ECEN4503 Random Signals Lecture #30 31 March 2014 Dr. George Scheets n Problems 8.7a & b, 8.11, 8.12a-c (1st Edition) n Problems 8.11a&b, 8.15, 8.16 (2nd Edition)

ECEN4503 Random Signals Lecture #31 2 April 2014 Dr. George Scheets n Problems: 8.15, 8.18, 8.24 (1st Edition) n Problems: 8.2, 8.19, 8.22 (2nd Edition) n Quiz #8, 4 April, 2nd Order PDF's & Autocorrelation n Exam #2, 18 April

2014 OSU ECE Spring Banquet n Hosted by Student Branch of IEEE n Wednesday, 16 April, at Meditations n Doors open at 5:30 pm, meal at 6:00 pm n Cash Bar n Sign up in ES201 to reserve your seat(s) n $20 value! n $5 if pay in advance and resume submitted to < 5:00 pm, 14 April. Otherwise $8. n Speaker: Dr. Legand Burge, USAF (retired) Dean of Engineering, Tuskegee University n Dress is Business Casual n Many door prizes available! +6 points extra credit n All are invited! Sponsored in part by:

Ergodic Process X(t) volts n E[X] = A[x(t)] volts u Mean u Average u Average Value n V dc on multi-meter n E[X] 2 = A[x(t)] 2 volts 2 u (Normalized) D.C. power watts

Ergodic Process n E[X 2 ] = A[x(t) 2 ] volts 2 u 2nd Moment u (Normalized) Average Power watts u (Normalized) Total Power watts u (Normalized) Average Total Power watts u (Normalized) Total Average Power watts

Ergodic Process n E[X 2 ] - E[X] 2 volts 2 n A[x(t) 2 ] - A[x(t)] 2 u Variance σ 2 X u (Normalized) AC Power watts n E[(X -E[X]) 2 ] = A[(x(t) -A[x(t)]) 2 ] u Standard Deviation σ X AC V rms on multi-meter volts

Histogram of Sinusoid Voltages

PDF of a 3vp Sinusoid Area under PDF E[X 2 ] Find PDF of voltage by treating Time as Uniform RV. Then map time → voltage. A[*] = E[*] when Ergodic.

Voltage PDF for Clean Sinusoid n If x(t) = α cos(2πβt + θ) then f X (x) = 1 / [π(α 2 - x 2 ) 0.5 ]; - α < x < α

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