Iterative Threshold Decoding of Majority Logic Decodable Codes on Rayleigh Fading Channels

Transcription

1 SETIT 7 4 th Iteratoal Coferee: Sees of Eletro, Tehologes of Iformato ad Teleommuatos Marh 5-9, 7 TUISIA Iteratve Threshold Deodg of Maorty og Deodable Codes o Raylegh Fadg Chaels Mohamed ahmer, Mostafa Belasm EST Mees Moroo & ESIAS Rabat Moroo lahmer@est-um.a.ma belasm@esas.ma Abstrat: Iteratve threshold deodg of produt ad parallel oateated blo odes based o oe step maorty log deodable (OSMD) odes has prove to perform remarably well o AWG haels. For these odes to be applable wreless evromet, ther performae o fadg haels must be examed. The purpose of ths wor s to study the performae of our teratve threshold deodg algorthm o the Raylegh fadg hael. Results have show that the slope of the bt-error rate () urve s as steep as for the Gaussa hael. We also preset a omparso betwee our results ad those for ovolutoal turbo ode terms of performae. Key words: Iteratve threshold deodg, OSMD odes, Raylegh fadg haels, produt odes, parallel oateated blo odes. ITRODUCTIO Se the troduto of turbo odes [] there has bee muh terest soft-put soft-output (SISO) deodg algorthms for oateated odes. However, for oateated shemes wth blo ompoet odes, the omputatoal omplexty of trells-based SISO deodg algorthms s ofte hgh. Ths has led to loo for ew SISO deodg algorthms wth low omplexty ad hgh performae, e.g., [3]-[5]. These algorthms alulate extrs formato usg lassal deoders suh as Chase algorthm [3], ordered statsts deodg algorthm [5] ad Hartma/Rudolph algorthm [4]. I ths perspetve we preset a ew teratve deodg algorthm based o a SISO exteso form of Massey algorthm []. The use of the threshold algorthm teratve deodg was trodued for the frst tme by Svrd [7] but for ovolutoal odes. Our teratve deodg proess follows that gve by Pydah [3]. However, stead of usg a exteso of Chase algorthm o BCH odes, we apply a exteso of Massey algorthm o oe step maorty log deodable (OSMD) odes. O the other had uas et al. [4] trodued a teratve deodg algorthm for several famles of odes (e.g. OSMD odes) but they use a approxmato of Hartma/Rudolph algorthm []. I ths paper we vestgate the performae of produt ad parallel oateated odes based o symmetr ad asymmetr (OSMD) ompoet odes o Raylegh fadg hael. The deodg algorthm used for AWG hael s uhaged ad oly the hael relablty fator eeds to be redefed. Ths study s a exteso of our wor [, 3] whh was lmted to AWG hael; The orgazato of the paper s as follows. I Seto, we start wth a desrpto of maorty log deodg algorthm, the we trodue the bas oept of SISO Threshold deodg algorthm. I Seto, we desrbe teratve deodg algorthm o Raylegh fadg hael. Seto 3 s dedated to smulato results ad aalyss for dfferet produt ad parallel oateated odes based o OSMD odes.. Threshold deodg.. Oe step maorty log odes Cosder a (, ) lear ode C wth party-he matrx H. The row spae of H s a (, -) ode, deoted by C, whh s the dual ode of C, or the ull spae of C. For ay vetor v C ad ay vetor w C, the er produt of v ad w s zero. ow suppose that a ode vetor C s trasmtted - -

2 SETIT7 over a bary symmetr hael. et e(e, e,., e ) ad R (r, r,, r ) be the error vetor ad the reeved bary vetor respetvely. The R = v + e. For ay vetor w the dual ode C, we a form the followg lear sum of the reeved dgts : A = r p w p = p () Whh s alled a party-he sum. Usg the fat that <w,v>=, we obta the followg relatoshp betwee the he sum A ad error dgts e: A = e p = p w p () Suppose that there exst J vetors the dual ode C, whh have the followg propertes:. The th ompoet of eah vetor w s a. For there s at most oe vetor whose th ompoet s a These J vetors are sad to be orthogoal o the th dgt posto. We all them orthogoal vetors. ow, let us form J party-he sums from these J orthogoal vetors, For eah {,.., J} A e + e = p we see that the error dgt e s heed by all the he sums above. Beause of the seod property of the orthogoal vetors, ay error dgt other tha e s heed by at most oe he sum. These J he sums are sad to be orthogoal o the error dgt e. If all the error dgts the sum A are zero for, the value of A s equal to e. Based o ths fat, the party-he sums orthogoal o e a be used to estmate e, or to deode the reeved dgt r. Table shows some examples of (OSMD) odes. I ths table we used the abbrevato DSC for Dfferee Set Cyl odes, EG for Euldea Geometry odes ad BCH for Bose Chaudhur ad Hoqueghem odes. The EG odes used ths study are -order ad aordg to [6], they are OSMD odes. J d m Rate ode DSC BCH DSC EG DSC EG DSC EG DSC Table : Set of OSMD Codes p.. Maorty log deodg prple The error dgt e s deoded as f at least oe-half of the he sums orthogoal o e, are equal to ; otherwse, e s deoded as le maorty rule. whe C s a yl ode, eah e a be deoded smply by ylally permutg the reeved word r to the buffer store. Example : let us osder the (7,3) ode, whh s the short ode dfferee set odes lass (see Table ). Ths ode s spefed by the perfet dfferee set P={,, 3} of order. From ths perfet set, we a form the followg three he sums orthogoal o e 7: A = e 4 + e 5 + e 7 A = e + e 6 + e 7 A 3 = e + e 3 + e 7 If a smple error e=() our, the we have A = A = A 3 =. If a double error our, as a example e 7 = ad oe value of e,..., e 6 s equal to, the two values of A are. So we a say that : - e 7 = f oly ad f at least values of A are - e 7 =, otherwse.3. Soft-put soft-output threshold deodg Threshold deodg s smply the logal exteso to soft desos of maorty deodg desrbed above. I Massey s orgal wor [], he osdered two dfferet varatos of the deodg algorthm. We osder here the method whh uses the B equatos that are obtaed from A by a smple trasformato [9]. et us osder a trasmsso of blo oded bary symbols {,} usg a BPSK modulato over AWG hael, the deoder soft output for the th bt posto of a gve soft put R ( r, r,.., r ) s defed as: P ( = l = / R) = / R ) (3) where C(,,..., ) s the trasmtted odeword. The hard deso vetor orrespodg to the reeved vetor R s deoted by H(h, h,..., h ). For a ode wth J orthogoal party he equatos, (3) a be expressed as: /{ B } /{ B } P ( = l (4) P ( = where B, =,..J are obtaed from the orthogoal party he equatos o the th bt as follows: The term B s defed to be B = h. Eah of the B =,..,J s omputed by droppg the term h from the th orthogoal party equato. By applyg BAYES rule, (4) beomes - -

3 SETIT7 ({ B } { } = ) = ) ( B = ) = ) P l (5) P Se the party he equatos are orthogoal o the th symbol the dvdual probabltes B = or ) are all depedet ad (5) a be rewrtte as J B l B = ) + l = ) = ) = = ) (6) Assume that the trasmtted symbols are equally lely to be + or, ad thus the last term (4) s ull. As a result, we obta J B l B = ) B + l = ) B = ) = = ) Aordg to [9], (7) a be expressed as J ) w + ( B ) = (7) ( B w (8) where the value of ( B ) s equal to + or ad w s a weghtg term proportoal to the relablty of the th party he. It s easy to show that: 4E ( B ) w = (9) ad w + = l = = s r tah( tah( / ) / ) () where s the total umber of terms the th orthogoal party equato wthout, represets the th elemet of the th party equato ad 4 E s = r. () Thus the soft output a be splt to two terms, amely to a ormalzed verso of the soft put r ad a extrs formato W represetg the estmates made by the orthogoal bts o the urret bt. Hee (8) beomes 4 E W () s = r + We use the followg otato: 4E = s, (3) Whh s alled the relablty value of the hael. The algorthm struture of the SISO threshold deodg a be summarzed as follows: For eah =,.., Compute the terms B ad,.., Calulate B ad w Compute the extrs formato W The Soft-output s obtaed by: = r + W w, { J }. Iteratve threshold deodg o Raylegh fadg hael.. Costruto of produt odes et us osder two lear blo odes C havg parameters (,, d ) ad C havg parameters (,, d ) where, ad d ( =,) stad for odeword legth, umber of formato bts ad mmum Hammg dstae respetvely. It s assumed that the formato symbols are the frst symbols of C ad the frst symbols of C. The produt ode PC = C C s a (,, dd ) ode whose odewords are ostruted by eodg formato symbols wth ode C ad the resultg symbols wth C (see Fg. ). A parallel oateated blo (PCB) ode a be ostruted by a parallel oateato of blo odes. The PCB ode s a PC but wthout the hes o hes part (see Fg. ). The rate of produt ode PC ad PCB odes are gve respetvely by R PC = R PCB ad =. ) [( ) ( )] ( Iformato Ches symbols Fg. : Costruto of a Produt ode. Ches symbols Ches O Ches Tables ad 3 show some examples of ostruted produt ad PCB odes by usg odes table

4 SETIT7 Costruted PC Compoet ode (C ) Compoet ode (C ) Rate (44,) DSC(,) DSC(,).7 (533,495) DSC(,) DSC(73,45).3 (495,337) BCH (5,7) DSC(73,9).3 (3969,369) EG (63,37) EG (63,37),34 (539,5) DSC(73,45) DSC(73,45).37 (999, 8595) DSC(73,45) DSC(73,9).43 (7459,3648) DSC(73,9) DSC(73,9).48 Table : Set of PC odes Costruted PCB ode Compoet ode (C ) Compoet ode (C ) Rate (6,49) BCH (5,7) BCH (5,7).3 (34,) DSC (,) DSC (,).35 (53,495) DSC (,) DSC (73,45).38 (393, 369) EG(63,37) EG(63,37),4 (4545,5) DSC (73,45) DSC (73,45).45 (7633,8595) DSC (73,45) DSC (73,9).48 (6785, 3648) DSC(73,9) DSC(73,9).53 Table 3 : Set of PCB odes.. Iteratve threshold deodg Algorthm The deodg proess of turbo odes s a suboptmal teratve proessg whh eah ompoet deoder taes advatage of the extrs formato produed by the other ompoet deoder at the prevous step. The way the extrs formato s oveyed ad how t s exploted by the ompoet deoders to mae ther deso has ot bee losed yet. The orgal wors ths otext are due to Berrou [] ad Robertso [8] for ovolutoal odes, Pydah [3] ad uas [4] for blo odes. Hageauer [6] gave a extrapolato of Robertso s sheme for blo odes by usg a trells deoder. The teratve deodg proess preseted here follows that gve by Pydah [3] (see Fg. ). W(q) α (q) R R (q) SISO- Threshold Deoder Fg. : The blo dagram of the ompoet deoder W ( q + ) ( q + ) The soft put respetvely the soft output of the q th ompoet deoder s gve by: R( q) = R + α( q) W ( q) (4) ( q + ) = R ( q) + W ( q + ) (5) where R represet les (or olums) of the reeved data, W ( q ) s the extrs formato omputed by the prevous ompoet deoder (see Seto ). I our proedure we use a fxed value /J for the parameter ad ths for all teratos. Whereas Pydah use a o ostat value for α ( q ). The value hose for α ( q ) reats as f oe too a average of all J estmators whh otrbute the omputg of W. Ths teratve deodg algorthm a be appled to PC ad PCB odes ostruted from OSMD odes (see Table ad Table 3)..3. Modfatos for Raylegh fadg hael I the hael model we use, eah reeved bt r a be expressed as : r = a ˆ + (6) I ths represetato, ĉ s a BPSK symbol assoated to the trasmtted bt, ad s a AWG. The Raylegh varable a s geerated as a = x + y (7) where x ad y are zero mea statstally depedet Gaussa radom varables eah havg a varae σ. We osder the power ormalzed to oe as [ ] = σ = E, (8) a Whh gve a varae of.5 for Gausse varables. O the Raylegh fadg hael, the avalablty of hael sde formato s the ey ssue determg the eessary modfato for the teratve threshold deodg algorthm. The threshold deodg algorthm has to be modfed slghtly by hagg equato (3) whh defe the relablty value of the hael by 4E s = a (9) Wth ths modfato we a use the same deoder struture whh was desrbed Fgure. 3. Smulato results Ths seto osders smulato results ad aalyss for some PC ad PCB odes, all of whh use oe step maorty log deodable (OSMD) ompoet odes. I all our smulatos we assume a aurate fade estmate at the reevg ed ad a depedet Raylegh dstrbuto of the fades. Due to omputatoal lmtatos, we have used a mmal resdual error of. The performae mproves wth eah terato all smulato results preseted. Fg. 3 depts the performae of (533, 495) asymmetr PC ode of rate.3 ostruted from (, ) ad (73, 45) DSC odes. We a see that the mprovemet s great for the frst teratos ad s - 4 -

5 SETIT7 eglgble after the 6 th terato terato () terato () terato (4) -6 terato (8) terato (6) terato () Fg. 3 : Performae of teratve deodg of (533, 495) produt ode the I Fg.4, we preset a omparso betwee our smulato results for (533, 495) produt ode ad results publshed [4] for ovolutoal turbo ode of rate /3 usg 6 states. We a observe that produt ode has worse performae at low SR ompared to ovolutoal ode, whereas at SR> 4.3 db, the produt ode s better. The urves the Fg.6 show the aheved bt error rates for the followg odes: (6, 49), (34,), (95,495) approxmately wth the same rate. We have used a umber of teratos suh that there s o sgfatly more to ga by more terato. We observe that the performae reases whe we rease the legth of the ode wth ode rates almost equals ter() ter(4) ter(8) -5 ter(6) ter() ter(3) GAUSS Fg. 5 : Performae of teratve deodg of the (4545,5) PCB ode (533,495)terato(8) Rate /3 Covoluoal ode terato(8) Fg. 4 : Performae of teratve deodg of (533, 495) produt ode. the Fgure 5, show the performae of PCB ode of rate.45 ostruted from (73, 45) DSC ode o both Raylegh ad Gauss haels. As observed for turbo odes[4], the performae of PCB odes a depedet Raylegh hael s worse tha that for the AWG hael by approxmately.5 db odg ga. It s worth metog that the umber of teratos eeded s about the same as for the AWG hael. (6,49) 5ter (34,) ter (53,495) ter Fg. 6 : Performae of teratve deodg of (6, 49), (34, ), (53, 495) PCB odes. the The urves Fg. 7 preset the smulato results of the asymmetr (53, 495) PCB ode (ostruted from (,) ad (73,45)) odes ad symmetr (539,5) PC (ostruted from (73,45) ode). It seems that otrary to the PCB odes, produt odes does t have error floor. Furthermore, produt odes are better tha PCB odes for larger SRs

6 SETIT PC (539,5) PCB (53,495) Fg. 7 : Performae of teratve deodg of the (95, 495) PCB ode ad (539,5) produt ode. 4. Coluso I ths paper we have vestgated performae of a ew teratve threshold deodg algorthm over Raylegh fadg hael. The deodg algorthm used for AWG hael [, 3] s uhaged ad oly the hael relablty fator eeds to be redefed. Ths algorthm has bee tested o several PC ad PCB odes based o OSMD odes ad good performaes have bee obtaed over the Raylegh fadg hael. It s terestg to apply ths teratve deodg algorthm o other haels models le Re or aagam Referees [] C. Berrou, A. Glaveux ad P. Thtmashma, ear Shao lmt error-orretg odg ad deodg : Turbo-odes, IEEE It. Cof. o Comm. ICC 93, Geeva, May 993, pp [] J. Massey, Threshold Deodg, Cambrdge, Ma, M.I.T. Press, 963. [3] R. Pydah, ear-optmum Deodg of Produt Codes: Blo Turbo Codes, IEEE Tras. Commu., Aug. 998, Vol. 46, 8, pp. 3-. [7] Yur V. Svrd ad Sve Redel, Threshold Deodg of Turbo-Codes, IEEE It. Symposum o Iformato Theory, 995, pp. 39. [8] P. Robertso, Illumatg the struture of ode ad deoder of parallel oateated reursve systemat (turbo) odes, Pro. IEEE Global Commu. Cof. (GOBCOM 94), Sa Fraso, CA, De. 994, pp [9] C. Clar ad B. Ca, Error-Correto Codg for dgtal ommuatos, Pleum Press, 98. [] C. R. P. Hartma ad. D. Rudolph, A optmum symbol by symbol deodg rule for lear odes, IEEE Tras. Iform. Theory, Sept. 973, Vol. IT-, pp [] O. Y. Taeshta, O. M. Colls, P.C. Massey ad D. J. Costello, Asymmetr Turbo-Codes, ISIT, Cambrdge, MA, USA, August 6, 998 [] M. Belasm, M. ahmer, ad M. Behrfa, Iteratve Threshold Deodg of Parallel Coateated Blo Codes, Proeedg Turbo Codg 6 Cof.,4-7 Aprl 6, Muh. [3] M. Belasm, M. ahmer, ad F. Ayoub, Iteratve Threshold Deodg of Produt Codes Costruted from Maorty og Deodable Codes, ICTTA 6 Cof., 4-8 Aprl 6, Damasus Syre. [4] S. A. Barbulesu, Iteratve deodg of turbo odes ad other oateated odes, thess, Uversty of South Australa, Februry 996, pp [5] E. K. Hall ad S. G. Wlso, Desg ad Aalyss of Turbo Codes o Raylegh Fadg Chaels," IEEE Jour. Selet. Areas. Comm., vol. 6, pp. 6-74, Feb [6] S. ad D. J. Costello, Error Cotrol Codg, Fudametals ad Applatos, Eglewood Clffs, J: Prete-Hall, 983. [4] R. uas, M. Bossert ad M. Bretbah, O Iteratve Soft-Deso Deodg of ear Bary Blo Codes ad Produt Codes, IEEE Joural o seleted areas ommuatos, February 998, Vol. 6,, pp [5] M. P. C. Fossorer ad S.. Soft-Iput Soft-Output Deodg of ear Blo Codes Based o Ordered Statsts, Pro. 998 IEEE Global Teleomm. Cof. (GOBECOM 98), Sydey, Australa ov pp , [6] J. Hageauer, E. Offer, ad. Pape, teratve deodg of bary blo ad ovolutoal odes, IEEE Tras. Iform. Theory, Mar. 996, Vol. 4, pp