Modrn Chnnl Coding Ingmr Lnd & Joss Sir Lctr 4: EXIT Chrts ACoRN Smmr School 27 Itrti Dcoding How dos th mtl informtion ol in n itrti dcoding lgorithm? W h lrnd tht it is possibl to optimiz LDPC cods so s to mimiz thir thrshold W will s tht w cn dsign cpcit-chiing, itrtil dcodbl fmilis of LDPC cods!! (i.., thrshold cpcit) Wht is th impliction in trms of mtl informtion?
Mtl Informtion Trjctor I(Xi;Y[it]) 2 3 4 5 6 7 8 9 Itrtions Mtl Informtion Trjctor Th L-ls clcltd in th tr r optiml in th sns of MAP-clcltor, i.., L(X i Y [it] ) is sfficint sttistic for Y [it] : I(X i ; L(X i Y [it] )) = I(X i ; Y [it] ) W cn lso drw th trjctor t hlf-itrtions (ftr ribl nods & ftr chck nods) Bt: th otpt mssgs of ribl nods nd chck nods r trinsic L-ls, whrs th mtl informtion trjctor w considr now is for -postriori L-ls
Mssg Pssing L ch L V A Vribl Nod L V E L C APP L C A Chck Nod L C E Trcking of Mssgs I () E This ssms tht th dcodr dpnds onl on mtl informtion! I (2) A I A () I E (2) Problm: How to compt th trnsfr fnctions f nd f 2?
Trcking of Mssgs Trcking of mssgs wold mn trcking of pdfs ( Dnsit Eoltion) Instd of trcking th pdfs w rdc th problm to trcking of mtl informtion btwn th mssgs nd th codword which r sclr qntitis I A,I E... rg smbolwis mtl informtion Etrinsic Chnnl Modl Enc Enc 2 comm. ch tr. ch pp A-priori mssgs r modld s indpndnt nois obsrtions of th ncodd sorc. Assmptions: - indpndnt obsrtions - modl for trinsic chnnl with qlit if th dcodr is optiml
Trnsfr Fnctions Enc Enc 2 comm. ch tr. ch pp Assming modl for th trinsic chnnl w cn r I A b ring th chnnl prmtr. At th otpt of th dcodr w cn msr/clclt I E I E = f(i A ) This is onl ct if th modl of th trinsic chnnl is corrct! EXIT Chrt of LDPC Cod.9.8.7 I A,chk.5.6 I E,r.4.3.2...2.3.4.5.6.7.8.9 I A,r I E,chk
Intrscting Crs.9.8.7 I A,chk.5.6 I E,r.4.3.2...2.3.4.5.6.7.8.9 I A,r I E,chk Etrinsic Informtion Trnsfr Chrts (Stphn tn Brink) (Stphn is th g on th right, not th clown on th lft) Photo b Joss Sir Stphn did his PhD t th U of Stttgrt, thn workd for Bll Lbs in th U.K., thn in Nw Jrs. H is crrntl with RlTk. H is rglr isitor of ftw. nd TU Win.
Trcking of Mssgs I () E This ssms tht th dcodr dpnds onl on mtl informtion! I (2) A I A () I E (2) Problm: How to compt th trnsfr fnctions f nd f 2? Trcking of Mssgs Trcking of mssgs wold mn trcking of pdfs. Instd of trcking th pdfs w rdc th problm to trcking of mtl informtion btwn th mssgs nd th codword which r sclr qntitis. I A,I E... rg smbolwis mtl informtion
Etrinsic Chnnl Modl Ch + + Dc + + + () pp () - + + - () π π + + (2) pp (2) + + - + Dc 2 + + - (2) π - Enc Enc 2 comm. ch tr. ch pp Etrinsic Chnnl Modl Enc Enc 2 comm. ch tr. ch pp A-priori mssgs r modld s indpndnt nois obsrtions of th ncodd sorc. Assmptions: - indpndnt obsrtions - modl for trinsic chnnl with qlit if th dcodr is optiml
Trnsfr Fnctions Enc Enc 2 comm. ch tr. ch pp Assming modl for th trinsic chnnl w cn r I A b ring th chnnl prmtr. At th otpt of th dcodr w cn msr/clclt I E I E = f(i A ) This is onl ct if th modl of th trinsic chnnl is corrct! Vribl Nods nd BEC rp d BEC q BEC p pp Etrinsic chnnl is modld s BEC (ct)..9.8.7.6.5 I E d.4 = 3, 4, 5.3.2...2.3.4.5.6.7.8.9 I A
Chck Nods nd BEC SPC d c BEC p pp SPC... singl prit chck.9.8.7 I A.6.5.4 d c = 6, 8,.3.2...2.3.4.5.6.7.8.9 I E EXIT Chrt of LDPC Cod.9.8.7 I A,chk I E,r.6.5.4.3.2...2.3.4.5.6.7.8.9 I A,r I E,chk
Othr Chnnls Modling th trinsic chnnl s BEC is ct if th commniction chnnl is BEC. For othr commniction chnnls, th ssmption of th trinsic chnnl is in gnrl n pproimtion. If th commniction chnnl is n AWGN chnnl, w modl th trinsic chnnl lso s n AWGN, bt this is onl n pproimtion! AWGN Chnnl ribl nods AWGN σ c rp d AWGN σ pp chck nods SPC d c AWGN σ pp
Conoltionl Cods Stphn tn Brink, Conrgnc Bhior of Itrtil Dcodd Prlll Conctntd Cods, IEEE Trns. Comm. Octobr 2 Sril / Prlll Conctntion Ch + + Dc + + + () pp () - + + - () π switchs opn sril conctntion π switchs closd prlll conctntion + + (2) pp (2) + + - + Dc 2 + + - (2) π - Sril conctntion: = pp - Prlll conctntion: = pp -
Informtion Combining BEC Wht is th ffct on mtl informtion whn w dd L-ls? SRC δ BEC LLR L δ 2 BEC2 LLR L 2 + I = I(X;L ) = - δ I 2 = I(X;L 2 ) = - δ 2 I(X;L L 2 ) = - δ δ 2 = (-I )(-I 2 ) Intrscting Crs.9.8.7 I A,chk I E,r.6.5.4.3.2...2.3.4.5.6.7.8.9 I A,r I E,chk
.. BER from EXIT Chrt (BEC) pp = + I(X;APP) = P b = (-I A )(-I E ).9.5...5...8.2..5.7.2.6.3.4.5.2..5.4.6.4.3.3.5.2..8.6.3..5.2.9.5.7..2.3.4.5.6.7.8.9.4 Informtion Combining AWGN σ SRC AWGN LLR L
Informtion Combining AWGN SRC σ AWGN LLR L σ 2 AWGN2 2 LLR L 2 + BER from EXIT Chrt (AWGN).9 E b /N =.db P b =.3 E b /N =.db P b =.7.8.7.6.5.4.3.5.6.7.8..2.3.4.5.9..6.7.8.2.9.3.4.5.5.6..2.7.8..9.5.6.8.7.2.6.7.8..9.5..2.9.3.2..6..2.3.4.5.6.7.8.9
Indpndnt Obsrtions Mssgs rcid from th trinsic chnnl r indpndnt obsrtions, which is onl flfilld if N Sttistics W s sttisticl qntitis, which r onl corrct if N P b thrshold E b /N
Smmr of Assmptions - Mssgs rcid from th trinsic chnnl r indpndnt obsrtions, which is onl flfilld if N - W s sttisticl qntitis, which r onl corrct if N - W modl trinsic mssgs with n trinsic chnnl. This cn onl b don ct for th BEC. Th Gssin ssmption is n pproimtion. Ar Proprt Enc comm. ch Enc 2 BEC p pp I E I A
Drition of Ar Proprt Drition of Ar Proprt 2
Drition of H(V Y) Vribl Nods rp d comm. ch BEC p pp
Chck Nods SPC d c BEC p pp Ar of LDPC Componnt Cods I E I A I A I E Ncssr condition for sccssfl dcoding:
Consqncs of Ar Proprt Srprising rslt: Th r proprt tlls s tht th dcodr cn onl conrg if th rt is smllr thn cpcit! Mor Consqncs... Sppos th condition for conrgnc is flfilld γ< Wht is th rslt of this inqlit?
Ar nd Rt Loss If γ w cn trnsmit t rts tht pproch cpcit. If γ < w r bondd from cpcit. γ mns tht - A = A c Frthrmor, th crs mst not intrsct. Th crs h to b mtchd. Cod dsign rdcs to cr fitting! Cr Fitting Cod Mitr W onl considrd rglr cods, whr r smbol hs th sm proprtis. Thrfor, rging or ll smbols is qilnt to th mtl informtion of n rbitrril smbol. If w prtition m into n grops j=...n ch with lngth l j, w cn writ I E s Th rslting EXIT fnction is th wightd rg of th EXIT fnctions of th grops.
Empl Vribl Mitr rp d BEC q BEC p pp 7% of th ribl nods h d =2 3% of th ribl nods h d =5 This is polnomil in p Not tht γ j = Empl Vribl Mitr.9 I E2.8.7 I E.6 I E.5.4.3.2...2.3.4.5.6.7.8.9 I A
Cr Fitting Lts fi th EXIT fnction of th chck nod dcodr. For cr fitting, w cn chng th following qntitis Thrfor, w cn writ th EXIT fnction of th ribl nod dcodr s th inrs EXIT fnction of th chck nod dcodr. Tlor Sris Epnsion Assming for mpl d c =5 w cn pnd I E s Tlor sris Trncting th Tlor sris nd normlizing th cofficints to rslts in Compr this with th trnsfr fnction of th mitr of ribl nods...
Cr Fitting.9.8.7.6.5.4.3.2...2.3.4.5.6.7.8.9 En mor Consqncs... Using th sm modl s for th ribl nd chck nod dcodr, it cn b shown tht th rs for sril conctntd cod with n otr cod R ot =k ot /n ot nd n innr cod R in =k in /n in r gin b Th sm ncssr condition -A ot < A in lds to If th innr cod hs rt <, i.. I(X;Y)/n in <C thn w cn not chi cpcit with sril conctntd cods!