# Lower bound

1. May 16, 2008

### brendan_foo

I apologize if this belongs in the EE section; I figured that I might get a better response if I posted it in the math section.

This question regards a lower bound on the channel capacity of an arbitrary channel (single channel, per use) by means of the 'power inequality'.

I really want to solve this myself so if someone could just nudge me in the right direction I would be very grateful. The upper bound in terms of the entropy power is trivial to show, but for some reason in employing the entropy power inequality I cannot arrive at the result that I seek.

The channel is noise additive using the relations that Y = X + Z, where Z is the noise source and X is the channel symbol.

By defining

$$Q \triangleq (2\pi e)^{-1} 2^{2h(z)}$$

and using the inequality

$$e^{2h(y)} \geq e^{2h(x)} + e^{2h(z)}$$

I wish to derive a lower bound on the channel capacity in terms of $P$ and $Q$. The upper bound on the capacity (in terms of the entropy power) is simply $C \leq \frac{1}{2}log(\frac{P+N}{Q})$, where N is the noise power.

Any shove in the right direction would be greatly appreciated!

All the best
-Brendan

Last edited: May 16, 2008