If DMS and the duo-binary channel is considered an information source
then the Markov process that exist within this information source is called Hidden Markov Process.
Its state diagram is
Difference equation for the state probabilities can therefore be written as
An alternative to the Markov process diagram is the Gallagher's diagram. The diagram represents the source and channel using state-dependent memoryless channel diagrams.
In general, the entropy rate is given by
The generation of same output sequence y̅ for two or more different sequences x̅ is called catastrophic sequence system.
This means that if we knew a priori about the initial state S(n−1) (i.e, either S0 or S1), we can decode y̅ into sequence x̅. This is called state dependent decoding.
One way of measuring–how much of the information in y̅ is from X? is by computing the entropy rate H'(Y). And if H'(Y) ≡ H(X), this means that H'(Y) is tracking .
Set–distinguishibility decoding may be implemented for initial state dependency.
One way is to introduce a de–energized state S2. This is a unique state representing relaxedness of the channel also called start state.
Another is by pre–coding. Non–Return–to–Zero–Inverted (NRZI) code is an example.
This results in
States S0 and S3 are said to form a closed irreducible recurrent non–null Markov chain also known as ergodic Markov chain.