Modified Q-functions and their use in detection analysis

The Marcum´s Q-function is extended to complex arguments. In particular, a set of modified generalized Q-functions for imaginary values of one or both arguments are defined and their properties are investigated. An extended probabilistic interpretation for the modified Q-functions is given appealing to a proposed generalization of the concept of random variable. Complex pseudo-random variables, pseudo-events, and pseudo-probability measures are defined, which formally obey the usual rules of composition of probability measures of independent events. The modified Q-functions are then used to derive the Rayleigh multivariate multidimensional distribution function and the error probability for partially coherent detection of equicorrelated signals