A New Iterative Threshold Decoding Algorithm for One Step Majority Logic Decodable Block Codes

Transcription

1 Internatonal Journal of Comuter Alcaton ( ) Volume 7 No.7, October A New Iteratve Threhold Decodng Algorthm for One Ste Maorty Logc Decodable Bloc Code M. Lahmer Hgh School of Technologe Km5, Route gouray B.P:33 Toulal Mene - Morroco M. Belam Natonal Inttute of Comuter Scence and Sytem Analy (ENSIAS) B.P 73 ENSIAS AGDAL RABAT-Morroco ABSTRACT The erformance of teratve decodng algorthm for one-te maorty logc decodable (OSMLD) code nvetgated. We ntroduce a new oft-n oft-out of APP threhold algorthm whch able to decode thee code nearly a well a belef roagaton (BP) algorthm. However the comutaton tme of the rooed algorthm very low. The develoed algorthm can alo be aled to roduct code and arallel concatenated code baed on bloc code. Numercal reult on both AWGN and Raylegh channel are rovded. The erformance of teratve decodng of arallel concatenated code (7633,8595) wth rate.5 only.8 db away from the Shannon caacty lmt at a of -5. General Term Informaton theory and Codng, Sgnal Proceng. Keyword One te maorty logc decodable code, OSMLD, teratve threhold decodng, arallel concatenated bloc code, roduct code.. INTRODUCTION TURBO code or LDPC code are currently the mot effectve oluton for Forward Error-Correcton (FEC) n alcaton requrng ether hgh code rate (R >.8), very low error floor, or low-comlexty decoder able to oerate at everal hundred of megabt er econd and even hgher. Practcal mlcaton of thee code are numerou namely 3G wrele hone, Dgtal Vdeo Broadcatng (DVB) ytem, or Wrele Metrooltan Area Networ (WMAN). Turbo decodng rele on the exchange of robabltc meage (the o-called extrnc nformaton) between two oft-nut oft-outut (SISO) decoder for the comonent convolutonal code. However, for concatenated cheme wth bloc comonent code, the comutatonal comlexty of trell-baed SISO decodng algorthm often hgh. Th ha led to loo for new SISO decodng algorthm wth low comlexty and hgh erformance, e.g., []-[4]. Thee algorthm calculate extrnc nformaton ung clacal decoder uch a Chae algorthm n [], ordered tattc decodng algorthm n [4] and Hartmann/Rudolh algorthm n [3]. In th erectve we reent a new teratve decodng algorthm baed on a SISO extenon form of threhold algorthm [8]. The rooed algorthm attractve for three reaon: () It can be mly mlemented; () the decodng delay hort; and (3) t ha good erformance. The ue of the threhold algorthm n teratve decodng wa ntroduced for the frt tme by Svrd [7] but for convolutonal code. Our teratve decodng roce follow that gven by Pyndah n []. However, ntead of ung an extenon of Chae algorthm on BCH code, we wll aly an extenon of Maey algorthm on one te maorty logc decodable (OSMLD) code. On the other hand Luca et al. [3] ntroduce an teratve decodng algorthm for everal famle of code (e.g. OSMLD code) but they ue an aroxmaton of Hartmann/Rudolh algorthm. Th aer focued on teratve decodng of OSMLD code. The organzaton of the aer a follow. In Secton, the bac concet of SISO Threhold decodng algorthm ntroduced. In Secton 3, we decrbe teratve decodng algorthm. Secton 4 dedcated to mulaton reult of erformance for dfferent arallel concatenated code baed on OSMLD code.. SOFT-IN/SOFT-OUT THRESHOLD DECODINGAGE SIZE. One Ste Maorty Logc Code Conder an (n, ) lnear code C wth arty-chec matrx H. The row ace of H an (n, n-) cyclc code, denoted by C, whch the dual code of C, or the null ace of C. For any vector v n C and any vector w n C, the nner roduct of v and w zero. Now uoe that a code vector n C tranmtted over a bnary ymmetrc channel. Let e (e, e,., e n) and R (r, r,, r n) be the error vector and the receved bnary vector reectvely. Then R = v + e. For any vector w n the dual code C, we can form the followng lnear um of the receved dgt: A n r w Whch called a arty-chec um. Ung the fact that <w,v>=, we obtan the followng relatonh between the chec um A and error dgt n e: n A e w

2 Internatonal Journal of Comuter Alcaton ( ) Volume 7 No.7, October Suoe that there ext J vector n the dual code C, whch have the followng roerte:. The th comonent of each vector w a. For there at mot one vector whoe th comonent a Thee J vector are ad to be orthogonal on the th dgt oton. We call them orthogonal vector. Now, let ue form J artychec um from thee J orthogonal vector, For each n {,.., J} A we ee that the error dgt e checed by all the chec um above. Becaue of the econd roerty of the orthogonal vector, any error dgt other than e checed by at mot one chec um. Thee J chec um are ad to be orthogonal on the error dgt e. If all the error dgt n the um A are zero for, the value of A equal to e. Baed on th fact, the arty-chec um orthogonal on e can be ued to etmate e, or to decode the receved dgt r. Table how ome examle of OSMLD code that can be decoded wth the rooed SISO decoder decrbed above. In th table we ued the abbrevaton DSC for Dfference Set Cyclc code, EG for Eucldean Geometry code and BCH for Boe Chaudhur and Hocquenghem code. The EG code ued n th tudy are -order and, for an extenve decrton of roectve geometry code and Eucldean geometry can be found n []. Table. Set of OSMLD Code N J d mn Rate code DSC BCH DSC EG DSC EG DSC OSMLD DSC DSC DSC. Maorty Logc Decodng Prncle The error dgt e decoded a f at leat one-half of the chec um orthogonal on e, are equal to ; otherwe, e decoded a le maorty rule. When C a cyclc code, each e can be decoded mly by cyclcally ermutng the receved word r nto the buffer tore. Examle: let u conder the (7,3) code, whch the hort code n dfference et code cla (ee Table ). Th code ecfed by the erfect dfference et P={,, 3} of order. From th refect et, we can form the followng three chec um orthogonal on e 7: e e A = e 4 + e 5 + e 7 A = e + e 6 + e 7 A 3 = e + e 3 + e 7 The If a mle error e=() occur, than we have A = A = A 3 =. If a double error occur, a an examle e 7= and one value of e,..., e 6 equal to, then two value of A are. So we can ay that : - e 7= f only and f at leat value of A are - e 7=, otherwe..3 Soft-nut Soft-outut Threhold Decodng Threhold decodng mly the logcal extenon to oft decon of maorty decodng. Soft-out decodng algorthm when aled to OSMLD code tated a follow: Let u conder a tranmon of bloc coded bnary ymbol {,} ung a BPSK modulaton over AWGN channel, the decoder oft outut for the th bt oton of a gven oft nut R(r,r,..,r ) defned a: n P(c / R) ln () P(c / R) where C(c,c,...,c n ) the tranmtted codeword. The hard decon vector correondng to the receved vector r denoted by H(h, h,..., h n). For a code wth J orthogonal arty chec equaton, () can be exreed a: P(c / B () ln P(c / B where B, =...J are obtaned from the orthogonal arty chec equaton on the th bt a follow: The term B defned to be B = h. Each of the B =,..,J comuted by drong the term h from the th orthogonal arty equaton. Thu, each of the B can be wrtten a: n B c e (3) Where e the th error term n the th arty chec equaton excludng e ( n the total number of term n the th orthogonal arty equaton wthout c ). By alyng BAYES rule, () become P( B ) P(c ) ln (4) P( B ) P(c ) Snce the arty chec equaton are orthogonal on the th ymbol the ndvdual robablte P(B or ) are all ndeendent and (4) can be rewrtten a 3

3 Internatonal Journal of Comuter Alcaton ( ) Volume 7 No.7, October J P(B ) P(c ) ln ln P(B ) P(c ) Aume that the tranmtted ymbol are equally lely to be or, and thu the lat term n (5) null. A a reult, we obtan J P(B ) P(B ) ln ln P(B ) P(B ) Accordng to [9], (6) can be exreed a J ( B )w ( B )w (7) where the value of ( B ) equal to + or and w a weghtng term roortonal to the relablty of the th orthogonal arty chec. It eay to how that: ( B )w r and w ln N n tanh(l / ) n tanh(l / ) where rereent the th element of the th arty equaton and L N r (5) (6) (8) (9). () Thu the oft outut can be lt nto two term, namely nto a normalzed veron of the oft nut r and an extrnc nformaton E rereentng the etmate made by orthogonal bt on the current bt c. Hence (7) become r E N We mae the followng notaton: L c the () N, () whch called the relablty value of the channel. The algorthmc tructure of the SISO threhold decodng can be ummarzed a follow: For each,.., n Comute the term B and w, Calculate B and W Comute the extrnc nformaton E,..,J The Soft-outut obtaned by: Lcr E.4 Modfcaton for Raylegh fadng channel For our algorthm to be alcable n wrele envronment, ther erformance on fadng channel mut be examned. In the channel model we ue, each receved bt r can be exreed a: r a cˆ n (3) In th rereentaton, ĉ a BPSK ymbol aocated to the tranmtted bt c, and n an AWGN. The Raylegh varable a generated a a x y (4) where x and y are zero mean tattcally ndeendent Gauan random varable each havng a varance. We conder the ower normalzed to one a E a (5) whch gve a varance of.5 for Gauen varable. On the Raylegh fadng channel, the avalablty of channel de nformaton the ey ue n determnng the neceary modfcaton for the teratve threhold decodng algorthm. The threhold decodng algorthm ha to be modfed lghtly by changng equaton () whch defne the relablty value of the channel by L c a (6) N 3. ITERATIVE THRESHOLD DECODING 3. Iteratve Threhold Decodng of OSMLD code Iteratve decodng roce (ee Fg. ) can be decrbed a follow: In the frt teraton, the decoder only ue the channel outut a nut, and generate extrnc nformaton for each ymbol. In ubequent teraton, a combnaton of extrnc nformaton and channel outut ued a nut E (q) (q) Fg. : The bloc dagram of the q th teraton. The oft nut reectvely the oft outut of the q th teraton gven by: R(q) R (q)e(q) (7) (q) L R(q) E(q ) (8) c R R(q) SISO- Threhold Decoder E (q (q where R rereent the receved data, E(q) the extrnc nformaton comuted by the revou teraton. In our rocedure we ue a fxed value /J for the arameter (q) and th for all teraton. The value choen for (q) react a an average of all J etmator whch contrbute n the comutaton of E. ) ) 4

4 Internatonal Journal of Comuter Alcaton ( ) Volume 7 No.7, October 3. Iteratve Threhold Decodng of roduct code The develoed algorthm can alo be aled to roduct code and arallel concatenated code baed on bloc code. Let u conder two lnear bloc code C havng arameter ( n,, d ) and C havng arameter ( n,, d ) where n, and d ( =,) tand for codeword length, number of nformaton bt and mnmum Hammng dtance reectvely. It aumed that the nformaton ymbol are the frt ymbol of C and the frt ymbol of C. The roduct code PC C C an ( n n,, dd ) code whoe codeword are contructed by encodng nformaton ymbol wth code C and the reultng nymbol wth C (ee Fg. ). A arallel concatenated bloc (PCB) code can be contructed by a arallel concatenaton of bloc code. The PCB code a PC but wthout the chec on chec art (ee Fg. ). The rate of roduct code PC and PCB code are gven reectvely by R and PC n n R PCB (n n ) [(n ) (n. ) The maor dadvantage of the arallel concatenated code the lo n mnmum dtance. It only (d d ) comared to d d for the roduct code. n Table Set of PC and PCB code baed on OSMLD Code Contructed code Comonent code (C ) Comonent code (C ) Rate PCB (6,49) BCH (5,7) BCH (5,7).3 PCB (34,) DSC (,) DSC (,).35 PCB (95,495) DSC (,) DSC (73,45).38 PCB (393, 369) EG(63,37) EG(63,37),4 PCB (4545,5) DSC (73,45) DSC (73,45).45 PCB (7633,8595) DSC (73,45) DSC (73,9).48 PCB (6785, 3648) DSC(73,9) DSC(73,9).53 PC (533,495) DSC(,) DSC(73,45).3 PC (495,337) BCH (5,7) DSC(73,9).3 PC (3969,369) EG (63,37) EG (63,37),34 PC (539,5) DSC(73,45) DSC(73,45).37 PC (999, 8595) DSC(73,45) DSC(73,9) SIMULATION RESULTS Th ecton conder mulaton reult and analy for ome OSMLD code, roduct code and arallel concatenated bloc code. We would le to notfy that for all our mulaton we have ued a mnmal redual error bt of and redual error bloc of 3. We have ued a number of teraton uch that there no gnfcantly more gan by more teraton. The erformance mrove wth each teraton n all mulaton reult reented. 4. Performance of OSMLD code Fgure 3 dect the erformance of teratve decodng of (57, 83) DSC code wth rate.76. We can ee that the erformance mrove wth each teraton. The frt teraton how the erformance of clacal threhold decodng [8]. n -4 Informaton ymbol Chec ymbol -5 ter () ter () Chec ter (4) ter (8) Chec ymbol -6 ter (5) On Chec Fg. : Contructon of a Product code The decodng rocedure of roduct code erformed by cacadng elementary decoder (row and column). On recevng matrx [R], the frt decoder erform the oft decodng of the row (or column) ung a nut matrx [R]. Soft Inut / Soft Outut decodng erformed ung the new rooed algorthm. Table how ome examle of contructed roduct and PCB code by ung code n table. Fg.3 : erformance of (57, 83) code on AWGN channel Fgure 4 how the frame error rate (FER) erformance of (73,45), (73,9), (89,447) and (57,83) code on both AWGN and Raylegh fadng channel. A t can be een the loe of the frame-error rate (FER) curve a tee a for the Gauan channel. It worth mentonng that the number of 5

5 FER Internatonal Journal of Comuter Alcaton ( ) Volume 7 No.7, October teraton needed for Raylegh fadng channel about the ame a for the AWGN channel. Gau 4. Normal or Body Text Raylegh Pleae ue a 9-ont Tme Roman font, or other Roman font wth erf, a cloe a oble n aearance to Tme Roman n whch thee gudelne have been et. The goal to have a 9- ont text, a you ee here. Pleae ue an-erf or nonroortonal font only for ecal uroe, uch a dtnguhng ource code text. If Tme Roman not avalable, try the -4 font (73,45)-5ter named Comuter Modern Roman. On a Macntoh, (73,9)-8ter ue the font named Tme. Rght margn hould be utfed, (89,447)-5ter not ragged. (57,83)-5ter Fg. 4: FER erformance of (73,45), (73, 9), (89, 447) and (57,83) code on AWGN and Raylegh fadng channel OSMLD code are LDPC(Low Denty Party Chec) code and can be decoded by belef roagaton (BP) algorthm []. Performance of teratve decodng for (73,45) and (73,9) code are hown and comared to thoe of BP[] n Fgure 5. A t can be een our reult are wore by. db at -5. Th codng gan neglgble comared to the requred comlexty (ee Table 3). Table 3 how mulaton tme to decode frame of (73, 45), (73,9), (89,447) and (57, 83) code wth our algorthm and BP algorthm ung 5 teraton (by ung a comuter wth Pentum 4, 3.6 GHz ). We can oberve that a the code length ncreae the comutatonal tme comlexty of BP ncreae comared to that of our teratve decodng algorthm. 4.3 Performance of roduct code Th ecton conder mulaton reult and analy for ome PC and PCB code, all of whch ue one te maorty logc decodable (OSMLD) comonent code (ee Table ). In Fg.6, we reent the mulaton reult for the (4545, 5) PCB code of rate.45 contructed from (73, 45) DSC code. We can ee that the mrovement great for the frt teraton and neglgble after the 6 th teraton. Here we recognze the Turbo effect. In Fg.7, we reent a comaron between our mulaton reult and reult ublhed n [3] for the (4545, 5) PCB code. Although they ue a modfed Gauan channel wth a tanh functon modulaton, ther reult are wore by.6 db at a of -4. The Fg. 8 how that erformance reult after 5 teraton for the (7633,8595) PCB code only.8 db away from the Shannon caacty lmt at a of -5. In Fg.9, we reent the mulaton reult for the (533,495) aymmetrc PC code of rate.3 contructed from (, ) and (73, 45) DSC code. A can be een, erformance ncreae wth each teraton (73,45)-5ter (73,45)-BP,5ter (73,9)-8ter -7 (73,9)-BP,5ter Fg. 5: Performance of (73,45) and (73,9) n comaron wth BP on AWGN channel Table 3 : Comaron n term of tme comutaton and error erformance Code Our algorthm BP algorthm Gan of BP at = -5 (73,45) 4[] 3[]. db (73,9) 58[] 3[]. db (57,83) 8[] 3[].45 db -4 teraton () teraton () -5 teraton (4) teraton (8) teraton (6) -6 teraton ().5.5 Fg. 6 Performance of teratve decodng of the (4545,5) code on AWGN channel The curve n Fg. reent the mulaton reult for the (53,495), (4545,5) PCB code and (539,5) PC. The PC comared to two PCB code. In the frt cae, the PC comared to the PCB code contructed from the ame comonent code ((73, 45) DSC code). A envaged, the PC outerform the PCB code. In the econd cae, the PC comared to a PCB code wth ame rate. A can be een, the PC better than PCB code. It eem that contrary to the PCB code, roduct code doen t have error floor. Furthermore, roduct code are better than PCB code for larger SNR. 6

6 Internatonal Journal of Comuter Alcaton ( ) Volume 7 No.7, October In Fg., we reent a comaron between our mulaton reult for (533, 495) roduct code and reult ublhed n [] for convolutonal turbo code of rate /3 ung 6 tate. We can oberve that roduct code have wore erformance at low SNR comared to convolutonal code, wherea at SNR> 4.3 db, the roduct code are better (4545,5) Luca (4545,5) ter teraton- -4 teraton- teraton-4 teraton-8 teraton-6-5 teraton SNR Fg. 9 : Performance of teratve decodng of the (533, 495) roduct code on AWGN channel Fg. 7 : Performance of teratve decodng of the PCB(4545, 5). The of the Luca et al.[3] algorthm gven a reference. -4 PCB(7633,8595) R=.48 Shannon Caatty Lmt R= (53, 495)-ter-8 (4545, 5)-ter-8 (539, 5)-ter Fg. : Performance of teratve decodng of the PC(539,5),PCB(4545,5) and PCB(53,495) code SNR Fg. 8 : Performance of the (7633,8595) PCB code comared to Shannon caacty lmt Fg. : Performance of teratve decodng of the (533, 495) roduct code on Raylegh fadng channel 7

7 Internatonal Journal of Comuter Alcaton ( ) Volume 7 No.7, October 5. CONCLUSION In th aer we have reented a new teratve threhold decodng algorthm for mle code, roduct code and arallel concatenated bloc code baed on one-te maorty logc decodable code. We ue an extenon of Maey algorthm [8] a a Soft nut/ Soft-outut comonent decoder. The tructure of our teratve decoder follow the model of Pyndah [] wth ome modfcaton. Th algorthm ha been teted on everal code baed on OSMLD code and good erformance have been obtaned over the Gauan and Raylegh fadng channel. It nteretng to extend th teratve decodng algorthm on quacyclc and mult-te maorty logc decodable code. 6. REFERENCES [] C. Berrou, A. Glaveux and P. Thtmahma, Near Shannon lmt error-correctng codng and decodng : Turbo-code (), IEEE Int. Conf. on Comm. ICC 93, Geneva, May 993, [] R. Pyndah, Near-Otmum Decodng of Product Code: Bloc Turbo Code, IEEE Tran. Commun., Aug. 998, Vol. 46, N 8,. 3-. [3] R. Luca, M. Boert and M. Bretbach, On Iteratve Soft-Decon Decodng of Lnear Bnary Bloc Code and Product Code, IEEE Journal on elected area n communcaton, February 998, Vol. [4] M. P. C. Foorer and S. Ln. Soft-Inut Soft-Outut Decodng of Lnear Bloc Code Baed on Ordered Stattc, Proc. 998 IEEE Global Telecomm. Conf. (GLOBECOM 98), Sydney, Autrala Nov , [5] J. Hagenauer, E. Offer, and L. Pae, teratve decodng of bnary bloc and convolutonal code, IEEE Tran. Inform. Theory, Mar. 996, Vol. 4, [6] R. Luca, M. P. C. Foorer, Yu Kou, and Shu Ln, Iteratve Decodng of One-Ste Maorty Logc Decodable Code Baed on Belef Proagaton, IEEE Tran. Commun, June, VOL. 48, NO. 6, [7] Yur V. Svrd and Sven Redel, Threhold Decodng of Turbo-Code, IEEE Int. Symoum on Informaton Theory, 995,. 39. [8] J.L Maey, Threhold Decodng, Cambrdge, Ma, M.I.T. Pre, 963. [9] C. Clar and B. Can, Error-Correcton Codng for dgtal communcaton, Plenum Pre, 98. [] S. Ln and D. J. Cotello, Error Control Codng, Fundamental and Alcaton, Englewood Clff, NJ: Prentce-Hall, 983. [] R. Luca, M. P. C. Foorer, Yu Kou, and Shu Ln, Iteratve Decodng of One-Ste Maorty Logc Decodable Code Baed on Belef Proagaton, IEEE Tran. Commun, June, VOL. 48, NO. 6, [] S. A. Barbulecu, Iteratve decodng of turbo code and other concatenated code, the, Unverty of South Autrala, Februry 996,