Applied Stochastic Processes Problem Set 7

Problem 6.25 Consider a wide sense stationary random sequence X[n] input to a linear filter with impulse response h[n] = 1/2, n = {0, 1} 0, otherwise. (1) Write the PSD of the output sequence S Y Y (ω) in terms of the PSD of the input sequence S XX (ω). (a) Show that the PSD is real-valued, even if X[n] is a complex-valued random sequence. (b) Show that if X[n] is real-valued, then S XX (ω) = S XX (−ω). (c) Show that S XX (ω) ≥ 0 for every ω, regardless of whether X[n] is complex-valued or not. Let's begin by determining the PSD of the output sequence S Y Y (ω) in terms of the PSD of the input sequence S XX (ω). First, we recall that the autocorrelation of the output sequence R Y Y [m] is given by Equation 6.4-1 on page 350 in [5]. R Y Y [m] = g[m] * R XX [m], for g[m] h[m] * h * [−m] Note that g[m], the autocorrelation impulse response, is given by g[m] = h[m] * h * [−m] = ∞ k=−∞ h[k]h * [m − k] =    1/4, m = {0, 2} 1/2, m = 1 0, otherwise. Also recall that the power spectral density (PSD) of the input sequence S XX (ω) is defined as the discrete-time Fourier transform (DFT) of the input autocorrelation function R XX [m]. S XX (ω) ∞ m=−∞ R XX [m]e −jωm , for − π ≤ ω ≤ π (2) Similarly, the PSD of the output sequence S Y Y (ω) is given by the DFT of R Y Y (ω). Substituting the previous expression for R Y Y [m], we find S Y Y (ω) = ∞ m=−∞ (h[k] * h * [−m]) * R XX [m]e −jωm , for − π ≤ ω ≤ π.