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ECEN3513 Signal Analysis Lecture #23 16 October 2006

ECEN3513 Signal Analysis Lecture #23 16 October 2006. Read 6.1, 6.2; Review 6.3 Problems 4.7-2, 6.2-1, 6.3-5 Test #2 on 27 October Quiz 5 results: Hi = 7.8, Lo = 2.9, Ave = 5.35 Standard Deviation = 1.35 Read 6.4, 6.5 (1st section), 6.7 (1st section) Problems: 6.4-3, 6.4-4, 6.5-1

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ECEN3513 Signal Analysis Lecture #23 16 October 2006

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  1. ECEN3513 Signal AnalysisLecture #23 16 October 2006 • Read 6.1, 6.2; Review 6.3 • Problems 4.7-2, 6.2-1, 6.3-5 • Test #2 on 27 October • Quiz 5 results: Hi = 7.8, Lo = 2.9, Ave = 5.35Standard Deviation = 1.35 • Read 6.4, 6.5 (1st section), 6.7 (1st section) • Problems: 6.4-3, 6.4-4, 6.5-1 • Test #2 on 27 October • Quiz 6 Results: Hi =8.0, Lo = 2.0, Ave. = 5.00Standard Deviation = 2.04

  2. yp(t): Base Periodic Function

  3. Fourier Transform of yp(t)

  4. y(t): Periodic Time Domain Waveform

  5. Fourier Series of y(t)

  6. Linear Time Invariant? yes • δ(t) → h(t)... an impulse response exists • convolution works • y(t) = x(t) * h(t) • transfer function exists • Y(f) = X(f) H(f) • Energy & Power transfer functions exist • |Y(f)|2 = |X(f)|2 |H(f)|2 J/Hz • SYY(f) = SXX(f) |H(f)|2 W/Hz

  7. Linear Time Invariant? no • δ(t) → h(t)... an impulse response exists • convolution gives you the wrong answer • y(t) ≠ x(t) * h(t) • transfer function does not exist • Y(f) ≠ X(f) H(f) • Energy & Power transfer functions don't exist • |Y(f)|2≠ |X(f)|2 |H(f)|2 J/Hz • SYY(f) ≠ SXX(f) |H(f)|2 W/Hz

  8. Autocorrelation (Power Signal) RXX(τ) = lim 1 x(t)x(t+τ)dt T→∞ T T • When in doubt, evaluate this one first • If = 0, evaluate energy signal autocorrelation • SXX(f) = lim |X(f)|2non-zero at same T→∞ T freqs as X(f)! • RXX(τ) ↔ SXX(f) W/Hz • Autocorrelation & Power Spectrum form a Fourier Transform pair

  9. Autocorrelation (Power Signal) +∞ RXX(τ) = lim 1 x(t)x(t+τ)dt = SXX(f) e -j2πfτ df T→∞ T T • RXX(0) = ?? • Average (normalized) power of x(t) • How can you find RXX(0) from SXX(f)? • Evaluate at τ = 0 • The area under power spectrum = average (normalized) power of x(t) • What's SXX(f) for x(t) = 3cos2π100t? • What's SXX(f) for x(t) = 3sin2π100t? -∞ units are W/Hz

  10. Autocorrelation (Energy Signal) RXX(τ) = lim x(t)x(t+τ)dt T→∞ T • RXX(τ) ↔ |X(f)|2 J/Hz • Autocorrelation & Energy Spectrum form a Fourier Transform pair • GXX(f) = |X(f)|2non-zero at same freqs as X(f)!

  11. Autocorrelation (Energy Signal) +∞ RXX(τ) = lim x(t)x(t+τ)dt = |X(f)|2 e -j2πfτ df T→∞ T • RXX(0) = ?? • Average (normalized) energy of x(t) • How can you find RXX(0) from |X(f)|2? • Evaluate at τ = 0 • The area under energy spectrum = average (normalized) energy of x(t) • What's |X(f)|2for x(t) = rect[(t-0.5)/1]? • What's |X(f)|2for x(t) = rect[t/1]? -∞ units are J/Hz

  12. Autocorrelation (Real World) ^ RXX(τ) = 1 x(t)x(t+τ)dt T T • RXX(τ) ↔ SXX(f) W/Hz • FT of autocorrelation yields estimate of Power Spectrum • SXX(f) = |X(f)|2, where X(f) = x(t) e-jωt dt T observation interval ^ ^ ^ ^ ^ T

  13. Autocorrelation (Real World) ^ RXX(τ) = x(t)x(t+τ)dt T • RXX(τ) ↔ GXX(f) W/Hz • FT of autocorrelation yields estimate of Energy Spectrum • GXX(f) = |X(f)|2, where X(f) = x(t) ejωt dt observation interval ^ ^ ^ ^ ^ T

  14. sinusoid + d.c. δ(f-0.1) δ(f+0.1) δ(f)

  15. Magnitude & Phase Plots

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