Closed-Form Output Statistics of MIMO Block-Fading Channels

Abstract

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 MIMO block-fading channels, for an arbitrary SNR 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: the i.i.d. Gaussian distribution and 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-limited and the 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 the 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.

Finite-SNR Bounds on the Sum-Rate Capacity of Rayleigh Block-Fading Multiple-Access Channels With No A Priori CSI

“…We also consider the simple capacity upper bound obtained by assuming that the receiver has perfect knowledge of the channel matrix S in (11). This bound is given by [20] C(ρ) ≤ E log det I nr + ρ n t SS † .…”

“…Contributions: We present nonasymptotic (i.e., finite-SNR) upper and lower bounds on the sum-rate capacity of Rayleigh block-fading MACs. Similarly to the nonasymptotic capacity lower bounds previously reported in [9]- [11], our bounds are not in closed form, but they can be evaluated numerically. Our upper bound is obtained by assuming that the users can perfectly cooperate, which turns the MAC into an equivalent MIMO pointto-point channel.…”

“…(3.4)], which allows one to obtain a closed-form expression for the conditional probability distribution of the channel output given the diagonal input matrix D in (1). The method employed so far to assess the tightness of the bounds obtained in [9]- [11] is to compare these lower bounds with the asymptotic expansion (3) (with the o(1) term omitted). Unfortunately, this method is not conclusive because the error incurred by omitting the o(1) term in (3) is not quantified.…”

“…Leaving the point-to-point case and moving to the MAC, we note that the independence constraint on the signals transmitted by the various users implies that the partial characterization of the capacity-achieving input distribution obtained for the pointto-point case in [6,Thm. 2], as well as the asymptotic capacity expansions in [7], [8] and the nonasymptotic capacity lower bounds in [10], [11] do not carry over to the MAC sum-rate capacity. The only exception is the i.i.d.…”

“…Specifically, an i.i.d. Gaussian input distribution is considered in [9], USTM in [10], and BSTM in [11], where the analysis is also extended to Rician block-fading and to land mobile satellite channels. A key tool in the derivation of these nonasymptotic lower bounds is the Itzykson-Zuber integral [12,Eq.…”