- 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.
Comments (0)
You don't have permission to comment on this page.