Detection - Statistical Signal Processing of Complex-Valued Data - page 192

v

U

P U v

Figure 7.4 The cylinder is the set of transformations of the whitened measurement that leave the
matched subspace detector invariant, and the energy in the subspace U constant.

to the transformation

e j β

H

P ⊥ e j β φ + w , g ( v ) = ( + ) v = (7.55)

where the rotated noise w remains white and proper and the rotated signal is just a true
phase rotation of the original signal. This is the natural transformation to be invariant to
when nothing is known a priori about the amplitude and phase of the parameter

φ .

Let us now extend our study of this problem to the case in which the measurement
noise n is improper with Hermitian covariance R and complementary covariance R .
To this end, we construct the augmented measurement y and whiten it with the square
root of the augmented covariance matrix R − 1 / 2 . The resulting augmented measurement
v

R − 1 / 2 y has identity augmented covariance, so v is white and proper, and a widely
linear function of the measurement y . However, by whitening the measurement we have
turned the linear model y

=

= H + n into the widely linear model

v = G + w , (7.56)

where

R − 1 / 2 H0
0H ∗

=

G (7.57)

and w is proper and white noise. This model is then replaced with

v = U + w , (7.58)

where the augmented matrix U is a widely unitary basis for the widely linear (i.e.,
real) subspace , which satisfies U H U .

It is a straightforward adaptation of prior arguments to argue that the real test
statistic

G = I , and is a reparameterization of

1

2 v H P U v

1

2 v H UU H v

= =

L (7.59)

is invariant with respect to rotations in the widely linear (i.e., real) subspace U and to

bias in the widely linear (i.e., real) subspace orthogonal to U . The statistic L may also