Page 457 - Fundamentals of Radar Signal Processing

Treatment of the case where the noise samples are not equal-variance, and the

colored noise case where S is not even diagonal, is beyond the scope of this
y
text. The reader is referred to Dudgeon and Johnson (1993) and Kay (1998) for
these more complex situations.
Equation (6.26) now reduces to

(6.28)

Further simplifications occur when all of the means under H are identical so
1
that m = m1 , where m can now be complex-valued. In this case Eq. (6.28)
N
reduces slightly further to

(6.29)

The LRT for the coherent version of the previous Gaussian example can be

obtained by repeating the steps in the example of Eqs. (6.10) through (6.25)
using the PDF of Eq. (6.28), with m = 0 under hypothesis H and m ≠ 0 under
0
N
N
H . The log-likelihood ratio is
1

(6.30)

where the second line of Eq. (6.30) applies only to the case where the means
are identical (m = m1 ).
N
H
Some interpretation of Eq. (6.30) is in order. The term m y is the dot
product of the complex vectors m and y. As seen in App. B, this dot product
represents an FIR filtering operation evaluated at the particular instant when the
H
equivalent impulse response m and the data vector y completely overlap.
Furthermore, since the impulse response of the filter is identical to the signal
whose presence is to be detected under hypothesis H , namely m1 , it is a
N
1
matched filter. The same reasoning applies if the elements of m are the samples
of a modulated waveform or any other function of interest.
The second term in lnΛ, which is the complex dot product of m with itself,
expands to . This is just the energy E in m. In the equal means

2
case E = N|m| .