Parallel Concatenated Convolutional Codes (PCCC) or Turbo Codes

Transcription

1 Conatnatd Coding Paralll Conatnatd Convoltional Cod PCCC or Trbo Cod Firt Papr: NEAR SHANNON LIMIT ERROR-CORRECTING CODING AND DECODING: TURBO CODES1 by Clad Brro, Alain Glavix and Pnya Thitimajhima Eol National Sprir d Tlommniation ENST d Brtagn, Fran Intrnational Commniation Confrn ICC, Gnva, Switzrland May 1993, pp ytmati od nonytmati on hav qivalnt form ytmati od with fd forward nodr prod l powrfl od. Convoltional od in thir ytmati fdba form ar qivalnt to th nonytmati form in ditan and nart nighbor path proprti. Howvr at low ignal to noi ratio SNR, th bit rror rat BER i lightly bttr. Figr of FB-CC and FF-CC for R=1/2 v=2. x,1 x,2 x,1 = x,2-9 -

2 X d C 1 2, 1, 2 rriv ytmati onvoltional od dlay lin L 1 Y 1 Intrlavi ng Y 2 Y C 2 2, 1, 2 rriv ytmati onvoltional od W an now dod th information qn or mov to th ond dodr whih will dod for th am information qn bt in th intrlavd form. Now w ll ma of th L i gnratd by th firt nodr. Thi an b givn a a priori L i a L I2 i -aftr intrlaving information to th ond dodr. So fftivly w an intrlav L y i and L i and fd it to th ond dodr. p ap i t =-1 and p ap i t =1 an b d aoiatd with γ, in th allation of α,β for th ond dodr. Th th otpt of th ond dodr will b of th form L app i = L y2 i L I2 i L 2 i

3 Thi finih th firt or initial fll doding tp- tp 0. L 2 i rflt th xtrini information gnratd by th ond dodr. Now in a imilar mannr a arlir w an giv - fdba th firt dodr with L 2 i γto allat α,β aftr dintrlaving a nw a priori L I1 i information to th firt dodr. Doding tp 1 Firt Dodr L app i = L y i L I1 i L 1 i Sond Dodr aftr intrlaving L 1 i L app i = L y i L I2 i L 2 i Thn from dodr two fdba L 2 i a bfor to th firt dodr aftr dintrlaving and th pro rpat. Thi i nown a itrativ doding or trbo doding. Rmmbr it i alway th xtrini information that w pa to th nxt dodr a nw a priori. W don t fdba a priori or intrini informationwhy? that wa alrady gnratd by th arlir dodr o no point in fding ba th am information

4 LEAVING x 16-STATE DECODER DEC 1 INTER- LEAVING 16-STATE DECODER DEC 2 y DEINTER- DEINTER- LEAVING DEMUX/ INSERTIO dodd otpt dˆ Th p I i t =±1 ndd to allat γan alo b obtaind from L I i whih will in trn b obtaind from L i of th arlir doding tag. Rfrn: Itrativ Doding of Binary Blo and Convoltional Cod by Joahim Hagnar, El Offr, and Ltz Pap, IEEE Tan. Info. Thory, Vol. 42, No.2, Marh 1996, pp Txt Boo Chaptr: Bort : Chaptr 8 on Convoltional Cod and Appndix A on SCC, PCC Fndamntal of Convoltional Coding: Chaptr 7- Itrativ Doding

5 PCCC Rfrn: A oft-inpt Soft-Otpt maximm A Potriori MAP Modl to Dod Paralll and Srial Conatnatd Cod S. Bndtto, D. Divalar, G. Montori, and F. Pollara TDA Progr Rport , Novmbr 1996 from JPL wb it Blo diagram: Enodr ENCODER 1 RATE=1/2 TO CHANNEL ENCODER 2 RATE=1/2 NOT TRANSMITTED TO CHANNEL Dodr. rprnt log probabiliti or woring in th log domain. From Dmod From Dmod ;I ;O Not d ;I ;ONot d SISO SISO 1 2 ;I ;O :I ;O 1 Diion SISO oft-inpt, oft-otpt Modl: implmnt MAP algorithm 13

6 Th gnral Trlli Enodr: INPUT TRELLIS OUTPUT ENCODER SCCC Srial Conatnatd Convoltional Cod Enodr: OUTER ENCODER RATE=1/2 p TO CHANNEL INNER ENCODER RATE=2/3 Dodr: From Dmod ;I SISO ;O Not d ;I INNER ;O :I 1 SISO ;O OUTER ;I ;O DECISION

7 Diffrn btwn PCCC and SCCC: PCCC: Updatd probabiliti xtrini of od ymbol ar nvr d by th doding algorithm SCCC: Both, pdatd probabiliti xtrini of th inpt and od ymbol ar d in th doding algorithm Doding algorithm i Additiv SISO algorithm A-SISO woring in th log domain ;I ;I SISO ;O ;O E S, An Edg of th Trlli Stion Th following fntion ar aoiatd with ah dg Th tarting tat S Th nding tat E Th inpt ymbol Th otpt ymbol 15

8 16 Th rlationhip btwn th fntion dpnd on th partilar nodr. A an xampl, in th a of ytmati nodr, E, alo idntifi th dg in i niqly dtrmind by. Hr it i only amd that th pair S, niqly idntifi th nding tat E thi amption i alway vrifid, a it i qivalnt to ay that givn th initial trlli tat, thr i a on-to-on orrpondn btwn inpt qn and tat qn. Th Additiv SISO Algorithm A-SISO n I I S E 1,2,..., ]} ; ] ; ] xp{ log : 1 = = = α α 1,..,0, ]} ; ] ; ] xp{ log : = = = n I I S E β β Th ar forward and baward rrion. At tim, th otpt xtrini probability ditribtion ar omptd a approximatly = = E S I O : 1 ]} ] ; ] xp{ log ; β α = = E S I O : 1 ]} ] ; ] xp{ log ; β α with initial val α 0 = 0 for =S 0 othrwi α 0 = -

9 imilarly βn = 0 for =S n othrwi β n = - So w rpla log and xp with maximm val. Th w hav th following t of qation. E α = max : = {α -1 S ;I] ;I]}, =1,2..,n S β = max : = {β 1 E 1 ;I] 1 ;I]}, =n-1,,0 ;O = max := {α -1 S ;I] β E ]} ;O = max := {α -1 S ;I] β E ]} Gnrally for rial onatnatd od: OUTER CODE- NON RECURSIVE or NON-Fd ba, Non ytmati INNER CODE- RECURSIVE SYSTEMATIC or Fd-ba Sytmati It i n that th prforman of SCCC i ally bttr than that of PCCC. BER SCCC PCCC Error Floor Efft in PCCC E b /N 0 17

10 Th mmation involvd in th algorithm ar allatd ing trlli dg, rathr than ing pair of tat. Thi ma th algorithm gnral and apabl of daling with paralll dg itabl for TCM. Th A-SISO algorithm at Bit Lvl Conidr a rat ½ onvoltional nodr. U inpt and C 1, and C 2, otpt bit at tim, taing val {0,1}. Thrfor on th trlli dg at tim w hav, 1,, 2,. Drop for impliity. Dfin th rliability LLR of a bit Z taing val {0,1} at tim a P Z = 1;.] λ Z;.] log = Z = 1;.] Z = 0;.] P Z = 0;.] E α = max : = {α -1 S λ ;I] 1 λ C 1 ;I] 2 λ C 2 ;I] } h α β = max : S = {β 1 E λ 1 ;I] 1 λ 1 C 1 ;I] 2 λ 1 C 2 ;I] } h β with initial val α 0 = 0 if =S 0 and α 0 = - othrwi and β n =0 if =S n and β n = - othrwi. h α and h β ar normalization ontant. For th innr dodr, whih i onntd to th AWGN hannl, w hav λ C 1 ;I]= 2A/σ 2 r 1, λ C 2 ;I]= 2A/σ 2 r 2, whr r i, = A2Z i - 1 i,, i=1,2 i th rivd ampl at th otpt of th mathd filtr, i {-1,1} and n i, i th zro man indpndnt idntially ditribtd i.i.d. Gaian noi ampl with varian σ 2. 18

11 Th xtrini bit information for U,C 1 and C 2 an b obtaind a λ U;O = max :=1 {α -1 S 1 λ C 1 ;I] 2 λ C 2 ;I] β E ]} - max :=0 {α -1 S 1 λ C 1 ;I] 2 λ C 2 ;I] β E ]} λ C 1 ;O = max :1=1 {α -1 S λ U;I] 2 λ C 2 ;I] β E ]} - max :1=0 {α -1 S λ U;I] 2 λ C 2 ;I] β E ]} λ C 2 ;O = max :2=1 {α -1 S λ U;I] 1 λ C 1 ;I] β E ]} - max :2=0 {α -1 S λ U;I] 1 λ C 1 ;I] β E ]} Paramtr for PCCC In gnral for a Convoltional od CC w min i dfind a th minimm information wight in th rror vnt of th CC. w min = 1 for non rriv od w min = 2 for rriv od. Th rror offiint in p b rror intrlaving gain for a PCCC with larg intrlaving lngth go a N 1-w min, whr N i th intrlavr lngth. 19

12 Th for rriv CC th intrlaving gain or BER rdtion go a 1/N. On th othr hand all nonrriv CC and blo od hav w min = 1, o h od ar not fl in Paralll Conatnatd Cod. Th nxt mot important ontitnt od paramtr i z min, th minimm parity-h wight in th od qn with w=2. For a larg rang of SNR, th bhavior of PCCC i dtrmind by th fftiv fr ditan d fr,ff = 2 2 z min. It i poibl to ahiv z min = n-12 v-1 2 with a rat 1/n rriv CC with mmory v. PCCC xhibit rror floor fft dfind a th hang of lop of BER rv for Trbo od at lowr BER. Initially th oding gain inra with th nmbr of itration. Howvr, aftr a nmbr of itration in AWGN th prforman improvmnt i marginal. Extnion to onidr: 1. Trbo od with mltilvl modlation 2. Trbo TCM 3. Itrativ Eqalization and Doding Trbo Eqalization. 20