Page 462 - Fundamentals of Radar Signal Processing

(6.38)

Expanding the exponent in Eq. (6.38) gives

(6.39)

where ϕ is the unknown but fixed phase of the inner product .

Notice that p (y|H ,θ) does not depend on θ after all (not surprising since
0
y
there is no target present in this case to present an unknown phase), so it is not
necessary to apply Eq. (6.37). However, in p (y|H1) the dependence on θ is
y
explicit. Assuming a uniform random phase, defining θ′ = ϕ – θ, and applying
Eq. (6.37) under H gives
1

(6.40)

Equation (6.40) is a standard integral. Specifically, integral 9.6.16 in Olver et
al. (2010) is

(6.41)

where I (z) is the modified Bessel function of the first kind. Using this result and
0
properties of the cosine function, Eq. (6.40) becomes

(6.42)

The log-LRT now becomes