*state*of the channel

In addition to defining **'state'** based on symbol input *X* (set of all *x* symbols) we can also define 'state' of the channel with respect to *Y* (set of all *y* symbols). Hence,

*g*is the total possible number of states and

*h*is the total possible number of

*y*symbols.

Thus for the above example

Referring to the state diagram

we see that given the state

*S*

_{0}probability that

*y*=

*y*

_{0}= −2 or

*p*(

*y*

_{0}|

*S*

_{0}) = 0.5. Similarly

*p*(

*y*

_{1}|

*S*

_{0}) = 0.5 but

*p*(

*y*

_{2}|

*S*

_{0}) = 0. Thus

Let,

_{0}+ Π

_{1}+ Π

_{2}= 1. We can therefore write the equation for the output probabilities as

*f*is the transition probability

_{ij}*f*=

_{ij}*p*(

*y*|

_{j}*S*). Since given a particular state, sum of all probabilities from this state (probabilities in a particular column) must equal 1,

_{i}*f*

_{00}+

*f*

_{01}+

*f*

_{02}=

*f*

_{10}+

*f*

_{11}+

*f*

_{12}= 1.

Hence for the above example

**P**, the

**F**elements in each column adds to 1 but the elements in each row may not.

The output probability at steady–state (*ss*) is derived from the limit

Note that the conditional probabilities shown in the state diagram are all equivalent–

*Y*= {

*y*

_{0}= −2,

*y*

_{1}= 0,

*y*

_{2}= +2}. ❷

*Why is the system called Hidden Markov Process?*

☛

Since

*H'*(

*Y*) is the error rate, then because of the above inequality we should suspect that the error rate should be

*H*(

*Y*) grows as the output grows, that is

Recall that for information source with memory,

*Markov process was developed to compute error rates.*

However, for channels with memory the output sequence (*y*_{0}, *y*_{1}, …, *y*_{n−1}) is not a Markov process but the *State Model* is (a Markov process). Thus

*channel is a function**y*

_{n−1}does not provide knowledge about the state of channel. Therefore the channel states are called hidden and the system is called

*Hidden Markov process*. The fact that the channel is a function is what makes channels with memory difficult to analyze. One way is to use Gallagher's diagram. ❸