Some Aspects of Finite State Channel related to Hidden Markov Process

We have no satisfactory capacity formula for most channels with finite states. Here, we consider some interesting examples of finite state channels, such as Gilbert-Elliot channel, trapdoor channel, etc., to reveal special characters of problems and difficulties to determine the capacities. Meanwhile, we give a simple expression of the capacity formula for Gilbert-Elliot channel by using a hidden Markov source for the optimal input process. This idea should be extended to other finite state channels. 1 Gilbert-Elliot channel Let us start to consider the commom Gilbert-Elliot channel as a typical example of finite state channel. The channel shown in Figure 1 is a standard model of burst error channel. Thus, it is a finite state channel with two states, good(G) and bad(B) states. In any state, the channel is the binary symmetric channel, but has different cross-over probabilities, δ and ε. It is usually assumed that the cross-over probability δ of good(G) state is significantly smaller than ε (≤ 1/2) of bad(B) state. The channel changes its state in Markovian way. The state transition probability matrix is expressed as T = [ 1− a a b 1− b ] , (1) and its stationary distribution is ( b a+ b . a a+ b ) . (2) The probability of state sequence s = s1s2 . . . sn, si ∈ {G,B} of length n is given by P (s) = ps1a GGaGBb NBB bBG , (3) where ps1 is the stationary probability of s1, and Nst is the number of transitions from state s to state t, a = 1− a, b = 1− b. Dagstuhl Seminar Proceedings 09281 Search Methodologies http://drops.dagstuhl.de/opus/volltexte/2009/2243