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ECEN4503 Random Signals Lecture #39 21 April 2014 Dr. George Scheets

ECEN4503 Random Signals Lecture #39 21 April 2014 Dr. George Scheets. Read 10.1, 10.2 Problems: 10.3, 5, 7, 12,14 Exam #2 this Friday: Mappings → Autocorrelation Wednesday Class ??? Quiz #8 Results Hi = 10, Low = 0.8, Average = 5.70, σ = 2.94.

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ECEN4503 Random Signals Lecture #39 21 April 2014 Dr. George Scheets

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  1. ECEN4503 Random SignalsLecture #39 21 April 2014Dr. George Scheets • Read 10.1, 10.2 • Problems: 10.3, 5, 7, 12,14 • Exam #2 this Friday: Mappings → Autocorrelation • Wednesday Class ??? • Quiz #8 ResultsHi = 10, Low = 0.8, Average = 5.70, σ = 2.94

  2. ECEN4503 Random SignalsLecture #40 23 April 2014Dr. George Scheets • Read 10.3, 11.1 • Problems 10.16:11.1, 4, 15,21 • Exam #2 Next Time • Mappings → Autocorrelation

  3. Standard Operating Procedurefor Spring 2014 ECEN4503 If you're asked to find RXX(τ)Evaluate A[ x(t)x(t+τ) ]do not evaluateE[ X(t)X(t+τ) ]

  4. 1.25 1 x i 0 1 1 0 20 40 60 80 100 0 i 100 You attach a multi-meter to this waveform& flip to volts DC. What is reading? • Zero

  5. 1.25 1 x i 0 1 1 0 20 40 60 80 100 0 i 100 You attach a multi-meter to this waveform& flip to volts AC. What is reading? • 1 volt rms = σ • E[X2] = σ2 +E[X]2

  6. 1.25 1 x i 0 1 1 0 20 40 60 80 100 0 i 100 Shape of autocorrelation? • Triangle

  7. 1.25 1 x i 0 1 1 0 20 40 60 80 100 0 i 100 Rxx(τ) Value of RXX(0)? 1 τ (sec) 0

  8. 1.25 1 x i 0 1 1 0 20 40 60 80 100 0 i 100 Rxx(τ) Value of Constant Term? 1 0 τ (sec) 0

  9. 1.25 1 x i 0 1 1 0 20 40 60 80 100 0 i 100 Rxx(τ) If 1,000 bps,what time τ does triangle disappear? 1 0 τ (sec) 0 -0.001 0.001

  10. Power Spectrum SXX(f) By Definition = Fourier Transforms of RXX(τ). Units are watts/(Hertz) Area under curve = Average Power = E[X2] = A[x(t)2] = RXX(0) Has same info as Autocorrelation Different Format

  11. Crosscorrelation RXY(τ) • = A[x(t)y(t+τ)] • = A[x(t)]A[y(t+τ)]iff x(t) & y(t+τ) are Stat. Independent • Beware correlations or periodicities • Fourier Transforms to Cross-Power spectrum SXY(f).

  12. Ergodic Process X(t) volts • E[X] = A[x(t)] volts • Mean, Average, Average Value • Vdc on multi-meter • E[X]2 = A[x(t)]2 volts2 = constant term in Rxx(τ) • = Area of δ(f), using SXX(f) • (Normalized) D.C. power watts

  13. Ergodic Process • E[X2] = A[x(t)2] volts2 = Rxx(0)= Area under SXX(f) • 2nd Moment • (Normalized) Average Power watts • (Normalized) Total Power watts • (Normalized) Average Total Power watts • (Normalized) Total Average Power watts

  14. Ergodic Process • E[(X -E[X])2] = A[(x(t) -A[x(t)])2] • Variance σ2X • (Normalized) AC Power watts • E[X2] - E[X]2 volts2 • A[x(t)2] - A[x(t)]2 • Rxx(0) - Constant term • Area under SXX(f), excluding f = 0. • Standard Deviation σXAC Vrmson multi-meter

  15. Discrete time White Noise & RXX(τ)

  16. Autocorrelation & Power Spectrum of C.T. White Noise Rx(τ) A 0 Rx(τ) & Gx(f) form a Fourier Transform pair. They provide the same info in 2 different formats. tau seconds Gx(f) A watts/Hz 0 Hertz

  17. Autocorrelation & Power Spectrum of Band Limited C.T. White Noise Rx(tau) A 2AWN 0 tau seconds 1/(2WN) Average Power = ? D.C. Power = ? A.C. Power = ? Gx(f) A watts/Hz -WN Hz 0 Hertz

  18. 255 point Noise Waveform(Low Pass Filtered White Noise) 23 points Volts 0 Time

  19. Autocorrelation Estimate of Low Pass Filtered White Noise Rxx 0 23 tau samples

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