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Keysight Technologies
Revisiting Mismatch Uncertainty with the
Rayleigh Distribution

White Paper

Abstract--This paper examines several important aspects of estimating mismatch
uncertainty, which is often a major component of the total uncertainty for RF and
microwave measurements. Expressions for mismatch correction and the correspond-
ing uncertainty are presented for reflection coefficients with known magnitude and
phase. For reflection coefficients with unknown phase, two scenarios are considered;
that is, when estimates exist for the reflection coefficient magnitude resulting in the
well-known U-shaped uncertainty distribution and when dealing with assumed reflec-
tion coefficient magnitude values. For the latter scenario, this paper demonstrates
that the reflection coefficient magnitude is best modeled with a Rayleigh distribution.
Measurement data is presented revealing a very good fit to the Rayleigh distribution.
Finally, this paper presents methods for estimating the Rayleigh distribution parameter
from information found in manufacturer's data sheets. The objective of this paper is to
provide comprehensive techniques for estimating realistic mismatch uncertainty, which
usually gives a three to six times lower estimate of mismatch uncertainty compared
with estimates from commonly used techniques.

2011 NCSL International Workshop and Symposium
Introduction
Speaker/Author: Michael Dobbert
Keysight Technologies
Santa Rosa, CA

Co-Author: Joe Gorin
Keysight Technologies
Santa Rosa, CA

Mismatch affects the accuracy of measurements made using RF and microwave equipment such as
power meters, signal analyzers, noise figure meters, network analyzers, high frequency oscilloscopes,
signal generators, attenuators, couplers, cables and adapters. The measurement uncertainty due to
mismatch is often a major component of the total uncertainty for RF and microwave measurements.

For RF and microwave systems, knowledge of the complex-valued quantities of source and load
reflection coefficients allows for correcting for mismatch with a corresponding uncertainty. However,
for many measurements, only reflection coefficient magnitude is known. The lack of phase informa-
tion precludes the ability to correct for mismatch and is a source of measurement error when making
power measurements. The distribution of errors due to mismatch when dealing with unknown phase
is often associated with the well-known U-shaped probability distribution. However, this component
of uncertainty is only part of the picture. The total mismatch uncertainty must also consider the errors
associated with reflection coefficient magnitude. The total uncertainty may be determined by using
an estimate and an associated uncertainty, or by assigning a probability density function (PDF) to the
reflection coefficient magnitude. This paper demonstrates both.

Assigning a probability density function to the reflection coefficient magnitude finds its use when
reflection coefficient magnitude data is available from a manufacturer's data sheet or a pooled data
set. The natural tendency is for reflection coefficient magnitude to take on a Rayleigh probability
distribution. This paper gives several methods for estimating the Rayleigh distribution parameter from
available data enabling the estimation of mismatch uncertainty.

2011 NCSL International Workshop and Symposium
2. Mismatch
Mismatch is the term used to describe the consequence of traveling waves
that reflect off the various structures within RF and microwave transmission
systems.

Transmission Line

Figure 1. A generator connected to a load
through a transmission line.

Consider a signal generator with output impedance Z s , connected to a load,
Z l , through a lossless transmission line with characteristic impedance
Z 0 , as shown in Figure 1. Of interest with this circuit are the forward and
reverse traveling voltage waves on the transmission line. From the generator, a
traveling voltage wave moves along the transmission line towards the load. If
the load impedance does not exactly match the characteristic impedance of the
transmission line, a second traveling voltage wave reflected off the load heads
back towards the generator. The magnitude of this reflected, reverse traveling
wave depends on the load impedance. For many microwave power measure-
ments, the magnitude of the reflected wave is small, ideally, relative to the
forward wave. The reverse traveling voltage wave, upon reaching the generator,
re-reflects back in the forward direction adding to the traveling voltage wave
from the generator. The magnitude of the re-reflected voltage wave depends
on the generator impedance. The voltage wave from the generator and the
re-reflected voltage wave add either constructively or destructively, depending
upon the relative phase of the two traveling waves. The net magnitude of the
forward and reverse traveling voltage waves are the result of multiple reflections
and rereflections. In a system such as this, the voltage at the load depends on
the load impedance, the impedance of the generator and propagation delay of
the transmission line.

Figure 2. Flow diagram for Figure 1.

2011 NCSL International Workshop and Symposium

3
The flow diagram in Figure 2 models the circuit from Figure 1 where,

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