Detection in the presence of spherically symmetric random vectors

Abstract

A random <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> -vector <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">X=(X_{1}, \cdots , X_{n})</tex> is said to be spherically symmetric (SS) if its joint characteristic function (CF) can be expressed as a function of the quadratic form <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">u \rho u \prime</tex> , where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">u = (u_{1}, \cdots , u_{n})</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\rho</tex> is an <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n \times n</tex> positive definite matrix. The investigation in this paper is concerned with the properties of such vectors and some detection problems involving them. We first prove a theorem characterizing the form of SS random vectors <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">X</tex> and use it to find the form of the probability density functions (pdf's) of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">X</tex> and of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">X + N</tex> , where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N \sim {\cal N (0, \sigma^{2}I)</tex> is an independent Gaussian vector and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">I</tex> is the identity matrix. Applying these results we look at the problem of detecting a known signal vector in the presence of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">X + N</tex> when <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\rho =I</tex> . For the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</tex> -ary detection problem we present two conditions under which the "minimum distance" receiver is optimum. Lastly, we discuss an application of our findings to the problem of coherent detection of binary phase-shift keyed (PSK) signals in the presence of multiple co-channel interferences and white Gaussian noise.