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
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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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