Revisiting Mismatch Uncertainty with the Rayleigh Distribution
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Transcript Revisiting Mismatch Uncertainty with the Rayleigh Distribution
Revisiting Mismatch
Uncertainty with the
Rayleigh Distribution
1
Michael Dobbert
Joe Gorin
August 24, 2011
Overview
• Mismatch Model.
• The Rayleigh distribution.
• Finding the Rayleigh parameter.
• Is the Rayleigh assumption reasonable?
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Mismatch Error
~
Zl
generator
load
Pa PgZ 0
3
1 l
2
1 g l
2
Power equation
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Mismatch Model
2
M 1 g l
Im
Mismatch error equation.
g .re
, g .im
?
Re
Circular symmetry about the origin.
uM 2 2 ul .re ug .re
Mismatch uncertainty equation.
(real components)
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Reflection Coefficient Uncertainty (fixed)
Im
u g .re
g
ul .re
l
Re
re
2
2
pdf
u M 2 l g
5
0
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.
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Reflection Coefficient Uncertainty (fixed)
Im
u g .re
g
ul .re
l
Re
re
2
2
pdf
u M 2 l g
7
0
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.
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Reflection Coefficient Uncertainty (measured)
Im
Re
re
8
u re
2u 2
2
2
pdf
0
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Reflection Coefficient Uncertainty (Uniform)
Im
Re
u re
2
re pdf
9
0
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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Reflection Coefficient Uncertainty (Rayleigh)
Im
ure
Re
σ is the Rayleigh distribution parameter.
re
pdf
0
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Finding the Rayleigh Parameter
Rx 1 e
11
x2
2
2
Rayleigh cumulative distribution function
R(x) parameter
Computation
Max (99.73rd percentile)
95th percentile
95
2 ln20
80th percentile
80
2 ln5
Mean
mean
Median (50th percentile)
max
3.439
ug .re g
ul .re l
2
median
2 ln2
uM 2 2 l g
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Is the Rayleigh Assumption Reasonable?
eq re s11.re i im s11.im
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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