1. Feedback capacity

     The channel with feedback is illustrated as follows:        

     We define a feedback code as a sequence of mappings , where each is a function only of and the previous received values, , and a sequence of decoding functions . Thus

when W is uniformly distributed over .

Definition: The capacity with feedback, , of a discrete memorylesss channel is the supermum of all rates achievable by feedback codes.

Theorem: = =

Proof:

  Since a non-feedback code is a special case of a feedback code, any rate that can be achieved without feedback can be achieved with feedback, and hence

  Instead of using , we will use the index  and prove a similar series of inequalities.

  Let W be uniformly distributed over Then:

  by Feno's inequality

Using the entropy bound:

Putting theses together, we obtain:

and divding by n and letting n tend to infinity, we conclude:

Thus we cannot achieve any higher rates with feedback than we can without feedback, and

As we have seen in the example of the binary erasure channel, feedback can help enormously in simplifying encoding and decoding, However, it cannot increase

the capacity of the channel.