Reduced The Complexity of Soft Input Soft Output Maximum A posteriori Decoder of Linear Block Code By Using Parallel Trellises Structure

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1 Journal of Babylon nvrsy/engnrng Scncs/ No.4/ Vol.25: 27 Rducd Th Complxy of Sof Inpu Sof upu Maxmum A posror Dcodr of Lnar Block Cod By sng Paralll Trllss Srucur Samr Abdul Cahm Khohr Hadr Jabbar Abd Elcrcal Engnrng Dparmn Collag of Engnrng Babylon nvrsy hadrlasr@yahoo.com Absrac Ths sarch prsn rlls srucurs of lnar block cod capabl of achvng hgh dcodng spd whl sasfyng a consran on h srucural complxy of h rlls n rms of h maxmum numbr of sas a any parcular dph. Frs w dscrb mnmal rlls of lnar block cod ha mnmzs on or mor masurs of rlls complxy for h cod. W dnfy h prmv srucurs ha can appar n a mnmal rlls hn w appld h sconalzd o rlls whr only unform sconalzaons of h cod rlls dagram ar consdrd.nx paralll and srucurally dncal subrllss for a cod whou xcdng h maxmum sa complxy of h mnmal rlls of h cod s dscussd. Th complxy of dcodr basd on a sconalzd rlls dagram for a cod s dscrbd. In Ths papr w dscrb how o apply SISsof npu sof oupu max-log-map dcodr usng paralll srucur of sconalzd rlls for block cod. Th 84 and 65RM Rd Mullr cod ar ncludd hr bcaus hy offrs paralll and srucurally dncal subrllss whou cross conncons among hm ha wll rducs h dcodng complxy and mprovs dcodng spd. Ths papr also gvs smulaon rsuls for rav dcodng of paralll concanad block cod of hs wo cods ovr AWGN channl by usng SIS max-log-map dcodr basd on paralll rllss srucur. Kywords: SISSNRBER MAP and RM cod SIS-Log 65RM 84RM MAP SIS-max-log الكلمات المفتاحية:-مشفر MPA هياكل الشجيرة المتوازيت ترميس MAP نسبت االشارة الى الضوضاء المرمس الخطي. I Inroducon Modrn powrful dcodng schm lk urbo cods ncssa h ulzaon of rav dcodng algorhms. Bndo al. 44; Dvsalar and Pollara 43 dscrb a gnral SIS MAP modul ha can b usd n rav dcodng of sral concanaon convoluonal cods SCCC and paralll concanaon convoluonal cods PCCC. For praccal applcaon hs dcodng algorhm mus b smplfd and s dcodng complxy and dlay mus b rducd. sually max-log-map usd o rduc h dcodng complxy as compar wh MAP dcodr wh nglgbl dgradaon n BER prformanc Y Ln.al. 42. Any lnar block cod can horcally b dcodd by applyng h max-log- MAP algorhm o a rlls for h cod. Dcodng complxy rmans howvr a srous mpdmn o h us of rlls dcodrs for block cods. In hs papr w focus on opmal sconalzaon of a rlls o mnmz compuaonal complxy ha mnmzng h oal numbr of dcodng opraons. 44

2 Journal of Babylon nvrsy/engnrng Scncs/ No.4/ Vol.25: 27 II Mnmal Trlls of Bnary Lnar Block Cods and s Prmv Srucurs. A rlls s a powrful mhod of dscrbng a block cod. For h consrucon of a rlls h ncodr sars from som nal sa usually all-zro sa dnod by s. A ach m nsan { N} h ncodr aks on xacly on sa s. Thr ar a fn numbr of allowd sas a m nsan dnod as C.As h ncodr procss movs from o a cod b c s md Pr 29. A-Mnmal Trlls of Block Cods. Th gnraor marx of a lnar block cod C s sad o b n a rlls ornd form f h followng condons wr hold Pr 29; dayan 26: - Th ladng of any row appars n a column bfor h ladng of any rows appars bfor. 2- No wo rows hav hr ralng s n h sam columns. Any gnraor marx can b pu n a rlls ornd gnraor marx TGM by usng Gaussan lmnaon on row or mad any row prmuaon Pr 29; dayan 26. L r r 2.r k b h k rows of TGM whr r =[ r r 2... r N ] k hn h span of a row r n TGM s dfnd as h smalls nrval {+ } whch conans all h non zro lmns of a row r. Ths s dnod by span r. r dnod by For a row r n TGM whos span s [] h acv span of a row aspan r s dfnd as aspan r =[-] for <. Th aspan r =ф for =Pr 29; dayan 26. No ha TGM of a gvn gnraor marx s no unqu hn h TGM of a gvn gnraor marx s calld mnmal span gnraor marx MSGM f all h spans ar as shor as possbl Pr 29; dayan 26. A mnmal rlls for h cod can b consrucd from h MSGM. Th rlls has n + lvls of vrcs and n lvls of dgs. Th vrx lvls calld dphs ar numbrd from o n; h dg lvls calld sags ar numbrd from o n Pr 29; dayan 26. Th vrx span acv span of h h row s h s of dphs a whch h h nformaon symbol can affc h ncodr sa Kly al. 25. B- Pas and Fuur Subcods Th h pas and fuur subcods dnod P and F can b dfn as:h ss of all codwords whos vrx acv spans ar conand n [; -] and [+; n] rspcvly. Th dmnsons of hs cods can b asly drmnd from h MSGM: f Dm F s h numbr of rows for whch h lfmos nonzro nry ls n column + or lar and p Dm P s h numbr of rows for whch h rghmos nonzro nry ls n column or arlr. Ths mpls ha p and f ar monoonckly al. 25. p p pn K f f fn and nvr chang by mor han from on ndx o h nx. For ach N h lf- and rgh-bass ndcaors ar dfnd as l p {} o dnfy h posons whr h fuur and pas dmnsons changkly al. 25: l f f r p p 2 For any l = f and only f h dg span of som row of h MSGM bgns n column. Smlarly r = f and only f h dg span of som row of MSGM nds 44

3 Journal of Babylon nvrsy/engnrng Scncs/ No.4/ Vol.25: 27 n column. Th columns whr l = and h columns whr r = ach forms a bass for h column spac of MSGM and hs ss ar calld h lf bass and h rgh bass rspcvly. Th posons of h lf and rgh bass columns can b rgardd as nformaon posons whn h gnraor marx s usd o ncod h nformaon lf o rgh or rgh o lf rspcvly Kly.al. 25; Luna.al. 28. C- Prmv Srucurs of a Mnmal Trlls Thr ar four basc buldng blocks ha can b usd o consruc h mnmal rlls for any block cod. A any gvn sag all prmv srucurs ar of h sam yp whch s drmnd by h valus of l and r. Th prmv srucurs arkly.al. 25: -Smpl xnson : Ths prmv srucur appars a sag whn l=; r=.smpl xnsons a sag mply a sngl dg ou of ach vrx a dph - and a sngl dg no ach vrx a dph ; hnc h numbr of vrcs rmans consan. 2-Smpl xpanson <: Ths corrsponds o l = ; r =. Thr ar 2 dgs ou of ach vrx a dph - and a sngl dg no ach vrx a dph hnc mulplyng by 2 h numbr of sas from on vrx dph o h nx. 3-Smpl mrgr >: Ths corrsponds o l = ; r =. A smpl mrgr s a mrvrsd smpl xpanson rducng h numbr of sas by a facor of 2. 4-Burfly : Ths corrsponds o l = ; r =. Thr ar 2 dgs ou of ach vrx a dph - and 2dgs no ach vrx a dph ; hnc h numbr of sas s consan. III Trlls Complxy of Bnary Block Cods Th complxy of a rlls dcodng for a block cod C follows almos mmdaly from h complxy of a rlls rprsnaon on whch s basd. Toal numbr of vrcs VC and oal numbr of dgs EC of N-scons rlls of bnary lnar block cod C ar dfnd as Kly.al. 25; Luna.al. 28: EC VC N N Whr 2 s h oal numbr of dgs n scon s h dg spac dmnson of scon. For mnmal rlls h vrx spac dmnson a dph skly.al. 25: v K f p 5 h dg spac dmnson a sag skly.al. 25: K f p 6 and h oal numbr of mrgrs for rlls of bnary block cod MC s Kly.al. 25: M C N 2N E C V C 7 IV Trlls Sconalzaon and Paralll Trllss Srucur. Equaon shows ha h dcodng complxy dpnd manly on h numbr of dg and numbr of vrxs so ha w mus rduc EC and VC. Th frs mhod ha usd o rduc h rlls complxy s usng mnmal rlls whch dscussd n scon II h scond mhod s usng rlls sconalzaon mhodology

4 Journal of Babylon nvrsy/engnrng Scncs/ No.4/ Vol.25: 27 A- Sconalzd Trlls In h scon II a rlls s consrucd whr ach sa ranson s assocad wh on cod b. Som ms s favorabl o group mor han on b oghr for a sa ranson hus h boundary locaons Q={q = q q 2 q V- q V =N} n h rlls whr q - <q V and V dnos h oal numbr of scons. A scon of h rlls consss sas q C and q C and all possbl ransons bwn any sa s q and any sa s q. Thr ar n = q -q - cod bs assocad wh scon formng h squnc [ c... c ]. A rlls s calld unformly q sconalzd f n = q -q - = N/V for all Pr and Danl 27; Pr 29. B- Paralll Trllss srucur. Th hrd mhod o rduc h dcodng complxy s usng rlls sconalzaon wh paralll rllss srucur. A V scons rlls for a lnar block cod can b dcomposd no paralll and srucurally dncal subrllss whou cross conncons among hm and whou xcdng h maxmum sa complxy of h rlls. Each subrlls has much smallr sa complxy and conncvy han h full cod rlls. Ths paralll dcomposon allows dvsng dncal smallr dcodrs o procss h subrllss n paralll ndpndnly whou communcaon bwn hm. Ths also smplfs IC mplmnaon and spds up h dcodng procss Moorhy.al. 26. V SIS max-log-map Dcodr for Bnary lnar Block Cods. In Bndo.al. 28; Dvsalar and Pollara 27 Bndo.al. dscrbd a chnqu for dcodng convoluonal cods basd on connuously updang h maxmum a posror probably of npu and oupu cod bs usng sof npu sof oupu MAP algorhm. Ths algorhm s a symbol-by-symbol rlls basd dcodng algorhm hrfor w wll modfd hs algorhm o mak work on h rlls of lnar block cod. For praccal applcaon max-log-map s prfrrd o MAP algorhm snc max-log-map uss log-lklhood rao of ach b. Log- Lklhood Rao LLR of a bnary b z {-+} s dfnd as: p z z ln 8 p z Th SIS max-log-map modul shown n Fg. s a hr por dvc ha accps as npus h LLR s of h nformaon bs and cod bs C and oupus as updas LLR s for h nformaon bs. q C SIS max-log- MAP Dcodr Fg. SIS max-log-map Dcodr Modul A- SIS max-log-map Dcodr Basd Sconalzd Trlls. Th dynamcs of a m varan V scon rlls of NKbnary lnar block cod wh sa spac dmnson{ v v. v v V } and dg spac dmnson{. V } s complly spcfd by a sngl rlls scon whch dscrbs h ranson dg bwn sas of h rlls a m nsans - and. A rlls scon a m s characrzd by h followng: 443

5 Journal of Babylon nvrsy/engnrng Scncs/ No.4/ Vol.25: 27 v - A s of 2 sarng sa. 2- A s of 2 dgs whch rprsn all possbl ranson bwn h rlls sas. v 3- A s of 2 ndng sa. Th followng funcons ar assocad wh ach dg as shown n Fg.2 S E S S - / C Tm - -Th sarng sa S S of dg. - 2-Th ndng sa S E of dg. 2 k 3-Th nformaon bs f any [ u u... u ] whr u {-+} s h h nformaon b assocad wh dg a m and k s numbr of nformaon bs f any a scon. 2 n 4- Th cod bs C [ c c... c ] whr c {-+} s h h cod b assocad wh dg a m and n s numbr of cod bs a scon. Th wo npu vcors of SIS max-log-map dcodr ar dfnd as: N - C [ c... c ] s a pror nformaon LLR of a codword. K 2- [ u... u ] s a pror nformaon LLR of nformaon word. To apply SIS max-log-map algorhm on V-scon rlls w mus dvd ach npu squnc no V-group as follow: C { C... C... C } whr C [ c... c Fg.2 An dg of a rlls scon n ] V and { } whr [ u... u k ] and V K h oupu vcor [ u... u ] s h xrnsc nformaon LLR of nformaon. Th SIS max-log-map basd on V scon rlls algorhm prforms frs h wo rcursons: -Th forward rcurson of sa s a m s gvn by: k n S s max E S u u c c =2 V 9 : S s Snc h ncodr sars wh a known sa s o h forward rcurson wll b nalzd as o s. 2- Th backward rcurson of sa s a m s: k n E s max S S u u c c I =V-V-2 : S s 444

6 Journal of Babylon nvrsy/engnrng Scncs/ No.4/ Vol.25: Snc h rlls s rmnad o known sa s o hn h backward rcurson wll b nalzd as s V. Th oupu of h dcodr s h xrnsc nformaon LLR of nformaon bs whch can b dvdd no V-group } { V whr ]... [ u u I k and u s h xrnsc nformaon of h nformaon b a -scon whch can b oband as: max max : : S I c c I u u S S I c c I u u S u E n k S u E n k S u =2..V ; =2.. k B- SIS max-log-map Dcodr Basd on Paralll Trllss Srucur. To apply max-log-map algorhm o such rlls h oupu LLR of SIS dcodr Eq.7can b pu n a form Y Ln Shu Lo and Marc Fossorr 26: 2 Suppos a V scons rlls of an NK bnary lnar block cod s dcomposd no W subrllss T T2 TW. Th max-log-map dcodr for h subrlls Tw W w fnd a pars of w and w for ach b. Th fnal survvng pars of h nr rlls ar found by[3]: } : max{ W w w 3 } : { max W w w 4 Fnally h log-lklhood rao of ach b s found by usng Eq.8 as shown n Fg3. From h sand pon of spd h ffcv compuaonal complxy of dcodng a rcvd squnc s dfnd as h compuaonal complxy of a sngl paralll subrlls plus h cos of h fnal comparson max among h survvors gnrad by ach of h subrlls dcodr. Th m rqurd for fnal comparson s gnrally small rlav o h m rqurd for procssng a subrlls. Furhrmor Fg.3 SIS max-log-map dcodr basd on paralll rllss srucur

7 Journal of Babylon nvrsy/engnrng Scncs/ No.4/ Vol.25: 27 snc all h subrllss procssd n paralll h dcodng spd s hrfor lmd only by h m rqurd o procss on subrlls. Consqunly hs approach dos no only smplfy h dcodng complxy bu also gan spd Y Ln Shu Lo and Marc Fossorr 26. VI Dcodng Complxy of 84 and 65 RM cods Basd on paralll Trllss. Rd Mullr cod s on of powrful cods ha usd n rror corrcon codng and s rlls can b dcompnsa no paralll rllss srucur asly. A- 84 RM cod. Th rlls ornd gnraor marx of 84RM cod s: TGM 5 By usng h analyss prsn n scon II -Acv span of ach row: aspanr =[3] aspanr 2 =[26] aspanr 3 =[35] and aspanr 4 =[57]. 2- Trlls complxy and prmv srucur of h rlls can b found by fndng h h lf- and rgh-bass l r hn apply Eq.2 o fnd Th h pas and fuur subcods hn apply Eqs.56 o fnd h vrx spac dmnson v a dph and h dg spac dmnson a sag s gvn n h abl2 Tabl 2 Complxy analyss of 84RM cod ndx l X r X f p v SCP X srucur < < < > < > > > By usng Eqs.347 oal numbr of vrcs VC =34 oal numbr of dgs EC = 44 and oal numbr of mrgrs MC=44-34+=. Th sconalzd rlls of 84 RM cod f h boundng locaon Q={2468} no nclud vrcs 35 whch hav maxmum sa spac dmnson s shown n Fg.4. 44

8 Journal of Babylon nvrsy/engnrng Scncs/ No.4/ Vol.25: 27 / / / / / / / / / / -/ -/ / / / / / / / / -/ -/ / / Whr h sa complxy profl SCP of hs rlls s 222 and SCP of ach subrlls s v ={} and dg spac dmnson={22}. Thrfor Numbr of vrcs for ach subrlls VC =8. Numbr of dgs for ach subrlls EC =2. Numbr of mrgrs for ach subrlls MC=5. To xplan h complxy of SIS max-log-map dcodr basd on paralll rllss srucurs rlls of 84 RM cod has a four-scon 4-sa mnmal rlls dagram whch consss of wo paralll and srucurally dncal gh-sa subrllss ach s 2 dgs and 5 mrgrs cross conncons among hm. As a rsul w can dvs wo dncal gh-sa SIS max-log-map dcodrs o procss h wo subrllss n paralll whou communcaon bwn hm. A h oupu hr ar four LLR's wo for ach subrlls dcodr and h wo comparaors and subracor wll fnd h fnal wll. Thrfor h complxy of SIS max-log-map for on subrlls s: - Compung h forward rcurson s rqurs: >>Numbr mmory =VC =8. No ha only 6 of hm nd o b calculad s and no usful.no ha ach mmory wll sor floang numbr. 4 s >>Numbr of max opraon = MC-=4 no nd o calcula 4 s. >>Numbr of mulplcaons and addons: k a-th rm u u nd k addons and mulplcaons for ach dg = = 2 addons and 2 mulplcaons. No k =[2] n b-th rm c c nd n addons and mulplcaons for ach dg = Fg.4 Sconalzd rlls of 84 RM cod =24 addons and 24 mulplcaons. No n = [2222] c- Thr addons bwn rms Eq.3 for ach forward rcurson=3 6=8 Thn h oal dlay of compung forward rcursons=t max +T ADD +T ML. In h modrn procssng sysm hs hr m almos qual hn oal dlay= T = 94 T. 444

9 Journal of Babylon nvrsy/engnrng Scncs/ No.4/ Vol.25: 27 2-Complxy for calculang backward rcurson B s s sam as h complxy of forward rcurson. 3- Compung h rqurs: >>Numbr mmory =4 K=4 >>Numbr of max opraon =4 on for ach b >>Numbr of mulplcaons and addons: k a-th rm u u nd = 2 = 2 addons and 2 mulplcaons. n k addons and mulplcaons for ach dg b- Th rm c c nd n addons and mulplcaons for ach dg= =24 addons and 24 mulplcaons. c- Four addons bwn rms Eq.7 for ach valu=4 4=6. Thn h oal dlay of compung forward rcursons=t max +T ADD +T ML hn oal dlay= T = 72T 4- Complxy for calculang s sam as h complxy of. Th oal dlay of SIS max-log-map dcod for ach subrlls= T= 332T and oal numbr of mmory s =2 6+ 4=2. Snc hr ar wo paralll subrllss hn: a-if h wo dcodr run a sam m hn w nd 4mmory un o sor and un plus mmory un o sor and C as npu for ach dcodr whl h dlay of h SIS max-log-map dcodr s 332T+2 4T+ 4T=344T h fnal wo max opraon and subracon for ach valus. b-if h dcodr of subrlls run succssvly hn w nd only 2 mmory un plus 4+8mmory un o sor and C h sam mmory locaons usd for ach dcodrs whl an ovr all dlay s 2 332T+2 4T+ 4T=676T. B- 65 RM cod Th rlls ornd gnraor marx of 65RM cod s: TGM 6 -Acv span of ach row: aspanr =[7] aspanr 2 =[24] aspanr 3 =[33] aspanr 4 =[5] and aspanr 5 =[95]. 2- Trlls complxy and prmv srucur of hs rlls s gvn n h abl

10 Journal of Babylon nvrsy/engnrng Scncs/ No.4/ Vol.25: 27 Tabl 3 Complxy analyss of 65RM cod ndx l X r X f p v SCP X srucur < < < - < - - > < - - > - > > > By usng Eqs.347 oal numbr of vrcs VC =5 oal numbr of dgs EC =72 and oal numbr of mrgrs MC=23 Th sconalzd rlls of 65 RM cod f h boundng locaon Q={4826} no nclud vrcs 5679 whch hav maxmum sa spac dmnson s shown n Fg.5. Fg.5 Sconalzd rlls of 65 RM cod Whr h sa complxy profl SCP of hs rlls s 444 and SCP of ach subrlls s v ={} and dg spac dmnson={22}. Thrfor Numbr of vrcs for ach subrlls VC =8 Numbr of dgs for ach subrlls EC =2 Numbr of mrgrs for ach subrlls MC=5 To xplan h complxy of SIS max-log-map dcodr basd on paralll rllss srucurs rlls of 65 RM cod has a four-scon 26-sa mnmal rlls dagram whch consss of four paralll and srucurally dncal gh-sa subrllss whou cross conncons among hm. As a rsul w can dvs four dncal gh-sa SIS max-log-map dcodrs o procss h four subrllss n paralll whou communcaon bwn hm. A h oupu hr ar gh LLR's 444

11 Journal of Babylon nvrsy/engnrng Scncs/ No.4/ Vol.25: 27 and h sx comparaor hr o fnd and hr o fnd wh subracor wll fnd. Thrfor h complxy of SIS max-log-map for on subrlls s: - Compung h forward rcurson s rqurs: >>Numbr mmory =VC =8. No ha only 6 of hm nd o b calculad s and no usful No ha ach mmory wll sor floang numbr. 4 s >>Numbr of max opraon = MC-=4 no nd o calcula 4 s >>Numbr of mulplcaons and addons: k a-th rm u u nd = =4 addons and 4 mulplcaons. n k addons and mulplcaons for ach dg b-th rm c c nd n addons and mulplcaons for ach dg = = 48 addons and 48 mulplcaons. c- Thr addons bwn rms Eq.9 for ach forward rcurson=3 6=8 Thn h oal dlay of compung forward rcursons=t max +T ADD +T ML. In h modrn procssng sysm hs hr m almos qual hn oal dlay= T = 46T 2-Complxy for calculang backward rcurson B s s sam as h complxy of forward rcurson. 3- Compung h rqurs: >>Numbr mmory =5 K=5 >>Numbr of max opraon =5 on for ach b >>Numbr of mulplcaons and addons: k a-th rm u u nd = 2 2 = 4 addons and 4 mulplcaons. n k addons and mulplcaons for ach dg b- Th rm c c nd n addons and mulplcaons for ach dg= =48 addons and 48 mulplcaons. c- Four addons bwn rms Eq.7 for ach valu=4 5=2 Thn h oal dlay of compung forward rcursons=t max +T ADD +T ML hn oal dlay= T = 29T 4- Complxy for calculang s sam as h complxy of. Th oal dlay of SIS max-log-map dcod for ach subrlls= T= 55T and oal numbr of mmory s 2 6+5=22. Snc hr ar four paralll subrllss hn: a-if h four dcodr run a sam m hn w nd 4 22mmory un o sor and un plus mmory un o sor and C as npu for ach dcodr whl h dlay of h SIS max-log-map dcodr s 55T+2 3 5T+ 5T=585T h fnal sx max opraon and on subracor for ach valus. b-if h dcodrs of subrlls run succssvly hn w nd only 22 mmory un plus 5+6mmory un o sor and C h sam mmory locaons usd for ach dcodrs whl an ovr all dlay s 4 55T T+ 5T=2235T. 44

12 Journal of Babylon nvrsy/engnrng Scncs/ No.4/ Vol.25: 27 VII Smulaon of Turbo Dcodng of Paralll Concanad Block Cod usng SIS max-log-map algorhm Turbo cods can b consrucd from wo dncal or wo dffrn componn cods C and C2 n hs papr w wll dscuss Paralll Concanad Block Cods PCBCs. I maks sns o ak only cods n nonsysmac form as componn cods do no allow a sparaon of nformaon bs and pary bs. Th nformaon squnc wh K2 row and K column ar ncodd usng of ncodr wh paramr NK o produc codword Cwh K2 rows and N of column hn h nformaon squnc pass hrow row-by-column nrlavr o produc 2 wh K row and K2 column hn ncodd usng of ncodr2 wh paramr N2K2 o produc codword C2wh K rows and N2 of columnpr 29; Png and Yug 999. Th ovr all ra of ncod s gvn by: K K2 Ra 7 K2 N K N2 N N2 K K Ra Ra 2 Afr h oupu cod words CC2 ar modulad and ransmd hrow 2 channl whch consdr AWGN channl wh A.C nos powr h rcvd 2 squnc s mulply by h facor 2/ and dmulplxd no C C2 hn passs hrow urbo dcodr. Durng h frs raon s s o zro snc no a pror nformaon s avalabl on h npu nformaon bs of h ncodr. Th xrnsc LLR s of h nformaon bs of h dcodr ar passd hrough h nrlavr o oban h LLR s of nformaon bs of h ncodr2 2. Th oupu of dcodr2 2 afr dnrlavng s mulply by Alfa whr s a scalng facor usd o rduc h ffcs of h xrnsc nformaon n sof dcodr n h frs dcodng sps whn h BER s rlavly hgh. I aks a small valu n h frs dcodng sp and ncrass as h BER nds o zros. Th valus of Alfa wh h dcodng sp ar chosn as Alfa= [ ].Thn h oupu of dnrlavr s fdback o h lowr nry corrspondng o nformaon bs of h cod of SIS dcodr of dcodr o sar h scond raon.th fnal dcson for ach raon s found by addng 2 and hn passd hrow hard lmr hn compar wh npu nformaon o fnd BER for ach raon a gvn SNR as shown n Fg.6 Pr 29; Png 29; Crozr and Hun

13 Sof oupu Dmodulaor Modulaor Mulplxr Dmulplxr Journal of Babylon nvrsy/engnrng Scncs/ No.4/ Vol.25: 27 ENC2 C Inrlavr C X Y 2 ENC2 C2 AWGN D-nrlavr 2 2/ Dcson Alfa DEC C Inrlavr 2 2 DEC2 C2 Fg.6 Turbo dcodng of paralll concanad block cod Ths sysm s sofwar mplmnd usng MATLAB7 programmng languag whr wo cod ar usd 84 and 65RM cods: Th prformanc of urbo dcodng paralll concanad of wo 84RM cods usng SIS max-log-map algorhm basd on paralll rllss srucur shown n Fg.4 s shown n Fg.7. Fg.7BER prformanc of 84RM urbo cod Th prformanc of urbo dcodng paralll concanad of wo 65RM cods usng SIS max-log-map algorhm basd on paralll rllss srucur shown n Fg.5 s shown n Fg.8. Fg.8BER prformanc of 65RM urbo cod 442

14 Journal of Babylon nvrsy/engnrng Scncs/ No.4/ Vol.25: 27 VIII Conclusons W hav prsnd an approach for dcomposng h mnmal rlls of a bnary lnar block cod no paralll rllss srucurs ha achvng low dcodng complxy. Ths approach allows paralll procssng of h subrllss and dos no ncras h maxmum numbr of sas. Hnc has sgnfcan spd advanag. Snc h applcaon of hs mhod dpnds only on h gnraor marx of h cod can b appld o arbrary lnar block cods. An rav rlls basd dcodng SIS max-log-map algorhm of block cods was prsnd. Th complxy of hs algorhm was rducd by usng paralll rllss srucur. Ths rsuls n rducd dcodng complxy ffr block cods wh smpl rlls rprsnaons. I was shown ha h rducd-complxy dcodr provds nar opmal prformanc whn a lm s placd on h numbr of raons allowd. Th compuaonal complxy shows h of us paralll dcodng of hs subrllss gvs hgh dcodng spd bu nd lagr numbr of mmors and h us of succssv dcodng of hs subrllss nd fw numbr of mmors bu dcodng spd s low as compar wh paralll procssng as gvn n h abl4: Tabl 4 Dcodng complxy of 84 and 65RM cods Paralll dcodng Succssv dcodng Cod Numbr of Mmory Dlay Numbr of Mmory Dlay 84RM T T 65RM T T Rfrnc Bndo S.; Dvsalar D.; Monors G. and Pollara F. 28. Sof Inpu Sof upu Maxmum A posror MAP Modul o Dcod Paralll and Sral Concanad Cods TDA Progrss Rpor PP.-2 Novmbr 5. Brgr Y. and B y Y. 24. Sof Trlls-Basd Dcodng for Lnar Block Cods IEEE Trans. on Inform. Thory Vol.4 No.3 PP May. Crozr S. Kn Grac and Hun A. 28. ''Effcn Turbo Dcodng Tchnqus" Commun. Rsarch Cnr pp Dvsalar D. and Pollara F.27. Hybrd Concanad Cods and Irav Dcodng TDA Progrss Rpor 42-3 PP.-23 Augus 5. Kly A. B.; Dolnar S. and Kroo L.E. 25. Mnmum Trlls for Block Cods and Thr Duals TDA Progrss Rpor 42-2 PP May 5. Luna A. A. ; Fonan F.M. and Wckr S.B. 28. Irav Maxmum-Lklhood Trlls Dcodng for Block Cods IEEE Trasn. n Commun. Vol.47 No.3 PP March. Moorhy H.T.; Ln S. and hara G.T. 26. Good Trlls for IC Implmnaon for Vrb Dcodrs for Lnar Block Cods IEEE Trans. Commun. Vol.45 No. PP January. Pr J. S. 29. Irav Dcodng of Paralll Concanad Hgh Ra Lnar Block Cods MSC. Thss nv. Nor Dam. Pr J. S. and Danl J.C. Jr. 27. MAP Dcodng of Lnar Block Cods Basd on a Sconalzd Trlls of h Dual Cod nv. of Nor Dam IN46556 PP.-8. Png L. and Yug K. L. 29. ''Symbol-by-Symbol APP Dcodng of Golay cod and Irav Dcodng of Concanad Golay Cods" IEEE Trans. Commun. Vol.45 No.7 PP Novmbr. Tapp T.L.; Luna A.A. ; Wang X.; and Wchr S.B.27. Exndd Hammng BCH Sof Dcson Dcodrs for Mobl Daa Applcaons IEEE Trans. Commun. Vol.47 No.3 PP March. 443

15 Journal of Babylon nvrsy/engnrng Scncs/ No.4/ Vol.25: 27 Tn Brnk S. 27. ''Convrgnc Bhavor of Iravly Dcodd Parlll Concanad Cods'' IEEE Trans. Commun. Vol.49 No. PP cobr. dayan D.G. 26. Low Complxy Tchnqus for Channl Dcodng and Equalzaon Ph.D Thss nv. of Txas A &M Augus. Y Ln Shu Lo and Marc Fossorr 26. MAP Algorhms for Dcodng Lnar Block Cods Basd on Sconalzd Trlls Dagrams nv. of Hawa a Monoa PP