Catastrophic Sequence System and State Dependent Decoding

Let us say that for an initial state S0 of the channel, the DMS X emits sequence Then in time n the output sequence is However for an initial state S1 of the channel with the DMS X emitting sequence the output sequence at time n is Thus for two different initial states emitting different sequences the output sequences are the same.

The generation of same output sequence for two or more different sequences is called catastrophic sequence system. Thus ambiguous .

In the above example if we knew a priori about the initial state S(n−1) (i.e, either S0 or S1), we can decode into sequence . This is called state dependent decoding.

Since the system has the property of state dependent decoding, the question arises

how much of the information in y̅ is from X?

if not all of the information in is from X, where did the extra information come from?.

Therefore

how do we get this 'something' which is a measure of I(X; Y)?

Example

Let us consider the same DMS and the duo-binary channel system but with above parameters
The state diagram is therefore