A REDUCED-COMPLEXITY LDPC DECODING ALGORITHM WITH CHEBYSHEV POLYNOMIAL FITTING

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1 Joural of Theoretical ad Applied Iformatio Techology st March. Vol. 49 No. 5 - JATIT & LLS. All rights reserved. ISSN: E-ISSN: A REDUCED-COMPLEXITY LDPC DECODING ALGORITHM WITH CHEBYSHEV POLYNOMIAL FITTING SHAOBO SHI, YUE QI, LEI LI, MIN YAO, QIN WANG PhD,School of Computer ad commuicatio egieerig, Uiversity of Sciece ad Techology Beijig, Beijig 8, Chia shaobo985@gmail.com ABSTRACT BP decodig algorithm is a high-performace low-desity parity-chec (LDPC) code decodig algorithm, but because of its high compleity, it ca t be applied to the high-speed commuicatios systems. So i order to further reduce the implemetatio compleity with the miimum affectio to the system performace,, we proposed a BP algorithm based o Chebyshev polyomial fittig for its good approimatio performace i this paper. This method ca trasform the complicated epoetial fuctio ito polyomial with the error at level i the process of decodig, which ca reduce the cosumptio of memory resources cosiderably. At the same time, a Chebyshev uit is proposed i the process of implemetatio with less multiplier; we implemeted a semi-parallel LDPC decoder with this method o Altera CycloeII FPGA. The simulatio results show that the degradatio of this method is oly.4db compared with covetioal BP algorithm. Keywords: low-desity parity-chec (LDPC) code;bp algorithm;chebyshev Polyomial Fittig;. INTRODUCTION Gallager proposed LDPC(Low Desity Parity Chec) code []. The performace of this code is very close to Shao limit, which has oly less tha.db differece []. For the better performace i decodig, LDPC codes are widely used i various curret commuicatios systems. Such as DVB-S, WLAN ad WiMAX [] ad other popular wireless commuicatio system. Belief Propagatio decodig algorithm is cosidered as a optimal iterative decodig algorithm []. However, a large amout of multiplicatio ad epoetiatio lead to high compleity of implemetatio i BP algorithm, ad it is ot suitable for high-speed commuicatio system. Curretly, there are two mai methods to reduce the implemetatio compleity i LDPC code decodig. The first oe is to trasform the comple formula i the origial BP algorithm itoa easy to implemet oe, thereby reduce the compleity of implemetatio. The famous Mi- Sum algorithm[4] is the typical method, which adopt the operatio of simple sig fuctio ad additio to replace the multiplicatio ad epoetiatio i origial BP algorithm. But, compared with the traditioal BP algorithm, the performace degradatio of Mi-Sum is about db[5][6]. While, the other method is the Loo-uptable based o ROM[7]. This method has a comprehesive rage of applicatio for its simple structure ad ice accuracy. However, there is a limit i improvig performace, sice to get the better accuracy eed much more memory. Therefore, we should fid a reasoable method that ca provide better performace tha Mi-Sum ad use lesser memory tha Loo-up-table. This paper proposed a Reduced-Compleity BP algorithm based o Chebyshev Polyomial Fittig, we called it CPF-BP. I this method the epoetial operatios are trasformed ito a high proimate polyomial i the covetioal BP algorithm. We use oly hadful of multipliers, shifters ad adders to implemet this method i the implemetatio, so this is a algorithm that is suitable for VLSI implemetatio.t. ad a simulatio ad implemetatio of ecodig/decodig with threeorder polyomial CPF-BP are showed i this paper. I sectio I the LDPC code ad BP algorithm are itroduced. I sectioⅡ,the CPF-BP algorithm is preseted Ad i sectio Ⅲ,the hardware implemetatio is provided. The sectioⅣadⅤ will show the simulatio result of this method ad the coclusio.. LDPC CODE AND BP ALGORITHM 89

2 Joural of Theoretical ad Applied Iformatio Techology st March. Vol. 49 No. 5 - JATIT & LLS. All rights reserved. ISSN: E-ISSN: The iput to the ANN is the value of epoet of reactive power load-voltage characteristic ( q ) ad the output is the desired proportioal gai (K P ) ad itegral gai (K I ) parameters of the SVC. Normalized values of q are fed as the iput to the ANN the ormalized values of outputs are coverted ito the actual value. The process of. LDPC Code LDPC code is oe id of liear bloc code, ad it ca be preseted by Taer graph or chec matri H, both of them is equivalet. As showed i figure ad figure: v v v v4 v5 v6 v7 Variable ode c c c Chec ode vv5v7 vv vv4v6 Figure: Taer graph of LDPC ecoder A Figure: The chec matri correspodig to Taer graph I figure, the left odes represet the variable odes V, correspodig to the colum of chec matri. The right odes represet the chec odes C, correspodig to the row of chec matri. Whe there is a li betwee i-th V ad j-th C i Taer graph, the value of [j,i] i chec matri is. The BP algorithm of LDPC code is oe decodig algorithm that based o message propagatio ad iteratio of Taer graph. The basic mathematic theory is: Bayes Priciple: P( X Y) PY ( X)* P( X)/ PY ( ) () Judgmet criteria: if: PX ( Y) > PX ( Y) the : X else : the : X (). BP algorithm I this paper we assumed the accepted vector is y, the legth of y is N, the Chec matri H is a M N, ad defied these variable: p is the probabilistic values whe the value of Nth bit is ; q is the probabilistic values whe m, the value of Nth bit is, but the probabilistic values is got from the chec fuctio ecept Mth oe; r m, is the probabilistic values whe the value of Nth bit is, ad this bit must meet the Mth chec fuctio; The basic process of covetioal BP algorithm is described as followig: Iitializatio: calculate the prior probability of every variable ode i the iitial chael. We have this formula: p () y / σ e p p, for every ode that H, i the m, m, chec matri, q p, q p Horizotal step: get the probability of every chec ode from the outcome of step. For every (m,) i the chec matri: r ( q q ) (4) m, m, ' m, ' ' Lm ( )\ m Ad, r ( r ) /, r ( r ) / m, m, m, m, Vertical step: get the posterior probability from the values of step. For every variable ode: m, α m',, m, α m', m' M( )\ m m' M( )\ m q p r q p r (5) α is the ormalizatio coefficiet of equatio q q m, m, 84

3 Joural of Theoretical ad Applied Iformatio Techology st March. Vol. 49 No. 5 - JATIT & LLS. All rights reserved. ISSN: E-ISSN: Soft Decisio: calculate the posterior probability of variable ode: m,, m, m M( ) m M( ) q p r q p r (6) is the ormalizatio coefficiet of equatio q q Now, accordig to the Judgmet criteria we ca get the value of the every bit by these: if : q >.5 the : else : the : (7) If the umber of iteratio ot achieves the maimum, the the algorithm will repeat the step to step 4 to get the decodig result. From the formula (), we ca see that the epoetial fuctio ca brig too much computig burde to iteratio process, ad is hard to implemet. I the practical applicatios, the solutios of this problem are the Loo-up-table ad fuctio approimatio. As metioed before, Loo-up-table has its boudedess i practical project. So the fuctio approimatio is aother importat solutio for this problem, ad we adopt the Taylor series usually. But Taylor series is based o desig poit, ad there will be a icreasig error whe the data is far away the desig poit. So i this paper we adopt the method of chebyshev polyomial fittig, ad this is a ew ad effective method of fuctio approimatio test by our simulatio.. Chebyshev Polyomials I the problem of Cotiuous fuctio with a polyomial approimatio, whe f( ) Cab [, ], it is impossible to mae P( ) a a... a, a, a,..., a are ad real umber, equal to f( ) absolutely, but the polyomial P * ( ) ca be foud to mae the differece * * f P mi f( ) P ( ), this is called a b the best uiform approimatio. If f( ) C[,], through Fourier series epasio: C f CT (8) * * ( )~ ( ) Here C * f( T ) ( ) d(,,...). (9) π T ( ) cos( arccos ), () Fittig fucthio replace the iitial probability fuctio i BP algorithm I this part, we will itroduce how to fittig the probability fuctio with our method. Suppose the is the iput data of decoder, ad SNR, σ ca be got through formula: σ SNR/, so [, ] with oise ca be got. Actually, i our test system, the received bits almost are [,-], uder the AWGN. We cosider the situatio that the is i [,], ad i this process of fittig we will use fuctio replacemet. Suppose f( ), ad the / δ e assig t ( t ), [, ] So, we have f( t) f( ), t [,], ad assig cos, [, ]. t θθ π Through the formula (8), (9), ad (), ad combie the formula above, we have this reductio: * π C f(cos θ)cos θdθ(,...) π () C π *cos( ada ),,,... *(cos a )/ σ π e The, we ca calculate the Chebyshev coefficiets C, ad get the fittig fuctio we wat: C f( t) C*cos(arccos t) C*cos(*arccos t) C *cos(*arccos t) () We replace the t through t i formula (), ad get the epressio of the : P C C C C () For the same priciple, we ca get aother fittig fuctio i the regio [, ], ad just pay attetio to oe poit that we must assig t ( t ). So the epressio of the : 84

4 Joural of Theoretical ad Applied Iformatio Techology st March. Vol. 49 No. 5 - JATIT & LLS. All rights reserved. ISSN: E-ISSN: P C C C C (4) From the figure we ca see that there is a egligible error ( level) betwee the chebyshev polyomial fittig fuctio (CPF) --formula ()(4) ad the iitial probability fuctio (Primitive) --formula (). So we ca cosider it is early equivalet, ad ca combie formula () ad (4) to replace the comple iitial probability fuctio: p (5) y / σ e fied. So i order to reduce the compleity of circuit desig, as show i Figure5, we simplifies coefficiets i formula with form of epoet, the a b c d e f P C ( )* ( )* ( )*, i biary domai, multiplyig with ca be replaced by shift operator, so we ca use these computatioal compoet to implemet the CPF fuctio i decodig circuit. Shift Shift Shift Shift4 Shift5 Shift6 C C C C / P Figure 5: Architecture of CPF Uit Figure: Compariso betwee CPF ad Primitive fuctio.. THE IMPLEMENTATION OF BP ALGORITHM BASED ON CPF METHOD I the process of BP decodig algorithm implemet[8][9], we classifies odes as Chec Node Uit () ad Variable Node Uit (), horizotal ad vertical processig treatmet i decodig algorithm are respectively used to process the probability updatig of Chec Node Uit ad Variable Node Uit, i the same time, this paper use Chebyshev polyomial to do iitial processig. I this paper, the overall bloc diagram of the decoder is show as follows: Iput Chebys -hev uit Iput Buffer Compare uit Output RAM Figure 4: Architecture of Decoder From above, whe use P C C C C to fit formula, implemet of this formula, we eed some multipliers, adders, ad shifters. Whe the scope of is determied, Chebyshev coefficiets C are A. Hardware Implemetatio To avoid eormous storage resources of parallel structure, this paper adopts semi-parallel structure, to istead odes parallel computig by serial computig, ad use RAM as buffer. From the figure5, we ca see that the ew CPF uit added to the system, cosumes multiplier, 6 shifter, ad 7 adder compared with the covetioal BP algorithm. But CPF-BP decodig algorithm saved ROM resource which is used i looup table method. This paper uses Verilog hardware descriptio laguage, based o Altera CycloeII series EPCF484C8, call its multiplier IP core, the to do sythesis, layout ad eperimetal verificatio. Legth of code is 56 bits, the iteratio umber is 6. The FPGA sythesis results are show as table: Table:FPGA sythesis result Logic resources (LEs) Memory uits(bits ) Maimum cloc frequecy( MHz),98 7, SIMULATION We simulate the LDPC code with covetioal BP algorithm ad CPF-BP algorithm respectively. Uder the AWGN chael, the Pacet legth is 84

5 Joural of Theoretical ad Applied Iformatio Techology st March. Vol. 49 No. 5 - JATIT & LLS. All rights reserved. ISSN: E-ISSN: bits, codig rate is /, ad with BPSK modulatio. Figure 6 Performace compariso As show i figure 6, the CPF-BP algorithm obtais good performace of decodig. With the icreased of value of Eb/N, the BER of BP-CPF is ot decreased sigificatly. Whe the level of BER 4 is, the degradatio is just.4db compared with covetioal BP algorithm, ad this is much better tha the db of Mi-Sum algorithm i the same level of BER. The reaso of this good performace is that the error that is betwee the iitial probability fuctio ad the CPF fuctio is cotrolled ideally. However, with the icreased of iteratio umber, the degradatio result caused by error caot avoid. 5. CONCLUSION I this paper, we proposed a reduced-compleity BP algorithm based o chebyshev polyomial fittig(cpf-bp) uder the LDPC decodig bacgroud. We used this fittig fuctio to istead the epoetial fuctio i order to reduce the compleity of implemetatio. From the result of simulatio ad FPGA sythesis, the degradatio of this algorithm is.4db, which ca be egligible compared with covetioal BP algorithm. Ad we use oly 6 computatioal uits to implemet this comple fuctio ad save lots of memory space. I coclusio, the CPF-BP decodig algorithm has a better performace ad low-compleity i VLSI implemetatio. REFRENCES: [] GallagerRG.low-desityparity-checcodes[M ].Cambridge,Mass:MITPress,96. [] T. J. Richardso, A. Shorollahi, ad R. Urbae, Desig of provably good low-desity parity-chec codes, IEEE Tras. Iform. Theory,Apr. 999, submitted for publicatio. [] Digital Televisio Terrest rial Broadcastig Stadard Special,Group. GB 66 : Framig St ructure, Chael Codig ad Modulatio for Digital Televisio Terrest rial Broadcastig System [ S]. Beijig : Chiese Stadard Epress, 6 [4] E. Eleftheriou, T.Mittelholzer, ad A. Dholaia, A reduced-compleity decodig algorithm for low-desity parity-chec codes, Electro. Lett.,Oct., submitted for publicatio. [5] Aastasopoulos. A. A compariso betwee the sum-product ad the mi-sum iterative detectio algorithms based o desity evolutio.global Telecommuicatios Coferece,. GLOBECOM. IEEE Volume, Issue, Page(s): - 5 vol. [6] Bhattad. K, LDPC Code Desig for Mi-Sum Based Decodig. Teas A\&M Uiversity, Techical Report WCL-TR-7-, 6 [7] T. Theocharides, G. Li, Implemetig LDPC Decodig o Networ-O-Chip. 8th Iteratioal Coferece o VLSI Desig, 5 Page(s):4 7 [8] Lee. WL ad Wu. A, VLSI implemetatio for low desity parity chec decoder. The 8th IEEE Iteratioal Coferece o Electroics, Circuits ad Systems ICECS, Page(s): - 6 [9] Tog Zhag,"Eficiet VLSI Architectures for Error-Correctig Codig", Ph.D thesis of Uiversity of Miesota, Page(s):9- chapter 4 84