Detection - Statistical Signal Processing of Complex-Valued Data - page 185

P D

P F

L

d

h − d 2
d

h + d 2
√ d

Figure 7.3 The false-alarm probability P F (dark gray) and detection probability P D (light and
dark gray) are tail probabilities of a normalized Gaussian. They are determined by the threshold
η and the deflection d .

, any design value of false-alarm probability may be
achieved, and this choice of threshold determines the detection probability as well. In
Fig. 7.3 the argument is made that the normalized zero-mean, variance-one, G aussian

So, by choosing the threshold η

density determines performance. The desi gn er is given a string of length √ d , whose
head end is placed at location ( / √ d , t o a chieve the false-alarm probability P F , η + d / 2)

√ d , to determine the detection probability

and whose tail end falls at location ( η − d / 2) /

P D . The length of the string is √ d , which is the output voltage SNR , so obviously the
larger the SNR the better.

7.4.2 Common mean and uncommon covariances

When the mean of an improper complex random vector is common to both the hypothesis
and the alternative, but the covariances are uncommon, we might as well assume that
the mean is zero, since the measurement can always be centered. Then the pdf for the
measurement y : C n

−→ under hypothesis H i is

exp

2 y H R − i y

1

1

p i ( y ) = − (7.29)

n det 1 / 2 R i

π

where R i is the augmented covariance matrix under hypothesis H i . For a measured y
this is also the likelihood that the covariance model R i would have produced it. The
log-likelihood ratio for comparing the likelihood of alternative H 1 with the likelihood
of hypothesis H 0 is then the real widely quadratic form

y H R − H / 2
0

S − 1 ) R − 1 / 2
0

y H ( R − 0 − R − 1

1

L = ) y = ( I − y , (7.30)

R − 1 / 0 R 1 R − H / 2

0 is the augmented signal-to-noise ratio matrix . The trans-
formed measurement R − 1 / 0 y has augmented covariance matrix I under hypothesis H 0
and augmented covariance matrix S under hypothesis H 1 . Thus L is the log-likelihood
ratio for testing that the widely linearly transformed measurement, extracted as the top
half of R − 1 / 0 y ,is proper and white versus the alternative that it is improper with aug-
mented covariance S . When the SNR matrix is given its augmented EVD S

=

where S

UΛU H

=