In Section 10.5.5, we considered the estimation of the power spectrum of a sinusoid plus...

In Section 10.5.5, we considered the estimation of the power spectrum of a sinusoid plus white noise. In this problem, we will determine the true power spectrum of such a signal. Suppose that

x[n] = Acos(ω₀n + θ) + e[n],

where θ is a random variable that is uniformly distributed on the interval from 0 to 2π and e[n] is a sequence of zero-mean random variables that are independent of each other and also independent of θ. In other words, the cosine component has a randomly selected phase, and e[n] represents white noise.

(a) Show that for the preceding assumptions, the autocorrelation function for x[n] is

(b) From the result of part (a), show that over one period in frequency, the power spectrum of x[n] is