### 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?
2
August 24, 2011
Mismatch Error
~
Zl
generator
Pa  PgZ 0
3
1  l
2
1  g l
2
Power equation
August 24, 2011
Mismatch Model
2
M  1  g l
Im
Mismatch error equation.

g .re
, g .im 
?
Re
uM   2 2  ul .re  ug .re 
Mismatch uncertainty equation.
(real components)
4
August 24, 2011
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.
August 24, 2011
6
August 24, 2011
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

August 24, 2011
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.
August 24, 2011
Reflection Coefficient Uncertainty (Rayleigh)
Im

ure   
Re
σ is the Rayleigh distribution parameter.
re
pdf
0
10
August 24, 2011
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
August 24, 2011
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.
12
August 24, 2011
Power Sensor
13
August 24, 2011
Signal Generator
14
August 24, 2011
Signal Analyzer
15
August 24, 2011
Signal Generator Reflection Coefficient Real
Component
16
August 24, 2011
Comparing Models
Mismatch uncertainty is roughly proportional to the reflection coefficient median value.
17
August 24, 2011
Rayleigh Conclusion
• Reasonable model,
• Straight forward to calculate,
• Significantly lower estimate of mismatch uncertainty.
18
August 24, 2011
19
August 24, 2011
```