Einstein-Wiener-Khinchin theorem, PSD applications, modeling filters - - PowerPoint PPT Presentation

Einstein-Wiener-Khinchin theorem, PSD applications, modeling filters 6.011, Spring 2018 Lec 19

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v/(2p) v/(2p) 0.2 0.4 0.6 0.8 1 1 2 3 4 M = 4, T = 50 0.2 0.4 0.6 0.8 1 1 2 3 4 M = 4, T = 200 1 2 3 4 M = 16, T = 50 1 2 3 4 M = 16, T = 200 v/(2p) v/(2p)

− −

Periodograms (e.g., a unit-intensity “white” process)

M = 1, T = 50 4 3 2 1 M = 1, T = 50 4 3 2 1 M = 1, T = 50 4 3 2 1 M = 1, T = 50 4 3 2 1 0.5 1 0.5 1 0.5 1 0.5 1 v/(2p) v/(2p) v/(2p) v/(2p)

|XT (jω)|2 Periodogram = 2T |XN (ejΩ)|2 Periodogram = 2N + 1 CT case: XT (jω) ↔ x(t) windowed to [−T, T] DT case: XN (ejΩ) ↔ x[n] windowed to [−N, N]

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• Einstein-Wiener-Khinchin theorem

1 sin2(!T ) E h |XT (j!)|2i = Sxx(j!) ? 2T ⇡!2T sin2(!T ) Since lim

T →∞

⇡!2T 1 lim E h |XT (j!)|2i = Sxx(j!)

T →∞ 2T

= δ(ω) ,

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0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 v/(2p) v/(2p) 1 2 3 4 M 16, T 50 1 2 3 4 M 16, T 200 v/(2p) v/(2p)

Periodogram averaging (illustrating the Einstein-Wiener-Khinchin theorem)

M = 1, T = 50 4 3 2 1 M = 1, T = 50 4 3 2 1 M = 1, T = 50 4 3 2 1 M = 1, T = 50 4 3 2 1 0.5 1 0.5 1 0.5 1 0.5 1 v/(2p) v/(2p) v/(2p) v/(2p) M = 4, T = 50 4 3 2 1 M = 4, T = 200 4 3 2 1 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

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Periodogram averaging (illustrating the Einstein-Wiener-Khinchin theorem)

M = 1, T = 50 4 3 2 1 M = 1, T = 50 4 3 2 1 M = 1, T = 50 4 3 2 1 M = 1, T = 50 4 3 2 1 0.5 1 0.5 1 0.5 1 0.5 1 v/(2p) v/(2p) v/(2p) v/(2p) M = 4, T = 50 4 3 2 1 M = 4, T = 200 4 3 2 1 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 v/(2p) M = 16, T = 50 4 3 2 1 v/(2p) v/(2p) M = 16, T = 200 4 3 2 1 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 v/(2p)

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Respiratory model

• cf. Khoo’s

textbook for N=1

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Heart rate variability

ECG signal (a) −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 1.2 ECG amplitude (mV) 54 56 58 60 62 64 66 Time (sec) Instantaneous HR signal 45 50 55 60 65 70 75 80 Time (sec) (b) 1.1 1.15 1.2 1.25 1.3 x(t) (beats/sec) Power spectrum (c) 10 20 30 40 50 60 70 80 90 D

xx (e ) jÆ

−1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 Frequency (Hz) 1

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σ − δ δ δ −

Modeling filters

e.g., generate sample functions of WSS y[·] that has specified µy ↓ More generally, H(z)A(z) for allpass A(z) , A(z)A(z−1) = 1 Need to add mean µy to the output and Cyy[m] = σy

2(ρδ[m − 1] + δ[m] + ρδ[m + 1])

Try h[n] = aδ[n] + bδ[n − 1] driven by unit-intensity white noise x[·], H(z) = a + bz−1 , |H(ejΩ)|2 = Dyy(ejΩ)

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6.011 Signals, Systems and Inference

Spring 2018 For information about citing these materials or our Terms of Use, visit: https://ocw.mit.edu/terms.

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