Exact Joint Distribution Analysis of Zero-Forcing V-BLAST Gains with Greedy Ordering

We derive the joint probability distribution of zero-forcing (ZF) V-BLAST gains under a greedy selection of decoding order and no error propagation. Unlike the previous approximated analyses, a mathematical framework is built by applying order statistics rules and an exact closed-form joint probability density function expression for squared layer gains is obtained. Our analysis relies on the fact that all orderings are equiprobable under independent and identical Rayleigh fading. Based on this idea, we determine the joint distribution of the ordered gains from the joint distribution of the unordered gains. Our results are applicable for any number of transmit and receive antennas. Although we present our analysis in a ZF V-BLAST setting, our analytical results can be directly applied for the dual cases of ZF V-BLAST. Under the assumption of a low rate feedback of decoding order to the transmitter, a benefit of having exact expressions is illustrated by the calculation of the cutoff value under optimal power allocation that maximizes the sum of the substream outage capacities under a given sum power constraint. We provide numerical results and verify our analysis by means of simulations.