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Power Spectral Density (PSD)
6.011, Spring 2018 Lec 18
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iid signal x[n], uniform in [-0.5,+0.5]
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Power Spectral Density (PSD) 6.011, Spring 2018 Lec 18 1 iid - - PowerPoint PPT Presentation
Power Spectral Density (PSD) 6.011, Spring 2018 Lec 18 1 iid - - PowerPoint PPT Presentation
Power Spectral Density (PSD) 6.011, Spring 2018 Lec 18 1 iid signal x[n], uniform in [-0.5,+0.5] 2 y[.] obtained by passing x[.] through resonant 2 nd -order filter H(z), poles at 0.95e^{j /3} 3 Extracting the portion of x(t) in a
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y[.] obtained by passing x[.] through resonant 2nd-order filter H(z), poles at ±0.95e^{jπ/3}
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v
Extracting the portion of x(t) in a specified frequency band
x(t) H(jv) y(t) 1 H(jv) ¢ ¢
- v0
v0
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δ
σ
Questions (warm-up for Quiz 2!)
WSS process x[·] with What is the largest magnitude ρ can have?
WSS process x(·) with mean µx and PSD Sxx(jω). What is its FSD?
Zero-mean WSS process x(·) with 1 Sxx(jω) = 1 + ω2 What are µy and Syy(jω)?
Cxx[m] = ρδ[m − 1] + δ[m] + ρδ[m + 1] .
and let y(t) = Z +x(t), where Z has zero mean, variance σ2, and is uncorrelated with x(·).
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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)
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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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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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6.011 Signals, Systems and Inference
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