I Let E(v! v 0 ) denote the event that v 0 is selected instead of v I The block error probability is the union of such events

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1 ED042 Error Conrol Coding Kodningseknik) Chaper 3: Opimal Decoding Mehods, Par ML Decoding Error Proailiy Sepemer 23, 203 ED042 Error Conrol Coding: Chaper 3 20 / 35 Pairwise Error Proailiy Assme ha v 2 C is ransmied and consider ML decoding of r A decoding error occrs if for some v 0 2 C a disance d = d H v,v 0 ) we have pr v 0 ) > pr v) he proailiy ha a code word a disance d is more likely han v is called he pairwise error proailiy p d BSC: r = v + e, e = w H e) errors a posiions where v 6= v 0 p d < d e=d d 2 e d e e e e) d e, d odd For even d he erm for e = d/2 adds wih proailiy /2 Boh cases can e pper onded y he Bhaacharyya parameer B: p d < 2 p d def e e) = B d Union Bond Le Ev! v 0 ) denoe he even ha v 0 is seleced insead of v he lock error proailiy is he nion of sch evens [ P B = P Ev! v 0 ) apple P Ev! v 0 ) v 0 2C v 0 6=v v 0 2C v 0 6=v Using he pairwise error proailiy p d we oain P B apple A d p d where A d denoes he nmer of codewords of weigh d Using he weigh enmeraor fncion AX)=A 0 + A X + A 2 X A X which always incldes he zero codeword, i.e., A 0 = ) we can wrie P B apple AX) X=pd ED042 Error Conrol Coding: Chaper 3 2 / 35 ED042 Error Conrol Coding: Chaper 3 22 / 35

2 Decoding Error Proailiy For he BSC he decoding error proailiy is pper onded y P B < A d 2 p d e e) = AX) p X=B=2 e e) For he AG channel he pairwise error proailiy ecomes r p d = Q 2dR E < exp 0 R E d 0 def = B d he ML decoding error proailiy for AG is hs pper onded y r P B < A d Q 2dR E 0 < AX) X=B=exp R E 0 eigh Enmeraor Examples Repeiion codes: C = {0,0,...,0),,,...,)} ) AX)= + X Reed-Mller codes R, m): = 2 m, K = m +, AX)= +2 m+ 2)X 2m + X heorem Macilliams deniy) he weigh enmeraor AX) of a code can e comped from he weigh enmeraor BX) of is dal code as follows: X AX)=2 K) + X) B + X Example: he weigh enmeraor of a single pariy-check SPC) code can e oained from ha of he repeiion code ED042 Error Conrol Coding: Chaper 3 23 / 35 ED042 Error Conrol Coding: Chaper 3 24 / 35 Error Bonds for Convolional Codes For convolional codes he pah weigh enmeraor )=n + n n w w + cons he nmer of pahs wih weigh w in he sae diagram ha sar and end in he zero sae A decoding error even corresponds o a deor from he correc pah in he rellis, called a rs nsead of he lock error proailiy we are ineresed in he rs error proailiy P rs apple n d p d < ) d=d X=B free wih B = 2 p e e) BSC) or B = exp R E AG) 0 Signal Flow Char he signal flow char can e oained from he sae diagram: 00 /0 0/00 he pah weigh enmeraor ) is hen comped from a se of linear eqaions: z = z z 2 = z + z 2 z 3 = z + z 2 ) )= 2 z 3 = 5 2 = k k+5 ED042 Error Conrol Coding: Chaper 3 25 / 35 ED042 Error Conrol Coding: Chaper 3 26 / 35

3 Recrsive Compaion of ) Example is possile o oain he weigh enmeraor y sccessive redcion of he signal flow char nsead of solving a sysem of eqaions he following elemenary operaions can e sed: 2 = + 2 ) = = + 2 ) ED042 Error Conrol Coding: Chaper Mason s gain formla provides a general sysemaic mehod for finding he ransfer fncion of a linear signal flow char Signal flow chars can e sed in many oher applicaions, e.g., elecrical neworks, aomaic conrol, = + ) = = 27 / 35 ED042 Error Conrol Coding: Chaper 3 28 / 35 MAP and APP Decoding MAP and APP Decoding A maximm a poseriori MAP) decoder delivers a is op he vecor pr ) P) = arg max P r) = arg max = arg max pr ) P) pr) can e implemened wih he Vieri algorihm y incorporaing he a priori proailiies Pi ) ino he ranch meric vales A symol-y-symol MAP decoder ops he esimaed symol i according o i = max Pi r) i 2{0,} An a poseriori proailiy APP) decoder provides sof ops for he op symols i insead of he hard decisions i Li ) = log ED042 Error Conrol Coding: Chaper 3 29 / 35 Pi = 0 r) :i =0 pr ) P) = log Pi = r) :i = pr ) P) signli )) 2 {+, } corresponds o a hard decision, Li ) delivers he reliailiy of he decision ED042 Error Conrol Coding: Chaper 3 30 / 35

4 mplemenaion of APP Decoding A recrsive calclaion in he rellis was proposed y Bahl, Cocke, Jelinek and Raviv BJCR) Consider all pahs ha pass hrogh he ranch s,s ): r < r r > = he proailiy of sch pahs can e facorized as follows = ps 0,s,r) =ps 0,r < ) ps,r s 0 ) pr > s) def = a s 0 ) g s 0,s) s) he Forward-Backward BCJR) Algorihm. Forward recrsion: =,...,L + m a s)= g s 0,s) a s 0 ), a 0 s)= s 0 if s = 0 0 s 6= 0 wih g s 0,s)=pr s 0,s) Ps s 0 )=pr v ) P ) 2. Backward recrsion: = L + m,...,2 s 0 )= s g s 0,s) s), 3. Sof op: =,...,L + m L i) )=log a s 0,s): i) =0 s 0,s): i) = a s)= if s = 0 0 s 6= 0 s 0 ) g s 0,s) s) s 0 ) g s 0,s) s) ED042 Error Conrol Coding: Chaper 3 3 / 35 ED042 Error Conrol Coding: Chaper 3 32 / 35 Log-Domain Formlaion he proailiies a he inp can also e represened as L-vales log-likelihood raios): L a i) = log P i) = 0 P i) =, L ch v j) = log p rj) v j) = 0 p r j) v j) = n case of an AG channel we have L ch v j) = 2 r j) s 2 Definiion he max operaor is defined as max x,y)=loge x + e y )=maxx,y)+log + e x y e can now simplify he compaions in he log-domain: log e loga e logg e log = max log a + log g + log) ED042 Error Conrol Coding: Chaper 3 33 / 35 he Log-Domain BCJR Algorihm Log-APP). Forward recrsion: =,...,L + m a s)=max s 0 g s 0,s)+a s 0 ), a 0 s)= wih g s 0,s)= K i= L a i) /2 i) 2. Backward recrsion: = L + m,...,2 s 0 )=max s 3. Sof op: =,...,L + m L i) )= max + j= g s 0,s)+ s), L+ms)= s 0,s): i) =0 max s 0,s): i) = a a 0 if s = 0 s 6= 0 L ch v j) /2 v j) s 0 )+g s 0,s)+ s) s 0 )+g s 0,s)+ s) ED042 Error Conrol Coding: Chaper 3 34 / 35 0 if s = 0 s 6= 0

5 Max-Log Approximaion he max operaor can e approximaed y max x,y) maxx,y) which neglecs an addiive erm in he range 0 < log + e x y apple log2 = Replacing he max y he max operaor resls in he Max-Log-APP or Max-Log-MAP algorihm he Max-Log-APP is sopimal simpler o implemen: forward and ackward recrsion are similar o Vieri algorihm can e shown ha he Max-Log-APP algorihm is eqivalen o he sof-op Vieri algorihm SOVA) y Hagenaer he opimal Log-APP algorihm can e simlified y approximaing he erm log + e z y piecewise linear fncions or y look-p ales ED042 Error Conrol Coding: Chaper 3 35 / 35