brink, G. Kramr, and A. Ashikhmin, Dsign of low-dnsity parity-chck cods for modulation and dtction, IEEE Trans. Commun., vol. 52, no. 4, pp , Apr [5] S. tn Brink, Convrgnc bhavior of itrativly dcodd paralll concatnatd cods, IEEE Trans. Commun., vol. 49, no. 10, pp , Oct [6] H. Wymrsch, H. Stndam, and M. Monclay, Log-domain dcoding of LDPC cods ovr GF(q, in Proc. IEEE Int. Conf. Communications, Paris, Franc, Jun. 2004, pp [7] M. C. Davy and D. MacKay, Low-dnsity paritychck cods ovr GF(q, IEEE Commun. Ltt., vol. 2, pp , Jun 1998 [8] T. J. Richardson and R. L. Urbank, Th capacity of low-dnsity parity chck cods undr mssagpassing dcoding, IEEE Trans. Inform. Thory, vol. 47, pp , Fb [9] Byrs, G.J.; Takawira, F., "EXIT charts for nonbinary LDPC cods," Communications, ICC IEEE Intrnational Confrnc on, vol.1, no.pp Vol. 1, May 2005 ISSN: ISBN:

1 Minimum Cut Problem

1 Minimum Cut Problem CS 6 Lctur 6 Min Cut and argr s Algorithm Scribs: Png Hui How (05), Virginia Dat: May 4, 06 Minimum Cut Problm Today, w introduc th minimum cut problm. This problm has many motivations, on of which coms