4.6 In applications, we will often observe series containing a signal that has been delayed by some unknown time D, i.e., xt = st + Ast−D + nt, where st and nt are stationary and independent with zero means and spectral densities fs(ω) and fn(ω), respectively. The delayed signal is multiplied by some unknown constant A.

(a) Prove fx(ω) = [1 + A2 + 2Acos(2πωD)]fs(ω) + fn(ω).

(b) How could the periodicity expected in the spectrum derived in (a)

be used to estimate the delay D? (Hint: Consider the case where fn(ω) = 0; i.e., there is no noise.)