A representation theorem and its applications to spherically-invariant random processes 论文

摘要

The <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> th-order characteristic functions (cf) of spherically-invariant random processes (sirp) with zero means are defined as cf, which are functions of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> th-order quadratic forms of arbitrary positive definite matrices <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</tex> . Every <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> th-order spherically-invariant characteristic function (sicf) is represented as a weighted Lebesgue-Stieltjes integral transform of an arbitrary univariate probability distribution function <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F(\cdot)</tex> on <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[0,\infty)</tex> . Furthermore, every <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> th-order sicf has a corresponding spherically-invariant probability density (sipd). Then we show that every <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> th-order sicf (or sipd) is a random mixture of a <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> th-order Gaussian cf [or probability density]. The randomization is performed on <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\nu^2 \rho</tex> , where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\nu</tex> is a random variable (tv) specified by the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F(\cdot)</tex> 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.