In Section 10.6.5, we considered the estimation of the power spectrum of a sinusoid plus white noise. In this problem, we will determine the true powet spectrum of such a signal. Suppose that?

x[n] =Acos(?₀n + ?) + e[n],

where ? is a random variable that is uniformly distributed on the onterval from 0 to 2? and e[n] is a sequence of zero-mean random variavles that are uncorrelated with each tohter and also uncorrelated with ?. In other words, the cosine component has a randomly selected phase, and e[n] represents white noise.

(a) Show that for the preceding assumption, 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

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Part A A? cos(wom) +o [m]. Pxx[m] = E{x[n]r[m+n]} where o El(eln])'). Part B Pr(w) = [8(w-wo) + 6(w + wy)]+ o.