Signal Processing for Cognitive Radios

Example 4.4 (A GLRT for the Hypothesis Testing Problem Defined in Example 4.3)

Recall Example 4.3 in which we showed that there is no UMP test for the following problem:

(4.80)

where y, s, and are all L-vectors and θ ≠ 0. The null hypothesis is simple with Θ₀ = {0}, and the alternative hypothesis is composite with . In Example 4.3, we showed that although we can find UMP tests if or , for the case of , there is no UMP test. Moreover, for , we cannot apply the LMP test formulation of (4.71) because it is not enough to consider a small neighborhood of θ such that either θ > 0 or θ < 0. Hence, we may instead attempt to derive a GLRT for this problem.

Since the null hypothesis is simple in problem (4.80), we only need to find the ML estimate of θ for the likelihood of y under H₁:

(4.81)

Thus, ...