Detection - Statistical Signal Processing of Complex-Valued Data - page 184

µ 1

µ

Figure 7.2 Geometrical interpretation of a coherent matched filter. The thick dashed line is the
locus of measurements such that [ R − 1 ( 1 − 0 )] H ( y − y 0 ) = 0 .

R − 1 (

to produce the real log-likelihood ratio L , which is compared against a threshold. The
geometry of the log-likelihood ratio is illustrated in Fig. 7.2 : we choose lines parallel to
the thick dashed line in order to get the desired probability of false alarm P F .
But how do we determine the threshold to get the desired P F ? We will see that the
deflection, or output signal-to-noise ratio, plays a key role here.

The centered complex measurement y − y 0 is resolved onto the line 1 − 0 )

Definition 7.4. The deflection for a likelihood-ratio test with log-likelihood ratio L is
defined as

E 0 ( L )] 2
var 0 ( L )

[ E 1 ( L ) −

d = , (7.25)

where var 0 ( L ) denotes the variance of L under H 0 .

We will first show that L given by ( 7.23 ) results in the deflection

1 − 0 ) H R − 1 (

d = ( 1 − 0 ) , (7.26)

which is simply the output signal-to-noise ratio . The mean of L is d / 2 under H 1 and

− 2 under H 0 , and its variance is d under both hypotheses. Thus, the log-likelihood

ratio statistic L is distributed as a real Gaussian random variable with mean value

d /

2
and variance d . Thus, L has deflection, or output signal-to-noise ratio, d . In the general
non-Gaussian case, the deflection is much easier to compute than a complete ROC curve.
Now the false-alarm probability for the detector

± d /

φ that compares L against a threshold

η is

η

exp 2) 2 d L

η +

1
√ 2

1
2 d ( L

d / 2

P F = 1 − − + d / = 1 − √ d , (7.27)

π d

−∞

where is the probability distribution function of a zero-mean, variance-one, real

Gaussian random variable. A similar calculation shows the probability of detection to
be

η

exp 2) 2 d L

η −

1
√ 2

1
2 d ( L

d / 2

√ d

P D = 1 − − − d / = 1 − . (7.28)

π d −∞