Page 448 - Fundamentals of Radar Signal Processing

(6.17)

Equation (6.17) is the integral of a Gaussian PDF, so the error function

erf(x) will appear in the solution. The standard definition is (Abramowitz and
Stegun, 1972) 7

(6.18)

Also define the complementary error function erfc(x) corresponding to erf(x)

(6.19)

One will generally be interested in finding the value of x that results in a certain

value of erf(x) or erfc(x); thus the inverse error and complementary error
–1
–1
functions, denoted by erf (z) and erfc (z), respectively, are of interest. It
–1
follows from Eq. (6.19) that the two are related by erfc (z) = erf (1 – z). 8
–1
With the change of variables , Eq. (6.17) can be written as

(6.20)

Finally, Eq. (6.20) can be solved to obtain the threshold T in terms of the
tabulated inverse error function:

(6.21)

Equations (6.20) and (6.21) show how to compute P given T and vice versa.
FA
All of the information needed to carry out the LRT in its sufficient statistic

form of Eq. (6.14) is now available. ϒ(y) is just the sum of the data samples,
while the threshold T can be computed from the number N of samples, the

variance of the noise, which is assumed known, and the desired false alarm
probability P .
FA
The performance of this detector is evaluated by constructing a receiver
operating characteristic (ROC) curve. There are four interrelated variables of