Let us say that for an initial state

*S*

_{0}of the channel, the DMS

*X*emits sequence

*n*the output sequence is

*S*

_{1}of the channel with the DMS

*X*emitting sequence

*n*is

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

*x̅*.

In the above example if we knew a priori about the initial state *S*(*n*−1) (i.e, either *S*_{0} or *S*_{1}), we can *decode**y̅* ** into** sequence

*x̅*. 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 *y̅* 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*

The state diagram is therefore

From the state diagram we also see that

*H'*(

*Y*) ≡

*H*(

*X*). ❸

*What does imply?*

☛

This means that *H'*(*Y*) is tracking . This therefore implies that the channel is not really losing any information from *X*. (Thus answering, *how much of the information in* *y̅* *is from X*?)

Being a catastrophic sequence system the input sequence *x̅* is ambiguous. Our example system also has the property of state dependent decoding. That is, with a priori knowledge of the initial state *S*(*n*−1) we can decode *y̅* to *x̅*. Thus making unambiguous identification of *x̅*. The question is therefore,

*how to deal with initial state dependency and decoding?*❹