Page 445 - Fundamentals of Radar Signal Processing

threshold to be exceeded, and therefore without affecting the performance (P D

and P ). A well-chosen transformation can sometimes greatly simplify the
FA
computations required to actually carry out the LRT. Most common is to take the
natural logarithm of both sides of Eq. (6.7) to obtain the log likelihood ratio
test:

(6.8)

To make these procedures more concrete, consider what is perhaps the
simplest example, detection of the presence or absence of a constant in zero-

mean Gaussian noise of variance . Let w be a vector of independent
identically distributed (i.i.d.) zero mean Gaussian random variables. When the
constant is absent (hypothesis H ) the data vector y = w follows an N-
0
dimensional normal distribution with a scaled identity covariance matrix. When

the constant is present (hypothesis H ) , y = m + w = m1 + w and the
N
1
6
distribution is simply shifted to a nonzero positive mean :

(6.9)

where m > 0 and 0 , 1 , and I are, respectively, a vector of N zeros, a vector of
N
N
N
N ones, and the identity matrix of order N. The model of the required PDFs is
therefore

(6.10)

The likelihood ratio Λ(y) and the log-likelihood ratio can be directly

computed from Eq. (6.10):

(6.11)