Closed-Form Output Statistics of MIMO Block-Fading Channels

The information that can be transmitted through a wireless channel, with multiple-antenna-equipped transmitter and receiver, is crucially influenced by the channel behavior as well as by the structure of the input signal. We characterize in closed form, the probability density function (pdf) of the output of multiple-input and multiple-output block-fading channels, for an arbitrary signal-to-noise ratio value. Our results provide compact expressions for such output statistics, paving the way to a more detailed analytical information-theoretic exploration of communications in presence of block fading. The analysis is carried out assuming two different structures for the input signal: 1) the independent identically distributed (i.i.d.) Gaussian distribution and 2) a product form that has been proved to be optimal for noncoherent communication, i.e., in absence of any channel state information. When the channel is fed by an i.i.d. Gaussian input, we assume the Gramian of the channel matrix to be unitarily invariant and derive the output statistics in both the noise- and interference-limited scenario, considering different fading distributions. When the product-form input is adopted, we provide the expressions of the output pdf as the relationship between the overall number of antennas and fading coherence length varies. We also highlight the relation between our newly derived expressions and the results already available in the literature, and, for some cases, we numerically compute the mutual information based on the proposed expression of the output statistics.