مرکز منطقه ای اطلاع رساني علوم و فناوري - A representation theorem and its applications to spherically-invariant random processes

Abstract :

The $n$ th-order characteristic functions (cf) of spherically-invariant random processes (sirp) with zero means are defined as cf, which are functions of $n$ th-order quadratic forms of arbitrary positive definite matrices $p$ . Every $n$ th-order spherically-invariant characteristic function (sicf) is represented as a weighted Lebesgue-Stieltjes integral transform of an arbitrary univariate probability distribution function $F(\\cdot)$ on $[0,\\infty )$ . Furthermore, every $n$ th-order sicf has a corresponding spherically-invariant probability density (sipd). Then we show that every $n$ th-order sicf (or sipd) is a random mixture of a $n$ th-order Gaussian cf [or probability density]. The randomization is performed on $\\nu^2 \\rho$ , where $\\nu$ is a random variable (tv) specified by the $F(\\cdot)$ function. Examples of sirp are given. Relations to previously known results are discussed. Various expectation properties of Gaussian random processes are valid for sirp. Related conditional expectation, mean-square estimation, semMndependence, martingale, and closure properties are given. Finally, the form of the unit threshold likelihood ratio receiver in the detection of a known deterministic signal in additive sirp noise is shown to be a correlation receiver or a matched filter. The associated false-alarm and detection probabilities are expressed in closed forms.