Question: The range (distance from transmitter) of a radar target is determined from the delay between the transmitted and return signals. For targets of unknown range, the delay will be a random variable. The decision problem in this case can be modeled as
H1: z(t) = y(t - τ) + v(t) 0 ≤ t ≤ T
H0: z(t) = v(t) 0 ≤ t ≤ T
where y (t) is a known waveform and τ is the unknown delay. Derive the likelihood ratio test for this problem under the assumption that vet) is zero mean, white, Gaussian, and τ is uniformly distributed in [τ0,τ1].