In the radar system of Example 8.4, the probability that a target is present is P[H₁] = 0.01. In the case of a false alarm, the system issues an unnecessary alert at the cost of C₁₀ = 1 unit. The cost of a miss is C₁₀= 10⁴ units because the target could cause a lot of damage. When the target is present, the voltage is X = 4 + N, a Gaussian (4, 1) random variable. When there is no target present, the voltage is X = N, the Gaussian (0, 1) random variable. In a binary hypothesis test, the acceptance sets are A₀ = {X ≤ x0} and A₁ = {X > x0}.

(a) What is x₀ = x_MAP, the decision threshold of the maximum a posteriori probability hypothesis test?

(b) What are the error probabilities P_FA and PMISS of the MAP test?

Example 8.4

The noise voltage in a radar detection system is a Gaussian (0, 1) random variable, N. When a target is present, the received signal is X = v + N volts with v ≥ 0. Otherwise the received signal is X = N volts. Periodically, the detector performs a binary hypothesis test with H₀ as the hypothesis no target and H₁as the hypothesis target present. The acceptance sets for the test are A₀ = {X ≤ x0} and A₁ = {X > x0}. Draw the receiver operating curves of the radar system for the three target voltages v = 0, 1, 2 volts.