State Dependent Decoding by Set Distinguishibility
how to deal with initial state dependency and decoding?

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This problem of initial state dependency with decoding can be tackled with set–distinguishability decoding. One way of achieving set–distinguishability is the introduction of a de–energized state S sub 2 . This is a unique state representing relaxedness of the channel. That is 'start state'.

Fig. 1. State diagram of de-energized Fig 2 system
Notice that since x2 = 0 the entropy is same H(X) = 0.7219281.

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At time index n, if

mu sub 2 at n = p(S sub 2) = p(x = 0 at time index n) = 1
Since μ0 + μ1 + μ2 = 1, this means
mu sub 0 at n = p(S sub 0) = p(x = x sub 0 at time index n) = 0 and mu sub 1 at n = p(S sub 1) = p(x = x sub 1 at time index n) = 0
Thus
vector of length 3, mu at n+1 is equal to [0.2, 0.2, 0.2; 0.8, 0.8, 0.8; 0, 0, 0] times [mu sub 0 at n; mu sub 1 at n; mu sub 2 at n]
and its steady-state
vector of length 2, steady-state of mu at n+1 is equal to [0.2(mu sub 0 + mu sub 1 + mu sub 2); 0.8(mu sub 0 + mu sub 1 + mu sub 2); 0(mu sub 0 + mu sub 1 + mu sub 2)] = [0.2; 0.8; 0]
Notice that the introduction of the relaxed state results in
Y = {-2, -1, 0, +1, +2} where -1 and +1 are the possible start symbols uniquely defining the start (channel) state

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Another way of achieving set-distinguishability is to initially send either a pair of xi = −1 or a pair of xi = +1. This ensures unambiguous identification of either S0 or S1 due to y = −2 and y = +2 respectively. This initial sending signal is called 'start signal' and the symbol x sub i at n = 0 is called 'start symbol'.

There are number of ways of pre–coding the channel inputs. One common one is the non–return–to–zero–inverted (NRZI) code.

The NRZI pre–coded duo–binary channel becomes

Fig. 2. NRZI Pre-coded Duobinary channel

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If the states are defined as S(n) = ⟨β(n−1), α(n−1)⟩

Fig. 3. Table of NRZI Pre-coded Duobinary channel
The state diagram is therefore
Fig. 4. State diagram of NRZI Pre-coded Duobinary channel
Notice that the states S1 and S2 cannot be entered. Hence they can be initial states. States S0 and S3 are said to form a closed irreducible recurrent non–null Markov chain. They are also known as ergodic Markov chain.

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The trellis diagram for the NRZI pre–coded duo–binary channel is

Fig. 5. Trellis diagram of the NRZI Pre-coded Duobinary channel
Thus as n ≥ 1
y(n) = 0 implies x(n) = 1 and y(n) = plus or minus 2 implies x(n) = 0

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Summary (p:6) ➽

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