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

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

Standard Operating Procedure for Spring 2014 ECEN4503 If you're asked to find R XX (τ) Evaluate A[ x(t)x(t+τ) ] do not evaluate E[ X(t)X(t+τ) ]

You attach a multi-meter to this waveform & flip to volts DC. What is reading? n Zero

You attach a multi-meter to this waveform & flip to volts AC. What is reading? n 1 volt rms = σ n E[X 2 ] = σ 2 +E[X] 2

Shape of autocorrelation? n Triangle

Value of R XX (0)? τ (sec) Rxx(τ) 0 1

Value of Constant Term? τ (sec) Rxx(τ) 0 1 0

If 1,000 bps, what time τ does triangle disappear? τ (sec) Rxx(τ)

Power Spectrum S XX (f) n By Definition = Fourier Transforms of R XX (τ). n Units are watts/(Hertz) n Area under curve = Average Power u = E[X 2 ] = A[x(t) 2 ] = R XX (0) n Has same info as Autocorrelation u Different Format

Crosscorrelation R XY (τ) n = A[x(t)y(t+τ)] n = A[x(t)]A[y(t+τ)] iff x(t) & y(t+τ) are Stat. Independent u Beware correlations or periodicities n Fourier Transforms to Cross-Power spectrum S XY (f).

Ergodic Process X(t) volts n E[X] = A[x(t)] volts u Mean, Average, Average Value n V dc on multi-meter n E[X] 2 = A[x(t)] 2 volts 2 = constant term in Rxx(τ) n = Area of δ(f), using S XX (f) u (Normalized) D.C. power watts

Ergodic Process n E[X 2 ] = A[x(t) 2 ] volts 2 = Rxx(0) = Area under S XX (f) 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 -E[X]) 2 ] = A[(x(t) -A[x(t)]) 2 ] u Variance σ 2 X u (Normalized) AC Power watts n E[X 2 ] - E[X] 2 volts 2 n A[x(t) 2 ] - A[x(t)] 2 n Rxx(0) - Constant term n Area under S XX (f), excluding f = 0. n Standard Deviation σ X AC V rms on multi-meter

Discrete time White Noise & R XX (τ)

Autocorrelation & Power Spectrum of C.T. White Noise Rx(τ)Rx(τ) tau seconds 0 A G x (f) Hertz0 A watts/Hz Rx(τ) & Gx(f) form a Fourier Transform pair. They provide the same info in 2 different formats.

Autocorrelation & Power Spectrum of Band Limited C.T. White Noise R x (tau) tau seconds 0 A G x (f) Hertz0 A watts/Hz -W N Hz 2AW N 1/(2W N ) Average Power = ? D.C. Power = ? A.C. Power = ?

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

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