3.3 Likelihood-ratio test

The fundamental problem for statistical decision theory when applied to radar is the detection of a signal out of noise and interference. We measure the received signal vector z. This may be composed of a target echo signal vector s and noise vector n or only noise alone. We define the two hypotheses:

We want to make our decision for H ₀ or H ₁ based on the measured z. In other words. we have to divide the set of possible realisations for z into two regions D ₀ and D ₁, one for H ₀ and one for H ₁ (see Figure 3.9). If we measure z in region D ₀ we accept H ₀ and for z in D ₁ we accept H ₁. The problem now is to find the borderline between D ₀ and D ₁. We now define decision rules, following e.g. Middleton [3]:

Figure 3.9: Decision areas between two hypotheses

If z is in D ₁ then we express the decision for H ₁ by:

If z is in D ₀ then we express the decision for H ₀ by:

We always have for all z:

We assume as known the conditional probability densities p( z/0) and p( z/ s) according to the hypotheses H ₀ and H ₁