The following will be discussed in this chapter: Einstein-Wiener-Khinchin theorem, periodogram averaging (illustrating the Einstein-Wiener-Khinchin theorem), respiratory model, modeling filters. | Lecture Signals, systems & inference – Lecture 19: Einstein-Wiener-Khinchin theorem, PSD applications, modeling filters Einstein-Wiener-Khinchin theorem, PSD applications, modeling filters , Spring 2018 Lec 19 1 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 DT case: XN (ej⌦ ) $ x[n]4 windowed to [ N, N ] 3 3 2 2 |XN (ej⌦ )|2 1 Periodogram 1 2 = 0 0 2N + 1 Einstein-Wiener-Khinchin theorem 1 h i 2 sin (!T ) 2 E |XT (j!)| = Sxx (j!) ? 2T ⇡! 2 T sin2 (!T ) Since lim 2 = (!) , T !1 ⇡! T 1 h i 2 lim E |XT (j!)| = Sxx (j!) T !1 2T 3 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 4 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) 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 1 1 0 5 0 0 1 0 1 v/(2p) v/(2p) Respiratory model cf. Khoo’s 6 textbook for N=1 Heart rate variability ECG signal 1 ECG amplitude (mV)
