Comparison of Turbo Codes and Low Density Parity Check Codes

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1 IOSR Journl of Electronics nd Communiction Engineering (IOSR-JECE) e-iss: ,p- ISS: Volume 6, Issue 6 (Jul. - Aug. 013), PP Comprison of Turo Codes nd Low Density Prity Check Codes Ahmd Hsn Khn 1, Dr K C Roy, 1 Electronics nd Communiction Engg. Suresh Gyn Vihr University, Jipur Principl SBCET Jipur Astrct-The most powerful chnnel coding schemes, nmely, those sed on turo codes nd LPDC (Low density prity check) codes hve in common principle of itertive decoding. Shnnon s predictions for optiml codes would imply rndom like codes, intuitively implying tht the decoding opertion on these codes would e prohiitively complex. A rief comprison of Turo codes nd LDPC codes will e given in this section, oth in term of performnce nd complexity. In order to give fir comprison of the codes, we use codes of the sme input word length when compring. The rte of oth codes is R = 1/. However, the Berrou s coding scheme could e constructed y comining two or more simple codes. These codes could then e decoded seprtely, whilst exchnging proilistic, or uncertinty, informtion out the qulity of the decoding of ech it to ech other. This implied tht complex codes hd now ecome prcticl. This discovery triggered series of new, focused reserch progrmmes, nd prominent reserchers devoted their time to this new re.. Leding on from the work from Turo codes, McKy t the University of Cmridge revisited some 35 yer old work originlly undertken y Gllgher [5], who hd constructed clss of codes dued Low Density Prity Check (LDPC) codes. Building on the incresed understnding on itertive decoding nd proility propgtion on grphs tht led on from the work on Turo codes, McKy could now show tht Low Density Prity Check (LDPC) codes could e decoded in similr mnner to Turo codes, nd my ctully e le to et the Turo codes [6]. As review, this pper will consider oth these clsses of codes, nd compre the performnce nd the complexity of these codes. A description of oth clsses of codes will e given. I. Turo Codes There re different wys of constructing Turo codes, ut the commonlity is tht they use two or more codes in the encoding process. The constituent encoders re typiclly recursive, systemtic convolutionl (RSC) codes [3]. Figure 1 shows generl set-up of the encoder. The numer of its in the code word vries with the rte of the code. For n overll code rte of R = 1/, it is norml to use two R = 1/ RSC codes, where every systemtic it is trnsmitted, ut for the coded its puncturing mtrix is used where only every second code it from ech of the constituent codes is included in the trnsmitted symol. The permuttion mtrix Π permutes the systemtic its, so tht the two constituent codes re excited y the sme set of informtion its, leit in different order. The permuttion mtrix is criticl design fctor for designing good codes. The code word consists o f C i 1 s the systemtic it (di) nd C i nd C i 3 which re vectors of encoded its from the two convolution codes. These vectors cn contin one or more its nd di is the informtion it, Π is the permuttion mtrix. Figure 1: Block Digrm of Turo Encoder 11 Pge

2 Comprison Of Turo Codes And Low Density Prity Check Codes 1.1. The BCJR lgorithm The BCJR lgorithm [1] is used for optiml decoding of the constituent codes of the turo codes. This lgorithm is similr to the messge pssing lgorithm used in the context of LDPC codes. The BCJR lgorithm is symol sed decoder, nd delivers proilities of the correctness of the symol decoding together with ech decoded it. In convolution code, we must seprte etween stte trnsitions cused y n informtion it hving vlue 1 nd those cused y n informtion it hving vlue 0. We denote the sttes linked together y 1 1 stte trnsition tht occurs etween time (i 1) nd time i S i 1 nd S i if this trnsition is cused y 1 s the 0 0 informtion it. Correspondingly, the nottion is S i 1 nd S i if the trnsition is cused y 0 s the informtion it. We denote the vector of received chnnel symols y, where y is pertured y AWG. We denote the informtion it t time i di. The log likelihood rtio for stte trnsition ssocited with di = 1 nd di = 0 is then given y λ i = log P(S 1 1 i 1,Si,y) P(S 0 i 1,Si 0 (1),y) In the cse of mny possile stte trnsitions, the expression gets more complicted: λ i = log P(S i 1,S i,y) S i 1,S i di =1 P(S i 1,S () i,y) S i 1,S i di =0 The vector y my e split into three prts: i=1 y = p=1 y p +y i + y k k=i+1 (3) s i And y i depend only on the stte s i 1 nd the it d i, wheres k=i+1 y k depend only on s i nd the its d k k ϵ{i+1, }. The proility of the stte trnsition s i 1 s i is then given y the expression in (4). P (s i 1, s i y,) = P (s i 1, s i=1 i, p=1 y p + y i + y k k=i+1 (4) i=1 = P (s i 1,, p=1 y p )(y i i+ k=i+1 y k, S i S i 1 ) i 1 = P (s i 1,, p=1 y p )(y i, S i S i 1 )( k=i+1 y k S i ) Eqn. 4 my then e written s P (s i 1, s i y) =α s i 1, γ s i 1, s i β S i (5) With the following definitions: γ s i 1, s i P (y i, s i 1 s i,) = P (y i s i 1, s i,) P (s i 1, s i,) (6) i 1 α s i 1, P (s i 1,, y p ) p=1 (7) β S i ( k=i+1 y k S i (8) Furthermore, we my express α s i 1, = s i, α s i, γ s i, s i 1 (9) And β S i = s i 1, β s i+1, γ s i, s i+1 ( 10) We sustitute Eqn. (5) into Eqn. () nd otin Eqn. (11). λ i = log S i 1 S i 1,S i di =1 α s i 1, γ s i 1, s i β Si α s i 1, γ s i 1, s i β S (11),S i i di =0 As shown in Eqn. (6), we hve: γ s i 1, s i = P (s i 1 s i,) P (y i s i, s i 1,) (1) Through knowledge of the nture of AWG, Eqn. (1) my e written s: j j ( y (c 1) ) γ s i 1, s i = P (s i 1 s i ) e i i n σ j=1 (13) We compute the logrithm of Eqn. (13) nd otin Eqn. (14). 1 Pge

3 Comprison Of Turo Codes And Low Density Prity Check Codes j j i ) log γ s i 1, s i =P(log(s i 1 s n i )) + (y i j=1 -log( ( σ ) (14) σ P (s i 1 s i ) is equl for ll permissile sttes S if we ssume n equl proility of 1 s nd 0 s in the informtion sequence. This prt my therefore e omitted in the decoding process. The metric of symol n is then given y Eqn. (43)(15). M i = n (y j i ) j +( i ) j j +y i i σ j=1 - (15) 1.. The Log-BCJR lgorithm The Log-BCJR lgorithm is simplifiction of the BCJR lgorithm in terms of the needed computtions. We strt with the following definitions: f ( ) log α ( ) r ( ) log β ( ) g ( ) log γ ( ) f (s i ) = log ( α s i 1, γ s i 1, s i (16) s i 1, log (α s i 1, = f(α s i 1, ) α s i 1, ) = e f s i 1, This gives: f (s i, ) = log( e f r(s i 1, +g(s i 1, s i 1, s i )) (18) Correspondingly for β : +g(si 1, r (s i 1) = log( e r(s i, s i )) (19) s i, Eqn. (0) shows the expression for log γ (s i 1, s i ). γ s i 1, s i = log (P (y i (s i s i 1,) + log (P(s i s i 1 ) ) (0) γ ow ecomes: γi = log(( s i 1,, s i e f(s i, 1 +r(s i )+g(si 1, di =1 s i )) - log (( e f(s i, 1 di=0 s i )) s i 1,, s i +r(s i )+g(si 1, (17) (1) This implies summtion of series of exponentils. This my e simplified(4) : Log (e x + e y ) = mx(x, y) + log (1 + e x-y ) () We denote this clcultion L(x, y). log I i=1 e xi = log( e xi +log I i= e xi = log ( e xi +log I 1 i=1 e xi = L( xi,log I 1 i=1 e xi ) = L( xi,log(e I 1 + I i=1 e xi ) = L( xi,l(xi-1,log( I i=1 e xi )) (3) This ecomes recursion which we my write s Eqn. (4). log I i=1 e xi =L(xI,L(xI-1,L( L(x1,x1).))) (4) In most cses, this my e further simplified s shown in Eqn. (5). I i=1 e xi mx (xi) (5) Log Through the use of this simplifiction, the computtion of f( ) nd r( ) ecomes forwrd nd ckwrd Viteri lgorithm respectively Decoding of Turo Codes The decoding lgorithms descried in the preceding sections re lgorithms for decoding the constituent RSC codes of turo codes. In order to get the turo structure the decoding lgorithms of ech of the constituent codes hve to e comined in similr mnner to the messge pssing lgorithm descried previously. Figure illustrtes multiple itertion decoder. Here, there is feedck loop, where the first decoder is fed new informtion from the lst decoder of the previous itertion. The informtion tht is pssed on to the next decoder is generlly denoted γ e. This is the extrinsic prt of γ tht ws generted in the given decoding. The sutrctions shown in the figure ensure tht only new informtion is used in susequent decoding stges. This messge pssing implies tht the posteriori proility from one decoder is used s priori informtion in the susequent decoder Pge

4 Comprison Of Turo Codes And Low Density Prity Check Codes Figure () Block Digrm of Turo Decoder, severl itertion II. Low Density Prity Check Codes The structure of the code is given y prity check mtrix H, illustrted elow. Different codes hve different prity check mtrices. H = (6) The code word x is constructed so tht Eqn. (7) is true. Hx = 0 (mod ) (7) A genertor mtrix G is used for constructing the code. The genertor mtrix my e found from the prity check mtrix H..1. Construction of G First we note tht H X = X T H T (8) The code word x my e split into one informtion prt i nd one prity check prt c. The code word my then e written s X T = [i ] [c] (9) Correspondingly, the prity check mtrix my e split into two mtrices: H = [A B] (30) From Eqn.(8), we note tht vector i is multiplied with Mtrix A, wheres vector c is multiplied with mtrix B. On the sis of Eqn. 9 nd30, Eqn.7 my e written s A i + B c = 0 (31) If the mtrix B is non-singulr, Eqn. 31 my e inverted nd the check its c my e found from Eqn. (3) c = B -1 A i (33) In prctice, it my e necessry to swp over some of the columns in H in order for B to ecome nonsingulr. The product B -1 A mkes out the genertor mtrix G. This mtrix is clculted only once, nd is used for ll encoding. The prity check mtrix is used for constructing grph structure in the decoder... Grph structure The decoding of LDPC codes my e efficiently performed through the use of grph structure. In this work, Tnner grphs will e used for the decoding [8]. The grph is constructed from the prity check mtrix H. Ech row in the mtrix is represented y check node, wheres ech 1 in the row is represented y n edge into it node. Ech column is represented y it node, nd ech 1 in the column corresponds to n edge into check node. This is illustrted in Figures 3 nd. In this mnner, grph is constructed which contins totl of it nodes nd M check nodes. The numer of edges is decided y the numer of 1 s in the prity check mtrix. All edges re connected to check node nd to it node. The numer of edges connected to node denotes the degree of the node Pge

5 Comprison Of Turo Codes And Low Density Prity Check Codes Figure 3: Check ode.3. Decoding In this context, the decoder is soft-decision input decoder, implying tht it opertes on the chnnel symols, denoted y r = x 1 + n (34) Where n is the AWG noise vector dded in the chnnel nd x is the code word..4. The prity of vector Finding the proility of the prity of vector is centrl concept in the decoding of LDPC codes. Ech prity check my e regrded vector of even prity []. First, we define the Likelihood Rtio (LR) s the rtio etween the two proilities P(x = 1) nd P(x = 0): LR = P(X=1) P(X=0) The symol γ is used for the Log Likelihood Rtio, γ = log P (x = 1) P(x = 0) (35) (36) If x is vector of its, the LLR of it i in tht vector is given y γ i. P (xi = 1) γ i = log (37) P(xi = 0) The nottion φ((x) is used for the vector prity. The LLR of the prity of the vector x is then given y: λ φ(x) = log P (ϕ(x i) = 1) P((ϕ(x i ) = 0) (38) In order to compute λ φ(x) Eqn. 39 is used: tnh( λ ϕ(x) n ) = tnh i=1 λi (39) Eqn. 13[39] is solved with respect to λϕ : λϕ (x) =- tnh -1( = n i=1 tnh λi (40) The posteriori LLR of it n is given y: λ n = log P(x n =1 r) (41) P(x n =0 r) The vector r my e split into two prts r n which refers to the systemtic prt of the code word, nd {ri n}, which refers to the prity its of the code word. Eqn. 41 my then e expressed s λn = log P(x n =1 r n,{ri 1 }) P(x n =0 r n {ri 1 }) (4) Byes rue is given y: P (/) = P(,) (43) P() We use this rule in order to re-express the numertor of Eqn. 4: f(rn,xn =1, ri 1 ) P(x n = 1 r n, {ri 1 )= f(rn,, ri 1 ) (44) Furthermore, using the equlity P (, ) = P ( )P(): 15 Pge

6 Comprison Of Turo Codes And Low Density Prity Check Codes f rn, xn = 1, ri 1 f(xn = 1, ri 1 ) P x n = 1 r n, {ri 1 = f rn., ri 1 f ri 1 (45) Given xn, rn is sttisticlly independent of { ri 1}: f rn. xn = 1) f(xn = 1, ri 1 ) P xn = 1 rn, ri 1 = f rn., ri 1 (46) Following the sme line of resoning, the denomintor of Eqn. 16 is expnded. Eqn. 4 then ecomes: λ n = log f rn. xn =1 P(xn =1, {ri 1 }) f rn. xn=0 P(xn=0, {ri 1 } f rn. xn =1 P(xn =1, {ri 1 }) = log + log f rn. xn =0 P(xn =0, ri 1 ) In n AWG chnnel the conditionl proility f(r n x n ) is given y f(r n x n) = 1 πσ e rn xn +1 σ (48) Sustituting Eqn. into Eqn. 1 we get Eqn. 49. λ n = P(xn =1, {ri 1 }) rn + log (49) σ P(xn =0, ri 1 ) (47) The first element of the right hnd side of the equl sign is denoted the intrinsic prt, wheres the second element is denoted the extrinsic prt. ote tht if it is known tht the prity of vector x is 0 (even prity), the proility tht it xn is 1, given the received vlues of the rest of the vector ri n, is the sme s the proility tht the rest of the vector { ri n} hs odd prity. λ n = P(ϕ xi=1for i=1.j, {ri n }) rn + log σ P(ϕ xi=0for i=0.j, ri 1n ) (50) If the grph is free of cycles, the vectors x1,x..xj re Independent, given{ ri 1n }, nd if the elements within the vector re independent, Eqn. 50 my e written s Eqn. 51. λ n = rn + log i=1 P(ϕ xi =1{ri n }) σ j = σ j P(ϕ xi=0 {ri n }) i=1 j i=1 P(ϕ xi =0 {ri n }) j rn + log P(ϕ xi=1{ri n } rn + λ σ i=1 ϕ(xi) (51) In the grph λ i, l is the messge (contriution) from it node i to check node l: P(ϕ xi=1{ri n } λ i, l = log (5) P(ϕ xi=0 {ri n }) We sustitute Eqn. 14 into Eqn. 5 λ n = rn j σ tnh 1 k λ i,l i=1 ( l= tnh( )) (53) Eqn. 54 is the expression of the LLR of it n. The first Prt is the intrinsic informtion, wheres the second prt is the extrinsic informtion from the j vectors tht contin it n. Messges into it node n ech it node hs prticulr LLR ssocited with itself. The LLR is computed from the contriutions from ll connected check nodes, which constitute the extrinsic informtion, nd the received vlues rn, which constitute the intrinsic informtion. When check node k receives messge from it node i, it sutrcts the vlue it pssed on to it node i in the previous itertion, i.e. λ i,l= λ i- u m,i. This is done in order to reduce the correltions with previous itertions Pge

7 Comprison Of Turo Codes And Low Density Prity Check Codes Figure 4: Messges into it node n III. Comprison Of Turo Codes And Low Density Prity Check Codes A rief comprison of Turo codes nd LDPC codes will e given in this section, oth in term of performnce nd complexity. In order to give fir comprison of the codes, we use codes of the sme input word length when compring. The rte of oth codes is R = 1/. Figures 5 nd 6 show the performnce of Turo codes nd the LDPC codes with informtion length 1784 nd 3568 respectively. The Turo codes defined in 6 were used for otining the performnce curves. For oth code lengths the LDPC codes re etter until certin S/ is reched, respectively E/0 = 1.15 nd 1.1 db. This chrcteristic error floor or knee ws lso typicl of Turo codes in erlier yers, ut this hs ecome less of prolem ltely due to the definition of good permuttion mtrices. In order to compre the complexity of the codes, the numer of multiplictions, dditions nd complex opertions like log, tnh, e nd tnh -1 were counted in oth codes. Figure 8 shows the numer of opertion per itertion for the LDPC nd Turo codes of length I = 1784 nd The numer of itertions used my vry in the cse of the LDPC codes, s the decoding qulity my esily e checked, i.e. tht if vlid code word is decoded, the decoding my e stopped. Hence, there will e less nd less itertions performed the higher the S/. Hence, in order to chrcterize the mcomplexity, representtive numer ws chosen in order to give fir comprison with Turo codes: With the ove prmeters the it error rte for the LDPC code ws nd for the I = 1784 nd I = 3568 codes respectively, wheres for the Turo code the it error rte ws nd respectively, i.e. the error rtes were sufficiently similr for fir comprison to e given in Figure 5 nd Figure 6. Figure 5: Turo code nd LDPC codes: I = Figure 6: Turo code nd LDPC codes: I = Pge

8 Comprison Of Turo Codes And Low Density Prity Check Codes Figure 7: Complexity: The figure shows Additions),Multiplictions. nd Complex opertions.)per itertion for LDPC nd Turo codes for I = 1784 nd I = IV. Conclusion The results in the previous section show tht LDPC codes hve significntly lower complexity thn Turo codes t the code lengths, performnce nd S/ tht were considered here. Figure 7: Complexity: The figure shows Additions (Add.), Multiplictions (Mult.) nd Complex opertions (Komp.) per itertion for LDPC nd Turo codes for I = 1784 nd I = Turo codes hve fixed numer of itertions in the decoder. This implies tht the time spent in the decoding nd the it rte out of the decoder, re constnt entities. In contrst, the LDPC decoder stops when legl code word is found, implying tht there is potentil for significntly reducing the mount of work to e done reltive to Turo codes. This lso implies tht the it rte out of the decoder will vry, nd uffer system must e designed in order to mke the it rte constnt. The LDPC decoder will ecome fster the higher the S/. An dvntge with LDPC codes is tht the decoders my e implemented in prllel. This hs significnt dvntges when considering long codes. References [1] L. R. Bhl, J. Cocke, F. Jelinek, nd J. Rviv, Optiml decoding of liner codes for minimizing symol error rte, IEEE Trnsctions on Informtion Theory, vol. 0, IEEE, 1974, pp [] John R. Brry, Low-density prity-check codes, Tech. report, Georgi Institute of Technology, 001. [3] C. Berrou, A. Glvieux, nd P. Thitimjshim, er shnnon limit error-correcting coding nd decoding: Turo-codes, Proceedings ICC (Genev), IEEE, My 1993, pp [4] Kjetil Fgervik, Itertive decoding of conctented codes, Ph.D. thesis, University of Surrey (UIS), UK, August [5] R. G. Gllger, Low density prity check codes, Reserch monogrph series, no. 1, MIT Press, Cmridge, MA, [6] Dvid J.C. McKy nd Rdford M. el, er Shnnon limit performnce of low density prity check codes, Electronics Letters 3 (1996), no. 18, , Reprinted in Electronics Letters, 33 (1997), no. 6, Pge