# Ergodic MIMO Mutual Information: Twenty Years After Emre Telatar

## TL;DR

This paper revisits Telatar's 1999 integral representation of MIMO mutual information, explicitly evaluates it as a finite sum involving exponential integrals, and connects it to recent quantum information theory developments.

## Contribution

It provides an explicit finite-sum evaluation of the expected mutual information for MIMO channels, enhancing the classical integral representation with closed-form expressions.

## Key findings

- Explicit finite sum representation derived
- Involves exponential integral functions
- Connects classical information theory with quantum information developments

## Abstract

In the celebrated work of Emre Telatar in the year 1999 (14274 citations to date), it was shown that the expected value of the mutual information \begin{equation*} \mathrm{I} = \ln\det\left( \mathbf{I}_m + \frac{1}{t} \mathbf{HH}^{\dagger} \right) \end{equation*} of an $m\times n$ MIMO Rayleigh channel matrix $\mathbf{H}$ with a SNR $1/t$ can be represented as an integral involving Laguerre polynomials. We show, in this work, that Telatar's integral representation can be explicitly evaluated to a finite sum of the form \begin{equation*} \mathbb{E}\!\left[\mathrm{I}\right]=\sum_{k=0}^{n+m-3}a_{k}t^{k}+\rm e^{t}~\text{Ei}(-t)\sum_{k=0}^{n+m-2}b_{k}t^{k},, \end{equation*} where $\text{Ei}(-t)$ is the exponential integral and $a_{k}$, $b_{k}$ are known constants that do not dependent on $t$. The renewed interest in this classical information theory problem came from, quite surprisingly, the recent development in quantum information theory.