Spectrum analysis is often applied to signals comprised of sinusoids. Sinusoidal signals are particularly interesting, because they share properties with both deterministic and random signals. On the one hand, we can describe them in terms of a simple equation. On the other hand, they have infinite energy, so we often characterize them in terms of their average power, just as with random signals. This problem explores some theoretical issues in modeling sinusoidal signals from the point of view of random signals. We can consider sinusoidal signals as stationary random signals by assuming that the signal model is s[n] = A cos(ω0n + θ ) for -∞

(a) Show that the autocorrelation function for such a signal is

(b) Using Eq. (11.34), write the set of equations that is satisfied by the coefficients of a 2nd-order linear predictor for this signal.

(c) Solve the equations in (b) for the optimum predictor coefficients. Your answer should be a function of ω0.

(e) Use Eq. (11.37) to determine an expression for the minimum mean-squared prediction error. Your answer should confirm why random sinusoidal signals are called "predictable" and/or "deterministic."