A Complexity Reduction Method for Extended Min-Sum based Nonbinary LDPC Decoder

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1 M Abuselvi et.al / Iteratioal Joural of Egieerig ad Techology (IJET) A omplexity Reductio Method for Exteded Mi-Sum based Nobiary LDP Decoder M Abuselvi #1, S Salivahaa * # Assistat Professor, Departmet of EE, SSN college of egieerig Kalavakkam, Tamiladu, Idia 1 abuselvim@ss.edu.i * Professor, Departmet of EE, SSN college of egieerig Kalavakkam, Tamiladu, Idia salivahaas@ss.edu.i Abstract Nobiary LDP codes are a class of liear block codes havig the performace closer to Shao s limit. odes defied over higher order of Galois field, have icreased computatio complexity ad thereby it put forth the challeges i efficiet hardware realizatio of the decoder. I this paper, modificatios i the No-biary LDP decoder to obtai reduced cofiguratio sets, aimed to reduce the decodig complexity is proposed. ofiguratio sets are the possible combiatios of the message sets that are ivolved i the parity check equatio of LDP decoder. Mi-sum algorithm based decoder is modelled with the specificatio of IEEE 8.11 stadard, with the codelegth of 648, rate ½ for GF (4) Galois field, over the AWGN chael. The mathematical formulatio of the proposed complexity reductio methodology is preseted. The decodig performace of reduced cofiguratio sets is aalyzed ad it is foud that the complexity is reduced by a average of 83%, with egligible degradatio i performace i terms of bit error rate. Keyword- Liear block code, Iterative decodig, No-biary LDP decoder, ofiguratio Set, Mi- Sum decodig algorithm I. INTRODUTION Error correctio codig (E) is the mea whereby errors ca be idetified based upo received iformatio. Low Desity Parity heck (LDP) codes are used i emergig wireless commuicatio techologies amely DVB-S ad IEEE 8.11 ad IEEE 8.11g. It is claimed that LDP codes will be the choice for future stadards, such as 5G, ad later versios of WiMax etc. Gallager iveted the LDP code i 196, which is superior i terms of decodig tha turbo codes decodig [1]. I additio LDP approaches the Shao s theoretical limit of chael capacity []. By the work of David Mackay, LDP s revisios i mathematical models are evolved [3]. Use of the LDP codes o the wireless sesor etwork shows that the system ca be more eergy efficiet, i additio to better reliability [4]. Oe of the desig issues of LDP decodig is to make a tradeoff betwee the performace ad complexity. Recetly research has bee cocetrated o reducig the algorithmic complexity with reasoable decodig performace. I [5], the complexity reductio of LDP decodig usig mi-sum algorithm, for DVB-S stadard is addressed. It proposes two differet approaches for reducig the computatio stregth of the decodig algorithm. The Parity check matrix (PM) for o-biary ca be costructed similar to biary LDP, with the elemets of the Galois Field (GF) over q. The quasi-cyclic LDP code is aother variat of LDP code where the PM structure is formed by cyclic shift, ad is represeted by a sub matrix correspodig to each o-biary elemet [6]. I compariso to No-biary LDP codes, the biary LDP code has the competig performace ad complexity for large block legths. However for the shorter block legth, the performace of biary LDP is poor [7]. Thus No-biary LDP codes are more suitable for shorter block legth. It has bee oted that about.3 db reductio i bit error rate at SNR=1, for o-biary codes [8]. Various methodologies for the reductio of the cofiguratios with respect to the decodig algorithm are proposed i [9]. No-biary LDP ca also be used for multiple ateas systems where it has bee show that [1], throughput of 6 Mbits/s ca be obtaied with four trasmit ad four receive ateas. I this paper, a complexity reductio methodology for o-biary LDP codes is proposed ad complexity reductio is aalysed for IEEE 8.11 wireless commuicatio stadard [11]. II. NONBINARY LDP DEODER The desig of decoder i a No-biary LDP codes ivolves computatios of check ode processig ad variable ode processig. The chael probabilities for each symbol of GF (q), is a array of q x ζ, correspodig to the codeword ζ. The check ode computatio ivolves iitially, the idetificatio of possible cofiguratio sets of the symbols of GF (q) with respect to the chael probabilities. The parity check matrix H ISSN : Vol 5 No 6 Dec 13-Ja

2 M Abuselvi et.al / Iteratioal Joural of Egieerig ad Techology (IJET) also defies the computatio complexity of the check odes ad variable odes. The umber of cofiguratio sets ad the associated matrix computatio decides the complexity of the decoder. For a o-biary LDP, the computatio complexity of the Belief Propagatio (BP) algorithm is O (q ). I [1] Declercq elaborates o reducig the complexity of BP algorithm with the log domai aalysis ad proposes Exteded Mi-sum (EMS) algorithm ad the decoder is less sesitive to quatizatio oise. The efficiecy of the MS algorithm has bee aalyzed i the differet domais, amely probability domai, log domai ad Loglikelihood ratio (LLR) domai [13]. Some modificatios to the EMS algorithm towards the complexity reductio are addressed i [14, 15]. The system block diagram for the No-biary LDP decoder is show i Fig 1, which uses the Mi-sum decodig algorithm. The parity check matrix structure is embossed with the available a posterior chael iformatio to form the Q-matrix, which are the iput values for the check ode processig. The check ode processig of the Q-matrix, the cofiguratio sets are defied over the elemets of GF (q). The Mi-sum decodig algorithm estimates the codeword. Fig. 1. System Block Diagram of No-Biary LDP Decoder A. Mi-sum Decodig The mathematical model of the Mi-sum decodig algorithm is as follows (a) Step 1: Iitializatio: Let represets the a priori iformatio of the th symbol from the chael. Q m (a) are the chael log likelihood ratios, iitially assiged for the computatio to the variable ode uit. Step : heck ode update: Let to a ode a i the Taer graph. ( a) = l(pr( c = s chael/ Pr( c = a chael)) (1) Qm ( a) = ( a) () R m (a) represets the message computed at the check ode uit, with respect R m ( a) = mi m ( a ) N ( m)\{ } Am ( a) N ( m)\{ } ( Q ( a)) (3) A a) : = { a. h a + h a } (4) m ( m m N ( m)\{ } Step 3: Variable ode update: From the check ode update uit, the messages variable ode update uit for the estimatio of codeword. m m M ( )\{ m} a GF ( q) Qm( a) R m (a) are passed to the Q ( a) = ( a) + R ( a (5) m Q Q m = mi m ( a) = Qm ( a) Qm Step 4: Tetative decodig: With the messages received from the check ode update uit, the codeword is tetatively decoded for the each symbol as m m M ( ) ) Q ( a) = L ( a) + R ( a) (8) cˆ = arg mi( Q ( a)) (9) (6) (7) ISSN : Vol 5 No 6 Dec 13-Ja

3 M Abuselvi et.al / Iteratioal Joural of Egieerig ad Techology (IJET) cˆ cˆ...ˆ c = [ 1 N 1 ] (1) Step 5: Decisio: The tetatively decoded symbols are cross verified with the parity check matrix, to validate the decoded codeword usig H T = or iter max_ iter (11) The flow diagram of the Mi-Sum algorithm is preseted i Fig i which each step of the algorithm is depicted with the correspodig iput ad output variables. The variable ode is updated with the prior chael iformatio durig the first iteratio. From the ext iteratio, the computatios for the variable ode update are carried out with the iputs from the check ode update uit. The validatio of the codeword is performed i the decisio block ad the iteratio cout is verified. If the codeword is ot valid, the umber of iteratios is icreased for the ext cycle of decodig procedure. If the maximum iteratio is attaied ad the codeword is ot validated, the decodig failure is declared. If the iteratio boud is ot reached, the cycle repeats. Fig.. Flow Diagram of Mi-sum Algorithm B. Performace Aalysis The aalysis of the decodig performace of o-biary LDP decoder usig Mi-sum algorithm is doe with the Bit Error Rate (BER) graphs. It is compared with biary LDP decoder of same specificatio. Fig 3 idicates that BER is better for the o-biary LDP, whe compared to biary LDP codig. Simulatio results foud to agree with Hassai s [7] work o the compariso of biary ad No-biary LDP codes. Simulatios results also show that the decodig performace of No-biary LDP improves with the order of Galois field. Fig. 3. ompariso of bit error rate for biary, GF (4) ad GF (8) LDP odes ISSN : Vol 5 No 6 Dec 13-Ja

4 M Abuselvi et.al / Iteratioal Joural of Egieerig ad Techology (IJET) III. FORMULATION OF REDUED ONFIGURATION SET I this sectio, we propose a ovel method of reducig the umber of cofiguratios to be verified at the parity check equatios. We itroduce a techique called variable threshold based cofiguratio reductio. The flow of reducig the cofiguratio set ad the formatio of M matrix are depicted i Figure 4. The cofiguratio set block i the system diagram show i Fig.1 ca be replaced with this flow diagram Fig. 4. Flow Diagram of ofiguratio Set Reductio Notatios used: GF (q) = {, 1., q-1}, the Galois field with q elemets, where q is a power of a prime umber l to deote the GF(q) symbols S to deote the vectors of GF (q) symbols H (m,), the parity check matrix of the code ζ, codeword set of the LDP code H (), umber of check odes coected to the variable ode. H (m), umber of variable odes coected to the check ode m. ξ (m), set of local cofiguratios verifyig the check ode m with the liear costrait. ξ(m l = l), local cofiguratio sets, S verifyig m, such that l=l; give ε H(m) ad l ε GF(q), a prior chael iformatio for the th symbol For a give GF (q), aprior iformatio havig l colums ad q rows is defied as, () () () 1 l = ( q 1) 1 l The Q-matrix is framed as follows. For each o-zero elemet i H-matrix, place the correspodig colum vector from. The elemets that are placed should be permutated accordig to the symbol q. The partitio of the Q matrix represets the likelihoods that the th received symbol belogig to. 1 ( q 1 Q = 1 ( q 1 () 1) () 1) () () The received probabilities of messages i are arraged i descedig order. First half of the values from the list are segregated ad the thresholds are defied for framig the M matrix that cotais reduced umber of elemets from. The framig of reduced cofiguratio set, ψ is give by the equatio () () ψ { ξ reduced cofiguratio set } (1) ISSN : Vol 5 No 6 Dec 13-Ja

5 M Abuselvi et.al / Iteratioal Joural of Egieerig ad Techology (IJET) The rage of thresholds, T is fixed to frame the M matrix. The formatio of the M matrix is as follows. M { T Matrix of reduced cofiguratio set } (13) M 1 = 1 ( q 1 () () 1) () osider a o-biary LDP code with the code rate of ½, ad (34,648) over GF (4). The LDP code with these specificatios has bee followed i IEEE 8.11 stadard. Let the chael probabilities for the codeword legth of 6, are give as = Let the parity check matrix defied over the GF (4) with the symbol set {, 1, α, α} be α 1 α = H α α 1 1 α α α 1 1 The permuted Q-matrix ca be formed with the values of ad also accordig to the elemets i H- matrix as follows, Q = With the formulatio of the reduced cofiguratio set, the T-matrix ca be give as follows T = With these rage of values, defied as the threshold, the modified Q-matrix (called as M-matrix) ca be give as ISSN : Vol 5 No 6 Dec 13-Ja

6 M Abuselvi et.al / Iteratioal Joural of Egieerig ad Techology (IJET) M = Hece, the computatios o the check odes are carried with this modified Q-matrix(or M-matrix) values A. BER Aalysis With the proposed method of the cofiguratios set reductio, the Mi-sum decodig algorithm is aalysed. The depedet parameter which decides o the reductio of cofiguratio set is the threshold ad its rage. The threshold varies with respect to the chael probabilities, which i retur depeds o the SNR values. Therefore the variatio i the SNR has the impact o the rage of message probabilities ad thereby the variable threshold rage is fixed for the reduced cofiguratio set. Hece the percetage of cofiguratio reductio with the proposed algorithm varies as a fuctio of SNR. The rage of threshold is dyamically fixed based o the chael probabilities ad for the SNR defied. Hece this method of decodig shows improvemets o the decodig performace. 1 without cofiguratio reductio with cofiguratio reductio 1-1 BER SNR(dB) Fig.5 Performace compariso of Mi-sum based LDP decoder, without ad with cofiguratio reductio The decodig performace of the Nobiary LDP decoder based o Mi-sum algorithm, with ad without cofiguratio reductio techique is show i figure 5. We ifer that with the proposed cofiguratio reductio method, the decodig performace is improved. The computatio blocks ivolved i mi-sum algorithm based obiary LDP decoder are the compariso block ad the adder block. The check ode processig uit cosists of a comparator ad a adder for the mi sum operatio. As referred i the earlier sectio, the equatio associated with check ode uit, depicts the umber of comparisos ad additios with respect to the cofiguratio set. I table I, the formulatio of the computatios usig mi-sum algorithm is preseted. ISSN : Vol 5 No 6 Dec 13-Ja

7 M Abuselvi et.al / Iteratioal Joural of Egieerig ad Techology (IJET) TABLE I omputatio formulatio for Mi-sum algorithm omputatio block Additios omparisos NU Ψ ( k 1) wrmq ( Ψ 1)w r mq VNU Ψ ( w c 1) q --- Where (, m) represets the matrix size ad (w r, w c ) correspods to the row ad colum weights respectively. q is the order of Galois field ad Ψ correspods to the umber of cofiguratios defied. For the aalysis, we took the values of these symbols as give i the case study. The reduced umber of cofiguratios for the differet sigal to oise ratio is preseted i the table II below: TABLE II Reduced Number of ofiguratios SNR (db) ofiguratios without reductio Reduced umber of cofiguratios We ifer from the table that the umber of cofiguratios beig reduced is a fuctio of sigal to oise ratio. Also, the reductio i the cofiguratio set is predomiatly more, makig this techique more attractive i terms of reduced computatio complexity. The table III below shows the umber of computatios ivolved i both the check ode processig uit ad variable ode processig uit. The calculatios are doe with respect to the equatios give i the table I. Number of cofiguratios Without reductio 14 With reductio TABLE IIII Number of omputatios with ad without ofiguratio Reductio Number of computatios without cofiguratio reductio Number of computatios with cofiguratio reductio NU VNU NU VNU Additio omp. Additio Additio omp. Additio From the above table, it is evidet that the umber of computatios at the check ode uit ad variable ode uit is greatly reduced. From the aalysis of the computatio reductio, with the techique of cofiguratio reductio, we ifer that i check ode uit, o average 83.6% of additios ad 83.7% of comparisos are reduced. Similarly i the variable ode uit, there is 83.6% of additio reductio, o average i the variable ode uit. It is clear that the cotributio of this computatio complexity reductio techique is domiat i the check ode uit tha i the variable ode uit. IV. ONLUSION I this paper, we have proposed modificatios i the o-biary LDP decoder algorithm to obtai the reduced cofiguratio set. Simulatios have bee performed to aalyze the efficiecy of the reduced cofiguratio set LDP decoder, for the specificatio of IEEE The performace of the decoder is ISSN : Vol 5 No 6 Dec 13-Ja

8 M Abuselvi et.al / Iteratioal Joural of Egieerig ad Techology (IJET) foud to reduce the complexity by 83% at a margial degradatio i the decodig performace. Further, the costructio of the parity check matrix of the decoder ca be varied, satisfyig the row-colum costraits, to improve the decodig performace. REFERENES [1] R. G. Gallager, Low-Desity Parity-heck odes. ambridge, MA: MIT Press, [] D.J.MacKay ad R. Neal, Near Shao-limit performace of low desity parity-check codes, IEEE vol. 3, pp , [3] M. Davey ad D. MacKay, Low-desity parity-check codes over GF (q), IEEE ommuicatio Letter, vol., o. 1, pp , [4] Mohammad Rakibul ad Jisag Kim, O the Use of Low-desity Parity heck code for apacity ad Bit Error Rate Sesitive Wireless Sesor Network at Nakagami- chael, IETE Techical Review, vol.5, issue 5, pp , 8. [5] Eu A hoi, Ji-Wo Jug, Nae-Soo Kim ad Deock-Gil Oh, omplexity-reduced algorithms for LDP decoder for DVB-S systems, ETRI Joural, vol.7, o.5, 5. [6] K. Lally ad P. Fitzpatrick, Algebraic structure of quasi-cyclic codes, Discrete Applied Mathematics, vol. 111, o. 1-, pp , 1. [7] El Hassai, S. Hamo, M. ad H. Péard, P. ompariso Study of Biary ad No-Biary LDP odes Decodig, Proc i IEEE iteratioal coferece, software, telecommuicatios ad computer Networks(softOM), pp a, 1 [8] Rolado Atoio arrasco, Marti Johsto, No-biary Error otrol odig for Wireless ommuicatio ad Data Storage, Joh Wiley & Sos, ISBN , 9 [9] David Declereq, Marc Fossorier, Exteded Mi sum algorithm for decodig LDP codes over GF(q), Iformatio Theory, ISIT 5, 5 [1] RoghuiPeg ad Rog-Rog he, Applicatio of Nobiary LDP cycle codes to MIMO chaels, IEEE trasactios o Wireless ommuicatios, vol.7, o.6, 8 [11] IEEE 8.11 Wireless LANs WWiSE Proposal: High Throughput Extesio to the 8.11 Stadards, IEEE 4. [1] D. Declercq ad M. Fossorier, Decodig algorithms for o-biary LDP codes over GF(q) IEEE Tras. ommuicatios, vol. 55, o. 4, pp , 7. [13] H. Wymeersch, H. Steedam, ad M. Moeeclaey, Log- domai decodig of LDP codes over GF (q), i Proc. IEEE It. of. ommu., Paris, Frace, pp , 4. [14] Ximiao Zhag ad Fag ai, Reduced-complexity Exteded Mi-sum check ode processig for No-biary LDP decodig, IEEE, pp , 1 [15] Xiao Ma, Kai Zhag, Haiqiag he ad BaomigBai, Low complexity X-EMS algorithms for No-biary LDP codes, IEEE trasactios o commuicatios, vol.6, o.1, 1. ISSN : Vol 5 No 6 Dec 13-Ja