Revisiting Mismatch Uncertainty with the

Rayleigh Distribution

Michael DobbertJoe Gorin

August 24, 20111

Overview

• Mismatch Model.

• The Rayleigh distribution.

• Finding the Rayleigh parameter.

• Is the Rayleigh assumption reasonable?

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Mismatch Error

~ Zl

2

2

01

1

lg

lgZa PP

Power equation

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generator load

3

Mismatch Model

21 lgM Mismatch error equation.

imgreg .. ,

?

regrel uuMu ..22

Circular symmetry about the origin.Re

Im

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Mismatch uncertainty equation.

(real components)

4

Reflection Coefficient Uncertainty (fixed)

2

.

g

regu

Harris, I. A. and Warner, F. L., Re-examination of mismatch uncertainty when measuring microwave power and attenuation, Microwaves, Optics and Antennas, IEE Proceedings ~ H, February 1981, Volume 128, Issue 1.

0

pdfre

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Re

Im

glMu 2

2

.l

relu

5

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Reflection Coefficient Uncertainty (fixed)

2

.

g

regu

Harris, I. A. and Warner, F. L., Re-examination of mismatch uncertainty when measuring microwave power and attenuation, Microwaves, Optics and Antennas, IEE Proceedings ~ H, February 1981, Volume 128, Issue 1.

0

pdfre

August 24, 2011

Re

Im

glMu 2

2

.l

relu

7

Reflection Coefficient Uncertainty (measured)

2

2 22

uu re

0

pdfre

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Re

Im

8

Reflection Coefficient Uncertainty (Uniform)

2

reu

0

pdfre

Fundamentals of RF and Microwave Power Measurements (Part 3) Power Measurement Uncertainty per International Guides. Agilent Application Note AN 1449-3, April 5, 2011, literature number 5988-9215EN.

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Re

Im

9

Reflection Coefficient Uncertainty (Rayleigh)

reu

0

pdfre

σ is the Rayleigh distribution parameter.

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Re

Im

10

Finding the Rayleigh Parameter

2

2

21 x

exR

Rayleigh cumulative distribution function

R(x) parameter Computation

Max (99.73rd percentile)

95th percentile

80th percentile

Mean

Median (50th percentile)

439.3max

20ln295

5ln280

2

mean

2ln2median

glMu 22

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gregu .

lrelu .

11

Is the Rayleigh Assumption Reasonable?

imimrereeq sis .11.11

Mullen, J. A. and Pritchard, W. L., The Statistical Prediction of Voltage Standing-Wave Ratio, Microwave Theory and Techniques, IRE Transactions, April 1957, Volume 5, Issue 2.

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Power Sensor

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Signal Generator

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Signal Analyzer

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Signal Generator Reflection Coefficient Real Component

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Comparing Models

Mismatch uncertainty is roughly proportional to the reflection coefficient median value.

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Rayleigh Conclusion

• Reasonable model,

• Straight forward to calculate,

• Significantly lower estimate of mismatch uncertainty.

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