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ECEN4503 Random Signals Lecture #39 21 April 2014 Dr. George ScheetsPowerPoint Presentation

ECEN4503 Random Signals Lecture #39 21 April 2014 Dr. George Scheets

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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) ]

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

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

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 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 σXAC Vrmson multi-meter

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

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

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