Determining Channel Capacity
For a particular Trellis diagram we define a connection matrix D. Hence for k states the size of the matrix is k × k. D is called the connection matrix because its entries dij = number of Trellis paths from Si to Sj. For example
Notice that elements in a particular ith row represent the number of paths from Si.
The connection matrix D is therefore a transition matrix from the present state to next state. Therefore after a certain time t and hence t steps, for a particular state Si can we compute the possible number of paths (and hence possible number of sequences) that can reach the same Si state? The possible number of sequences at t steps can be computed by applying D. This is illustrated below. ❶
Then, at t = 1
Similarly at t = 2
We know that and hence which implies there are four possible ways (same as four possible Trellis paths) from state S0 at t = 0 to S0 at t = 3. Graphically,
gives the possible number of distinct messages (sequences) that can be sent by the channel in exactly t steps.Since the number of bits required to represent symbols in sequences (in t steps) is
Alternative to determining for computing channel capacity
In practice it is hard to compute and , in . Therefore it is hard to compute the channel capacity by the approach of determining . An easier approach is to diagonalize D.
If λm is the largest positive eigenvalue of D, then for very large t
Summary (p:3) ➽