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# ECEN4503 Random Signals Lecture #39 21 April 2014 Dr. George Scheets - PowerPoint PPT Presentation

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 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

Shape of autocorrelation?

• Triangle

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

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

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).

• 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

• 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

• 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

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

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 Noise(Low Pass Filtered White Noise)

23 points

Volts

0

Time

Rxx

0

23

tau samples