1 / 49

ECEN4503 Random Signals Lecture #27 11 March 2013 Dr. George Scheets

ECEN4503 Random Signals Lecture #27 11 March 2013 Dr. George Scheets. Read 8.5 Problems: 5.22, 8.3, 8.5, 8.6 (1st Edition) Problems: 5.51, 8.8, 8.9, 8.10 (2nd Edition). ECEN4503 Random Signals Lecture #29 15 March 2013 Dr. George Scheets.

mattox
Download Presentation

ECEN4503 Random Signals Lecture #27 11 March 2013 Dr. George Scheets

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. ECEN4503 Random SignalsLecture #27 11 March 2013Dr. George Scheets • Read 8.5 • Problems: 5.22, 8.3, 8.5, 8.6(1st Edition) • Problems: 5.51, 8.8, 8.9, 8.10 (2nd Edition)

  2. ECEN4503 Random SignalsLecture #29 15 March 2013Dr. George Scheets • Problems 8.7a & b, 8.11, 8.12a-c (1st Edition) • Problems 8.11a & b, 8.15, 8.16 (2nd Edition)

  3. Random Process X(t) Set of possible time domain waveforms Associated with some experiment x(t) = a specific waveform from X(t) Freeze Time at, say, t1X(t1) = set of points = X (Random Variable) one point from each waveform taken at same time (t1) Set of points has... Histogram → PDF, E[X], σ2X, etc.

  4. Random Process Stat Averages taken ┴ to time axis Using a point from every single waveformNot practical in Real World Time Averages taken along time axis Using one waveform x(t) Time Averages = Statistical Averages?Process is said to be Ergodic. Cautiously treat all processes in this class as ergodic.

  5. Some Waveforms Aren't Ergodic • Random DC voltage • x(t) might be, say, +2.3556 volts DC • X(t) might be the set of all DC waveformsfrom, say, -5 vdc to +5 vdc • E[X(t)] would = 0 vdc • If waveforms equally likely • A[x(t)] would = 2.3556 vdc

  6. 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)]2volts2 • (Normalized) D.C. power watts

  7. Ergodic Process • E[X2] = A[x(t)2] volts2 • 2nd Moment • (Normalized) Average Power watts • (Normalized) Total Power watts • (Normalized) Average Total Power watts • (Normalized) Total Average Power watts

  8. Ergodic Process • E[X2] - E[X]2 volts2 • A[x(t)2] - A[x(t)]2 • Variance σ2X • (Normalized) AC Power watts • E[(X -E[X])2] = A[(x(t) -A[x(t)])2] • Standard Deviation σXAC Vrms on multi-meter volts

  9. Histogram of Sinusoid Voltages

  10. PDF of a 3vp Sinusoid Area under PDF E[X2] Find PDF of voltage by treating Time as Uniform RV. Then map time → voltage. A[*] = E[*] when Ergodic.

  11. Voltage PDF for Clean Sinusoid If x(t) = α cos(2πβt + θ)thenfX(x) = 1 / [π(α2 - x2)0.5]; - α < x <α

  12. Given some waveform x(t)... • To find voltage PDF • Randomly sample waveformVisualize resulting Histogram • Mapping from 1 R.V. to AnotherMap time (t) → voltage (x)Treat time as Uniformly Distributed R.V.Treat waveform x(t) as mapping g(t) x (voltage) = g(t)

  13. Stationary Process • Statistics (such as Mean, Variance, PDF, etc.) collected at different times are ≈ the same. • There are actually several "sub definitions" of stationarity, but we'll only worry about this one.

  14. 1.25 1 x i 0 1 1 0 20 40 60 80 100 0 i 100 100 bps Continuous Random Bit Stream with S.I. bits. P(+1 volt) = P(-1 volt) = 0.5 Several hundred bits collected at different times? Stationary Process. Voltage statistics should be similar. Several hundred samples collected over a nsec? Not stationary. Statistics may differ.

  15. Internet Packet Traffic (Daily) Amsterdam Internet Exchange Internet traffic rate logged several thousand times over one hour periods? Not a stationary process. A process somewhat periodic is called cyclo-stationary.

  16. Internet Packet Traffic (Annual) Amsterdam Internet Exchange Internet traffic rate logged several thousand times over monthly periods? Not a stationary process. Mean is increasing. Looks like variance is increasing too (swings get wider).

  17. Annual Rainfall in SW USA Source: November 2007 National Geographic Daily rainfall amounts noted over a one year period? Not Stationary. This is probably a stationary process if the inputs to this experiment, sun output and environment factors, are constant.

  18. MELP Voice Coder • NATO Standard • Developed Using American Voices • Quality degrades a bit in other languages • Phoneme statistics aren't the same • Smallest contrastive unit of sound

  19. Review of PDF's & Histograms Probability Density Functions (PDF's), of which a Histogram is an estimate of shape, frequently (but not always!) deal with the voltage likelihoods Volts Time

  20. Discrete Time Noise Waveform255 point, 0 mean, 1 wattUniformly Distributed Voltages Volts 0 Time

  21. 15 Bin Histogram(255 points of Uniform Noise) Bin Count 0 Volts

  22. 15 Bin Histogram(2500 points of Uniform Noise) Bin Count When bin count range is from zero to max value, a histogram of a uniform PDF source will tend to look flatter as the number of sample points increases. 200 0 0 Volts

  23. 15 Bin Histogram(2500 points of Uniform Noise) Bin Count But there will still be variation if you zoom in. 200 140 0 Volts

  24. 15 Bin Histogram(25,000 points of Uniform Noise) 2,000 Bin Count 0 0 Volts

  25. Bin Count Volts Time Volts The histogram is telling us which voltages were most likely in this experiment. A histogram is an estimate of the shape of the underlying PDF. 0

  26. Discrete Time Noise Waveform255 point, 0 mean, 1 wattExponentially Distributed Voltages Volts 0 Time

  27. 15 bin Histogram(255 points of Exponential Noise) Bin Count 0 Volts

  28. Discrete Time Noise Waveform255 point, 0 mean, 1 wattGaussian Distributed Voltages Volts 0 Time

  29. 15 bin Histogram(255 points of Gaussian Noise) Bin Count 0 Volts

  30. 15 bin Histogram(2500 points of Gaussian Noise) Bin Count 400 0 Volts

  31. Autocorrelation Statistical average E[X(t)X(t+τ)] using Random Processes & PDF's Time average A[x(t)x(t+τ)] using a single waveform How alike is a waveform & shifted version of itself? Given an arbitrary point on the waveform x(t1), how predictable is a point τ seconds away at x(t1+τ)? RX(τ) = 0? Not alike. Uncorrelated. RX(τ) > 0? Alike. Positively correlated. RX(τ) < 0? Opposite. Negatively correlated.

  32. Time Average vs Statistical Average t1 + T ∞ A[ ? ] = lim (1/T) ? dt T →∞ t1 -∞ E[ ? ] = ? fX(x) dx

  33. Need to find the Autocorrelation? t1 + T = t2 • Take the time average!!! t1 RXX(τ) = lim (1/T) x(t)x(t+τ) dt T →∞

  34. Volts Time PDF's & Histograms • Voltage Probability Density Functions (PDF's), of which a Histograms is an estimate of shape, deal with the voltage likelihoods

  35. 1.25 1.25 1 1 x x i i 0 0 1 1 1 1 0 50 100 150 200 250 300 350 400 0 20 40 60 80 100 0 i 400 0 i 100 These waveforms have same Voltage PDF's

  36. Volts time Review of Autocorrelation • Autocorrelations deal with predictability over time. I.E. given an arbitrary point x(t1), how predictable is x(t1+τ)? τ t1

  37. Review of Autocorrelation t1 + T = t2 Rx(τ) = lim (1/T) x(t)x(t+τ) dt T →∞ Take x(t1)*x(t1+ τ), x(t1+ε)*x(t1+ τ + ε)... ....x(t2)*x(t2+ τ)... Add these all together, then average. t1

  38. Review of Autocorrelation N-τ ∑ Rx(τ) = 1/(N-τ) x(i)x(i+τ) i=1 Example: Rx(0) for discrete time signal ... ... x(i+0) x(1) x(2) x(3) x(100) x(i) x(1) x(2) x(3) x(100) Sum up x(1)2 + x(2)2 + ... + x(100)2, Then take a 100 point average.

  39. Review of Autocorrelation Example: Rx(1) for discrete time signal x(t+1) x(1) x(2) x(3) x(100) x(t) x(1) x(2) x(3) x(99) x(100) Sum up x(1)x(2) + x(2)x(3) + ... + x(99)x(100), Then average. If average is negative, paired numbers must tend to have opposite sign.

  40. Review of Autocorrelation Example: Rx(2) for discrete time signal x(t+2) x(1) x(2) x(3) x(100) x(t) x(1) x(2) x(98) x(99) Sum up x(1)x(3) + x(2)x(4) + ... + x(98)x(100), Then average. If average is near zero, paired numbers must tend to have unpredictable signs.

  41. 255 point discrete time Exponentially Distributed Noise Waveform(Adjacent points are independent) Vdc = 0 v, Normalized Power = 1 watt Volts 0 time

  42. 255 point discrete time Gaussian Distributed Noise Waveform(Adjacent points are independent) Vdc = 0 v, Normalized Power = 1 watt Volts 0 time

  43. 255 point discrete time Uniformly Distributed Noise Waveform(Adjacent points are independent) Vdc = 0 v, Normalized Power = 1 watt Volts 0 time

  44. Autocorrelation Estimate of Discrete Time White Noise Rxx The previous 3 waveforms all have the same theoretical autocorrelation function. 1 0 tau (samples)

  45. 255 point Noise Waveform(Low Pass Filtered White Noise) 23 points Volts 0 Time

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

  47. 1.25 1 x i 0 1 1 0 20 40 60 80 100 0 i 100 40 32 20 rx j 0 3 20 0 10 20 30 40 50 60 0 j 60 Autocorrelation of Random Bit StreamEach bit randomly Logic 1 or 0 1

  48. 1.25 1 x i 0 40 32 20 1 1 0 50 100 150 200 250 300 350 400 rx j 0 i 400 0 6 20 0 10 20 30 40 50 60 0 j 60 Bit Stream #2Logic 1 & 0 bursts of 20 bits (on average) 1

More Related