Detection in the presence of spherically symmetric random vectors

A random
n
-vector
X=(X_{1}, \cdots , X_{n})
is said to be spherically symmetric (SS) if its joint characteristic function (CF) can be expressed as a function of the quadratic form
u \rho u \prime
, where
u = (u_{1}, \cdots , u_{n})
and
\rho
is an
n \times n
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
X
and use it to find the form of the probability density functions (pdf's) of
X
and of
X + N
, where
N \sim {\cal N (0, \sigma^{2}I)
is an independent Gaussian vector and
I
is the identity matrix. Applying these results we look at the problem of detecting a known signal vector in the presence of
X + N
when
\rho =I
. For the
k
-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.