The following will be discussed in this chapter: iid signal x[n], uniform in []; extracting the portion of x(t) in a specified frequency band; questions (warm-up for Quiz 2!); periodograms (., a unit-intensity “white” process); periodogram averaging (illustrating the Einstein-Wiener-Khinchin theorem). | Lecture Signals, systems & inference – Lecture 18: Power Spectral Density (PSD) Power Spectral Density (PSD) , Spring 2018 Lec 18 1 iid signal x[n], uniform in [] 2 y[.] obtained by passing x[.] through resonant 2nd-order filter H(z), poles at ±{jπ/3} 3 Extracting the portion of x(t) in a specified frequency band x(t) H(jv) y(t) H(jv) ¢ ¢ 1 -v0 v0 4 Questions (warm-up for Quiz 2!) WSS process x[·] with Cxx [m] = ⇢ [m 1] + [m] + ⇢ [m + 1] . What is the largest magnitude ⇢ can have? WSS process x(·) with mean µx and PSD Sxx (j!). What is its FSD? Zero-mean WSS process x(·) with 1 Sxx (j!) = 1 + !2 2 and let y(t) = Z +x(t), where Z has zero mean, variance , and is uncorrelated with x(·). What are µy and Syy (j!)? 5 Periodograms (., a unit-intensity “white” process) M = 1, T = 50 M = 1, T = 50 M = 1, T = 50 M = 1, T = 50 4 4 4 4 3 3 3 3 2 2 2 2 1 1 1 1 0 0 0 0 0 1 0 1 0 1 0 1 v/(2p) v/(2p) v/(2p) v/(2p) M = 4, T = 50 M = 4, T = 200 4 4 3 CT case: XT (j!) $ x(t) 3 windowed to [ T, T ] 2 2 1 1 |XT (j!)|2 Periodogram = 0 0 2T 0 1 0 1 v/(2p) v/(2p) M = 16, T = 50 M = 16, T = 200 4 4 j⌦ DT case: XN (e ) $ x[n] windowed to [ N, N ] 3 3 2 2 |XN (ej⌦ )|2 1 Periodogram 1 6 = 0 0 2N + 1 Periodogram averaging (illustrating the Einstein-Wiener-Khinchin theorem) M = 1, T = 50 M = 1, T = 50 M = 1, T = 50 M = 1, T = 50 4 4 4 4 3 3 3 3 2 2 2 2 1 1 1 1 0 0 0 0 0 1 0 1 0 1 0 1 v/(2p) v/(2p) v/(2p) v/(2p) M = 4, T = 50 M = 4, T = 200 4 4 3 3 2 2 1 1 0 0 0 1 0 1 v/(2p) v/(2p) M 16, T 50 M 16, T 200 4 4 3 3 2 2 7 1 1 Periodogram averaging (illustrating the Einstein-Wiener-Khinchin theorem) M = 1, T = 50 M = 1, T = 50 M = 1, T = 50 M = 1, T = 50 4 4 4 4 3 3 3 3 2 2 2 2 1 1 1 1 0 0 0 0 0 1 0 1 0 1 0 1 v/(2p) v/(2p) v/(2p)
