B6)=∑pS.=sIS=s0 =∑p0lS=p=5ylS=s) = (4.16) with boundary condition B(s)=P(Sx=s). The term(s',s)can be further decomposed into r(s',s)=P(u=ilS-=s)p(S.=s.y:IS-=s'.ue =i) =P(u =ilS=s)-P(S==s',u =i)p(,=5,S=s'.u=i) P(u =1)-P(S,=s1S-=s',u =i)-p(y:le) (4.17) where P(ug)is the a priori probability of us=i,and P(S=sS=s',u=i)is the probability that a state transition SS with input=iexists,which is equal to eitherI or 0.p(y)=p(yS=s,S=s'u=i)is the probability of receiving y&given the specific trellis branch,which is determined by the channel transition probability.For the discerte memoryless AWGN channel,it can be calculated as p(y:lc)=p(vi.ylci.cf) p(lc)p(ylcf) o脚票[i-4-列 om是[时-d-时 y-on (-g-1. 1 2σ 2o1 2 exp2EOvici+) 4-21 4-21 1 1 ( ') ( , | ') N k k kk s  s pS s S s        y  1 1 ( | ) ( , | ') N k k k kk s p S s pS s S s         y y  ( ) ( ', ) k k s   s ss      (4.16) with boundary condition ( ) ( ) N N  s PS s   . The term ( ', ) i k  s s can be further decomposed into ( ', ) i k  s s 1 1 ( | ') ( , | ', )       Pu i S s pS s S s u i k k k kk k   y 11 1 ( | ') ( | ', ) ( | , ', )           Pu i S s PS s S s u i p S sS s u i k k k k k kk k k   y 1 ( ) ( | ', ) ( | )      Pu i PS s S s u i p k k k k kk  y c (4.17) where P uk ( ) is the a priori probability of uk = i, and 1 ( | ', ) PS s S s u i kk k     is the probability that a state transition k k 1 S S   with input uk = i exists, which is equal to either 1 or 0. 1 ( | ) ( | , ', ) kk k k k k p p S sS s u i     yc y is the probability of receiving yk given the specific trellis branch, which is determined by the channel transition probability. For the discerte memoryless AWGN channel, it can be calculated as ( |) (, |,) s p sp k k k k kk p y c  py y c c (|)( |) s s pp kk k k   p y c py c 2 2 2 2 1 exp (2 1) 2 2 1 exp (2 1) 2 2 s s s k k s p p k k E y c E y c                               2 2 22 2 1 ( ) 2 (2 1) 1 ( ) 2 (2 1) 1 exp exp 22 2 s ss p pp k kk k k k s s y yc y y c E E                   2 2 22 2 2 1 () ( ) 2 ( ) 2( ) exp exp exp 2 2 s p s p ss pp k k s k k s kk k k s y y E y y E yc y c E                      2 2( ) exp ss pp s kk k k k E yc y c A         