Ecen4503 random signals lecture 39 21 april 2014 dr george scheets
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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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Ecen4503 random signals lecture 39 21 april 2014 dr george scheets

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 signals lecture 40 23 april 2014 dr george scheets

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 procedure for spring 2014 ecen4503

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

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


You attach a multi meter to this waveform flip to volts ac what is reading

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


Shape of autocorrelation

1.25

1

x

i

0

1

1

0

20

40

60

80

100

0

i

100

Shape of autocorrelation?

  • Triangle


Value of r xx 0

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


Value of constant term

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


If 1 000 bps what time does triangle disappear

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


Power spectrum s xx f

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 r xy

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

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

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 process1

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


Discrete time white noise r xx

Discrete time White Noise & RXX(τ)


Autocorrelation power spectrum of c t white noise

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


Autocorrelation power spectrum of band limited c t white noise

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


255 point noise waveform low pass filtered white noise

255 point Noise Waveform(Low Pass Filtered White Noise)

23 points

Volts

0

Time


Autocorrelation estimate of low pass filtered white noise

Autocorrelation Estimate of Low Pass Filtered White Noise

Rxx

0

23

tau samples


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