Event Type:

Department Colloquium

Date/Time:

Monday, November 30, 2009 - 08:00

Location:

Kidd 364

Guest Speaker:

Mark Kelbert

Institution:

Swansea University

Abstract:

This talk addresses the issue of the proof of the

entropy power inequality, an important tool in the analysis of

Gaussian channels of information transmission, proposed by

Shannon. This inequality has many relations with different

problems in geometry, linear algebra, analysis, etc.

We analyze continuity properties of the mutual entropy

of the input and output signals in an additive memoryless channel

and show how this can be used for a correct proof of the entropy-power

inequality.

To introduce the entropy power inequality,

consider two independent random variables $X_1$ and $X_2$

taking values in $\R^d$, with probability density functions

$f_{X_1}(x)$ and $f_{X_2}(x)$, respectively, where

$x\in\R^d$. Let $h(X_i)$, $i=1,2$ stand for the differential

entropies

$$h(X_i)=-\int_{\R^d}f_{X_i}(x)\ln\;f_{X_i}(x){\rm d}x:=

-\E\ln\;f_{X_i}(X_i),$$

and assume that $-\infty

e^{\frac{2}{d}h(X_2)}.)$$