P$_{X}$$_{0}$$_{|}$$_{Y}$ 0 $_{−∞}$, first define the log-likelihood ratio: where m is the number of the states of Markov Chain, X ( i ) ∈ X is the i th state and X is the state space of Markov Chain. It is obviously that L (0) n ≡ 0 . Then given L$_{n}$ = { L (0) n ,L (1) n , · · · ,L ( m − 1) n } , P$_{X}$$_{n}$$_{|}$$_{Y}$ n ( X$_{n}$ | Y $^{n}$) can be calculated as and when n → ∞ , (133) converges to P$_{X}$$_{0}$$_{|}$$_{Y}$ 0 $_{−∞}$( X ( i $^{)}$| Y 0 $_{−∞}$) . In addition, L ( i ) n +1 can be calculated from L$_{n}$ iteratively as where Detail deduction of (134) is shown in (137) For the hidden Markov processes observed through the AWGN channel (1), the entropy of HMPs can be computed as [24] 27 h ( Y ) = lim n →∞ −$^{∫ ∫}$ r ( y, l$_{n}$ ) log r ( y, l$_{n}$ ) dy dP$_{L}$$_{n}$ ( l$_{n}$ ) (141) L ( i ) n +1 = log P$_{X}$$_{n}$$_{+1}$$_{|}$$_{Y}$ n +1 1 ( X ( i $^{)}$| Y n +1 1 ) P$_{X}$$_{n}$$_{+1}$$_{|}$$_{Y}$ n +1 1 ( X $^{(0)}$| Y n +1 1 ) (137) = log ∑ m − 1 k =0 P$_{Y}$$_{n}$$_{+1}$$_{|}$$_{X}$$_{n}$$_{+1}$ ( Y$_{n}$$_{+1}$ | X ( i $^{)}$) P$_{X}$$_{2}$$_{|}$$_{X}$$_{1}$ ( X ( i $^{)}$| X ( k $^{)}$) P$_{X}$$_{n}$$_{|}$$_{Y}$ n 1 ( X ( k $^{)}$| Y n 1 ) ∑ m − 1 k =0 P$_{Y}$$_{n}$$_{+1}$$_{|}$$_{X}$$_{n}$$_{+1}$ ( Y$_{n}$$_{+1}$ | X $^{(0)}$) P$_{X}$$_{2}$$_{|}$$_{X}$$_{1}$ ( X $^{(0)}$| X ( k $^{)}$) P$_{X}$$_{n}$$_{|}$$_{Y}$ n 1 ( X ( k $^{)}$| Y n 1 ) (138) = log P$_{Y}$$_{n}$$_{+1}$$_{|}$$_{X}$$_{n}$$_{+1}$ ( Y$_{n}$$_{+1}$ | X ( i $^{)}$) P$_{Y}$$_{n}$$_{+1}$$_{|}$$_{X}$$_{n}$$_{+1}$ ( Y$_{n}$$_{+1}$ | X $^{(0)}$) + log ∑ m − 1 k =0 P$_{X}$$_{2}$$_{|}$$_{X}$$_{1}$ ( X ( i $^{)}$| X ( k $^{)}$) e L ( k ) n ∑ m − 1 k =0 P$_{X}$$_{2}$$_{|}$$_{X}$$_{1}$ ( X $^{(0)}$| X ( k $^{)}$) e L ( k ) n (139) = R ( i $^{)}$( Y$_{n}$$_{+1}$ ) + F ( i $^{)}$( L$_{n}$ ) . (140) R ( i $^{)}$( Y$_{n}$$_{+1}$ ) = ( X ( i ) − X $^{(0)}$) Y$_{n}$$_{+1}$ − 1 2 (( X ( i $^{)}$)2 − ( X $^{(0)}$)$^{2}$) (135) F ( i $^{)}$( L$_{n}$ ) = log ∑ m − 1 k =0 P$_{X}$$_{2}$$_{|}$$_{X}$$_{1}$ ( X ( i $^{)}$| X ( k $^{)}$) e L ( k ) n ∑ m − 1 k =0 P$_{X}$$_{2}$$_{|}$$_{X}$$_{1}$ ( X $^{(0)}$| X ( k $^{)}$) e L ( k ) n . (136) L ( i ) n +1 = R ( i $^{)}$( Y$_{n}$$_{+1}$ ) + F ( i $^{)}$( L$_{n}$ ) (134) P$_{X}$$_{n}$$_{|}$$_{Y}$ n ( X ( i $^{)}$| Y $^{n}$) = e L ( i ) n ∑ m − 1 i =0 e L ( i ) n (133) L ( i ) n = log P$_{X}$$_{n}$$_{|}$$_{Y}$ n ( X ( i $^{)}$| Y $^{n}$) P$_{X}$$_{n}$$_{|}$$_{Y}$ n ( X $^{(0)}$| Y $^{n}$) , i = 0 , 1 , · · · ,m − 1 (132)