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

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

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) ]

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

0 20 40 60 80 1001

0

11.25

1

xi

1000 i

Zero

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

0 20 40 60 80 1001

0

11.25

1

xi

1000 i

1 volt rms = σ E[X2] = σ2 +E[X]2

Shape of autocorrelation?

0 20 40 60 80 1001

0

11.25

1

xi

1000 i

Triangle

Value of RXX(0)?

0 20 40 60 80 1001

0

11.25

1

xi

1000 i

τ (sec)

Rxx(τ)

0

1

Value of Constant Term?

0 20 40 60 80 1001

0

11.25

1

xi

1000 i

τ (sec)

Rxx(τ)

0

1

0

If 1,000 bps,what time τ does triangle disappear?

0 20 40 60 80 1001

0

11.25

1

xi

1000 i

τ (sec)

Rxx(τ)

0

1

0

0.001-0.001

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

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).

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

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

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 σX

AC Vrms on multi-meter

Discrete time White Noise & RXX(τ)

Autocorrelation & Power Spectrum of C.T. White Noise

Rx(τ)

tau seconds0

A

Gx(f)

Hertz0

A watts/Hz

Rx(τ) & Gx(f) form a Fourier Transform pair.

They provide the same infoin 2 different formats.

Autocorrelation & Power Spectrum of Band Limited C.T. White Noise

Rx(tau)

tau seconds0

A

Gx(f)

Hertz0

A watts/Hz

-WN Hz

2AWN

1/(2WN)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