**Originally posted Oct 23, 2013**

**Hello, Lord Rayleigh; Goodbye, Herr Doktor Gauss**

Joe Gorin and Michael Dobbert have done some impressive work that addresses the subject of **better measurements** from an unusual but very important direction. The result of their work is good news for RF and microwave engineers everywhere.

Here’s the short version of the story: for all of us, amplitude accuracy is a fundamental measurement goal that is often uncertain. Going deeper: in many RF and microwave signal measurements, the biggest single element of measurement uncertainty is due to impedance mismatch and its effect on power delivery from the DUT to the analyzer.

Mismatch error is calculated from the reflection coefficient specified by equipment manufacturers. That reflection coefficient is usually specified for frequency ranges in terms of a single magnitude value. ** The traditional approach to calculating mismatch accuracy using that reflection coefficient overestimates mismatch error by a factor of three to six. **This unduly pessimistic approach has been accepted practice for decades.

In the real world of RF/microwave engineering, the benefits of reduced uncertainty can lead to tightened DUT specifications or can be traded for productivity benefits such as increased yield and improved throughput.

Gorin and Dobbert have taken a statistical approach to accurately estimating uncertainty bounds, basing their model on a better understanding of the way the reflection coefficient behaves in analyzers, power sensors and signal generators. They then validated this approach by making a very large number of measurements to see how the measurements fit their model.

The new statistical model of reflection coefficient—and thus mismatch error—is based on two points:

- Represented in a complex I/Q plane, the real and imaginary parts of the reflection coefficient will have Gaussian distributions
- Because the probability density of the magnitude of both complex parts of the reflection coefficient is Gaussian, the probability density of the magnitude of the reflection coefficient will have a Rayleigh distribution instead of a Gaussian shape

This reflection coefficient behavior and the resulting Rayleigh distribution is shown below, annotated with the 95^{th} percentile measurement uncertainty bounds:

A complex I/Q representation of reflection coefficient with Gaussian distribution in each I/Q part (left) and the resulting Rayleigh distribution of the probability density of the magnitude of the reflection coefficient.

Of course, the test of a good theory or model is experimental evidence. There is room here for only a single example, but the real-world data fit the model well in all tested cases involving signal analyzers, signal generators and power sensors.

Comparing predicted Rayleigh (red) and measured (blue) values of the cumulative distribution function for the Agilent PXA signal analyzer preamplifier from 3.5 to 26.5 GHz. The fit error (green) is very low, validating the assumption of the Rayleigh distribution.

All this discussion of estimated error bounds can obscure the fact that this new work does not change the actual measurement accuracy of instruments or sensors. Mismatch uncertainties have not changed; however, traditional methods have overestimated them by a factor of three to six.

It is also worth noting that while Gorin and Dobbert validated their predictions on Agilent equipment, the predictions should apply broadly to RF and microwave signal generators, analyzers and power sensors.

For more information, see the *Microwave Journal* article “A New Understanding of Mismatch Error”and Agilent application note 1449-3 *Fundamentals of RF and Microwave Power Measurements (Part 3) Power Measurement Uncertainty per International Guides*. A spreadsheet-based uncertainty calculator is also available from Agilent to correspond to the calculations in the application note.