Page 476 - Fundamentals of Radar Signal Processing

threshold test. There are two practical problems with this equation. First, it is

desirable to avoid computing the function ln[I (·)] possibly millions of times
0
per second. Second, both the target amplitude and the noise power must
be known to perform the required scaling. The test can be simplified by using
the results of Sec. 6.2.3. Applying the square law detector approximation of Eq.
(6.55) to Eq. (6.68) gives the test:

(6.69)

Combining all constants into the threshold gives us the final detection rule:

(6.70)

Equation (6.70) states that the squared magnitudes of the data samples are

simply integrated and the integrated sum compared to a threshold to decide
whether a target is present or not. The integrated variable z is the sufficient
statistic ϒ for this problem.
The performance of the detector must now be determined. It is convenient
to scale the z , replacing them with the new variables and thus
n
replacing z with ; such a scaling does not change the
performance, but merely alters the threshold value that corresponds to a

particular P or P . The PDF of is still either Rayleigh or Rician as in Eqs.
D
FA
(6.62) and (6.63), but now with unit noise variance:

(6.71)

(6.72)

where is the SNR. Since a square law detector is being used, define
; then z′ = Σ r . The PDF of r is exponential under H and a generalized
0
n
n
noncentral chi-squared density under H :
1