Final answer:
The autocorrelation of the harmonic process is given by R_x(m) = Σ |A_k|² cos(w_k * m), where the summation is over all values of k.
Step-by-step explanation:
In the provided expression for the harmonic process x(n), the autocorrelation R_x(m) is obtained by taking the expected value of the product x(n) * x(n - m). Since the phases θ_k are statistically independent with uniform density over [0, 2π], the cross-terms involving different phases average out to zero. Thus, the autocorrelation simplifies to the sum of the squares of the amplitudes multiplied by the cosine of the frequency difference.
Mathematically, this is expressed as R_x(m) = E[x(n)x(n-m)] = E[Σ A_k * e^(j(θ_k * n + w_k * n)) * Σ A_l * e^(j(θ_l * (n - m) + w_l * (n - m)))].
Simplifying further and using the independence of amplitudes and phases, we get R_x(m) = Σ E[A_k * A_l] * cos(w_k * m), where the sum is over all values of k and l. Since the amplitudes are assumed to be random and mutually independent of the phases, E[A_k * A_l] = 0 for k ≠ l and E[A_k * A_k] = |A_k|².
Therefore, the final expression for the autocorrelation is R_x(m) = Σ |A_k|² * cos(w_k * m), where the sum is over all values of k. This result characterizes the correlation between the harmonic process values at different time instances m.