=lna)-ln∑a4s) Here naiw)=ln∑z-6.) =lh∑e.ouem） =lh∑eta-nan】 三max{4-(s)+「4(s,s} and n∑e=ln∑esm]mx ind.() Hence,the forward recursion of(4.24)can be expressed in the log domain as A三max{46)+rss-max{max{4-()+「(s,s以，or Ind(s)=max (ind(s)+In(s',s))-maxmax (in(s)+in(ss) Similarly,the backward state metric in(4.25)can be expressed as In月(s)三max{n月.(s)+lny(s;,s}-max{max{ha-(s)+lny(s,s} or equivalently, B.(三max{B,s)+「s,s-max{max{46)+r,ss月 Clearly,the forward and backward state metrics can be computed recursively with initial conditions: A(0)=0,A(s)=-0fors≠0 By(0)=0,By(s)=o for Vs0 Analogously,the final LLR calculation in (4.27)is approximated by oha-(s+h+血A) L(u)=In ehi+nca4h底间 In -Ir ehe+()in店o 4-31 4-31 ' ' ln ( ) ln ( ) k k s   s s          Here ' ln ( ) k  s 1 ' ln ( ') ( ', ) k k s   s ss           1 ln ( ') ln ( ', ) ' ln k k s ss s e              1 ( ') ( ', ) ' ln A s ss k k s e            1  ' max ( ') ( ', ) k k s A s ss    and   ' ' ' ln ( ) ln ( ) ln max ln ( ) k s k k s s s se s                 Hence, the forward recursion of (4.24) can be expressed in the log domain as Ak kk kk ( ) max ( ') ( ', ) max max ( ') ( ', ) s A s ss A s ss     s ss ' '      1 1   , or ln ( ) max ln ( ) ln ( ', ) max max ln ( ') ln ( ', )       k kk k k s s ss s ss    s ss ' '      1 1   Similarly, the backward state metric in (4.25) can be expressed as ln ( ') max ln ( ) ln ( ', ) max max ln ( ') ln ( ', )     k kk k k 1 s s ss s ss    s ss    '   1     , or equivalently, Bk kk k k   1 1 ( ') max ( ) ( ', ) max max ( ') ( ', ) s Bs ss A s ss     s ss     '  Clearly, the forward and backward state metrics can be computed recursively with initial conditions: 0 0 A As s (0) 0, ( ) for 0      (0) 0, ( ) for 0 B Bs s N N      Analogously, the final LLR calculation in (4.27) is approximated by 1 1 1 0 1 0 ln ( ') ln ( ', ) ln ( ) ( ', ) ln ( ') ln ( ', ) ln ( ) ( ', ) ( ) ln kk k k kk k k s ss s ss B k s ss s ss B e L u e                            1 0 1 1 1 0 ln ( ') ln ( ', ) ln ( ) ln ( ') ln ( ', ) ln ( ) ( ', ) ( ', ) ln ln kk k kk k k k s ss s s ss s ss B ss B e e                           