Revisiting Mismatch Uncertainty with the Rayleigh Distribution

Report
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.
uM   2 2  ul .re  ug .re 
Mismatch uncertainty equation.
(real components)
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Reflection Coefficient Uncertainty (fixed)
Im

u g .re  
g
ul .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
ul .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.
August 24, 2011
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

ure   
Re
σ is the Rayleigh distribution parameter.
re
pdf
0
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Finding the Rayleigh Parameter
Rx  1  e
11
x2
 2
2
Rayleigh cumulative distribution function
R(x) parameter
Computation
Max (99.73rd percentile)

95th percentile

95
2 ln20
80th percentile

80
2 ln5
Mean
  mean
Median (50th percentile)

max
3.439
ug .re    g
ul .re    l
2

median
2 ln2
uM   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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