Page 464 - Fundamentals of Radar Signal Processing

FIGURE 6.6 Structure of optimal detector when the absolute signal phase is
unknown.

The performance of this detector will now be established. Let .

The detection test becomes simply ; thus the distribution of z under each of
the two hypotheses is needed. As in the known phase case, under hypothesis H 0

(target absent) ; thus the real and imaginary parts of are
independent of one another and each distributed as . It follows (see
Chap. 2 or App. A) that z is Rayleigh distributed

(6.46)

The probability of false alarm is

(6.47)

It is convenient to invert this equation to obtain the threshold setting in terms of
P :
FA

(6.48)

Now consider hypothesis H , i.e., target present. In this case .
1
Since E is real-valued, the real part is distributed as N(E, Eβ /2) while the
2
2
imaginary part is distributed as N(0, Eβ /2). It again follows that the PDF of z is

(6.49)

where I (z) is again the modified Bessel function of the first kind. Equation
0
(6.49) is the Rician PDF. The probability of detection is obtained by integrating
it from T’ to + ∞.
In normalized form, the required integral is