## Gallagher's diagram

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.

For our above example, below-left shows the Gallagher's diagram is at time index *n*−1 the state is *S*_{0} while the right hand side shows the diagram if at *n*−1 the state is *S*_{1}.

Note that the output *y*^{(n)} and state *S*(*n*) [or *S*^{(n)} for short] **updates simultaneously.** That is

From Gallagher's diagram we have

In other words,

Since

*y*

^{(n)}and

*S*

^{(n)}are

**state independent.**❷

## Entropy rate

Let index *i* = 0 or 1 but fixed. Then for a fixed *S*^{(n−1)} we can get the conditional entropy *H*(*Y* | *S*^{(n−1)} = *S _{i}*). That is

**matrix are**

*F**f*=

_{ij}*p*(

*y*|

_{j}*S*). Thus

_{i}*S*, the elements

_{i}*f*goes down the respective column in

_{ij}**F**. That is

Hence, for fixed state *S _{i}* =

*S*

_{0}

*S*=

_{i}*S*

_{1}

Following determination of the conditional probability *H*(*Y* | *S*^{(n−1)} = *S _{i}*), the steady-state entropy rate of the output sequence is therefore