content_image/321707.jpg content_image/321706.jpg content_image/321705.jpg content_image/321703.jpg content_image/321704.jpg content_image/321702.jpg content_image/321701.jpg
P$_{X}$$_{0}$$_{|}$$_{Y}$ 0 $_{−∞}$, first define the log-likelihood ratio: 

<img src='content_image/321707.jpg'>

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 

<img src='content_image/321706.jpg'>

<img src='content_image/321705.jpg'>

<img src='content_image/321703.jpg'>

Detail deduction of (134) is shown in (137) 

<img src='content_image/321704.jpg'>

<img src='content_image/321702.jpg'>

For the hidden Markov processes observed through the AWGN channel (1), the entropy of HMPs can be computed as [24] 

<img src='content_image/321701.jpg'>

