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.
Categories: cs.IT math.IT