Question: In radar applications the transmitted waveform is often a segment of a sinusoidal signal of known frequency. The frequency of the return signal from the target, however, is shifted by an amount called the Doppler frequency, which is approximately equal to ωd = 2vωo/c, where v is the radial velocity of the target and c is the velocity of propagation of the electromagnetic wave. For targets of unknown velocity, ωd can be modeled as a random variable with known density function. The received observations corresponding to whether a target is present or not will thus be

H₁: z(t) = (√2)E sin(ω_dt + θ) + v(t)          0 ≤ t ≤ T

H₀: z(t) = v(t)                                       0 ≤ t ≤ T

where E and θ are assumed known. Determine the likelihood ratio test for this problem under the assumption that vet) is zero mean, Gaussian, white.