Generalized Framework for the Level Crossing Analysis of Ordered Random Processes

This paper investigates the following general problem relating to ordered random processes: given
n
independent but not necessarily identical random processes, how frequently, on average, does any given process become one of the
p
th (
p
=1,2,...,
n
-1) largest processes? This is a fundamental problem arising in the design and analysis of contemporary multidimensional wireless communication systems (e.g., multiantenna, multiuser) employing opportunistic selection. We formulate this problem as one involving the level crossing rate (LCR) of a carefully defined ordered random process across the zero threshold, which we solve by developing a new mathematical framework based on the theory of permanents. For the case where the processes correspond to time-varying Rayleigh fading channels, we present exact closed-form formulas for the LCR, simplified tight upper bounds, as well as asymptotic results for
n
and
p
approaching infinity but with fixed ratio. These results reveal interesting fundamental limits for the LCR, and are shown to given meaningful insight even for small values of
n
and
p
. We further use our mathematical framework to characterize the required per-branch and overall switching rate of a generalized selection combining diversity receiver, allowing for different average powers for each branch. With the aid of majorization theory, we demonstrate that the overall switching rate is maximized when the power delay profile is uniform.