Second-order description of complex random vectors - Statistical Signal Processing of Complex-Valued Data - page 78

proper. So how do we classify a problem as proper or improper? In this section, we
present a hypothesis test for impropriety that is based on a generalized likelihood-ratio
test (GLRT), which is a special case of a more general class of tests presented in
Section 4.5 . A general introduction to likelihood-ratio tests is provided in Section 7.1 .
In a GLR, the unknown parameters ( R xx and R xx in our case) are replaced by
maximum-likelihood estimates. The GLR is always invariant with respect to trans-
formations for which the hypothesis-testing problem itself is invariant. Since propriety
is preserved by strictly linear, but not widely linear, transformations, the hypothesis test
must be invariant with respect to strictly linear, but not widely linear, transformations. A
maximal invariant statistic under linear transformation is given by the circularity coef-
ficients. Since the GLR must be a function of a maximal invariant statistic the GLR is a
function of the circularity coefficients.
Let x be a complex Gaussian random vector with probability density function

= π − n (det R xx ) − 1 / 2 exp

xx ( x − x )

1

2 ( x − x ) H R − 1

p ( x ) − (3.87)

with augmented mean vector x = E x and augmented covariance matrix R xx = E [( x −

− x ) H ]. Consider M independent and identically distributed (i.i.d.) random sam-
ples X

x )( x

=

[ x 1 , x 2 ,...,

=

[ x 1 , x 2 ,...,

x M ]
denote the augmented sample matrix. As shown in Section 2.4 , the joint probability
density function of these samples is

x M ] drawn from this distribution, and let X

= π − Mn (det R xx ) − M / 2 exp

xx S xx )

M
2

tr( R − 1

p ( X ) − , (3.88)

where S xx is the augmented sample covariance matrix

S xx S xx
S xx S xx

M

1
M

1
M XX H

m x ) H m x m x

S xx = = ( x m − m x )( x m − = − (3.89)

m = 1

and m x is the augmented sample mean vector

M

1
M

m x = x m .

(3.90)

m = 1

We will now develop the GLR test of the hypotheses
H 0 : x is proper ( R xx = 0 )

,

H 1 : x is improper ( R xx =

0 ) .

The GLRT statistic is

max
R xx

p ( X )

R xx = 0

λ =

p ( X ) .

(3.91)

max
R xx

This is the ratio of likelihood with R xx constrained to have zero off-diagonal blocks,
R xx = 0 , to likelihood with R xx unconstrained. We are thus testing whether or not R xx
is block-diagonal.