# Uncertainty Analysis of a Hypothetical RF Noise Measurement System

By David Archer

## Introduction

The purpose of this article is to develop an uncertainty analysis for the noise measurement system described in the course MEAS403 – Basic to RF and Microwave Noise Measurement. In that course a hypothetical noise measurement system was described that used the Y-factor method to measure the noise temperature of an unknown noise source.  This system is based on a system that was actually implemented, with some of the details omitted. We will go into a fair amount of detail in the analysis, but there still will be some questions, such as details concerning the receiver, that will be left for a future article.

Also, after presenting most of the uncertainty analysis, we will leave one part of the analysis unfinished as an exercise so that experience with uncertainty analysis can be gained by the reader.  We may provide the solution as a future article if there is some interest.

Before starting in on the uncertainty analysis, we will review some of the contributing factors to the noise measurement.

## Noise Temperature and ENR

In the course we learned that the definition of noise temperature is the power-per-unit-bandwidth divided by Boltzmann’s constant, i.e. the relationship of the power received by a receiver with bandwidth B and the noise temperature is where k=Boltzmann’s constant=1.38044 x 10-23 joules per Kelvin.

We also pointed out that noise temperature is not quite the same as the physical temperature of a load that would deliver the same power and that the relationship between noise temperature and physical temperature for a load is where and h=Planck’s constant=6.62517 x 10-34 joule second.

If the uncertainty that is desired for the measurement system is such that this correction is necessary, it should be included in the analysis of the system.

Many noise sources are not calibrated in noise temperature, but in what is called Excess Noise Ratio or ENR. The definition of ENR is where 290K is a reference temperature that is used in the definition, ENR is typically expressed in dB .

## The Hypothetical System

The following diagram schematically describes the hypothetical noise measurement system. in the course we derived the following measurement equation for the noise temperature of the device under test: where Tr, the noise temperature of the receiver is for completeness, the “noise efficiencies” , i=DUT,h,c are and the mismatch terms Ni are where and is the reflection coefficient of the receiver.

These equations describe how to determine the noise temperature of the device under test. Of course all the reflection coefficients and S-parameters of the system must be measured in some manner before hand, with something like a network analyzer, and the noise temperature of the “hot” and “cold” standards, as well as the ambient temperature, must be known.

Obviously this would be an automated system, nobody would want to calculate the results by hand, but the advantage of such a system is that a complete uncertainty analysis could also be automated into the measurement. We will no turn to the subject of the uncertainty analysis.

## Uncertainty Analysis

In general when one has a measurement equation where M is the measureand and the xi are the component measurements that are required to determine its value, and each of the xi has an associated uncertainty , the RSS uncertainty in the measureand can be written This the technique that will be used to determine the various terms in the uncertainty analysis. We will not go into a great detail on how the individual partial derivatives are performed, but just state the results.

### Uncertainty in the Noise Temperature of the DUT

Performing this type of analysis on the measurement equation, the following expression for the uncertainty in the DUT noise temperature is obtained. where        This then is the expression for the uncertainty of the device under test’s noise temperature, unfortunately we are not finished. There are still eight terms in the above equations that we do not know. Each of these eight terms will require a separate analysis, these eight terms are Of these terms, the uncertainty in the cold standard noise temperature is probably available as some sort of calibration certificate, or the result of some other uncertainty analysis, and the uncertainty in the ambient temperature is a function of the device used to measure the ambient temperature.

The uncertainty of the Y-factor measurement is a function of the linearity of the receiver, and so can be easily determined as the power ratio measurement uncertainty of the receiver.

The remaining five terms each require an uncertainty analysis of their own. That will be the task of the remaining sections of this document.

### Uncertainty in the Receiver Noise

Using the same analysis technique, the uncertainty associated with the receiver noise is where        Again the uncertainty in the Y-factor is the receiver power ratio measurement accuracy,  and the uncertainty in the noise temperature of the “hot” noise standard either comes from a calibration certificate or a separate uncertainty analysis. There are four uncertainties associated with the hot standard we have not seen yet. At this point it is worth noting that in the course there was a tradeoff between the time it takes to make the measurement and the amount of drift in the receiver noise temperature. If the receiver noise drifts say a factor of 10 less than the calculated uncertainty over the course of a measurement, it probably can be ignored. Otherwise it is a tradeoff of measurement uncertainty due to the drift in the noise temperature vs the uncertainty associated with not having enough samples to get a good average. This tradeoff can only be determined empirically.

### Uncertainty Analysis of the Mismatch Terms

At this stage we have all but six uncertainty terms, the three ’s and the three ’s. These are actually only two additional analysis, but there is some difficulty. Since these parameters are measured typically with some sort of network analyzer, and network analyzer uncertainties are usually expressed in terms of magnitude and phase uncertainties, we need to express these in terms of magnitude an phase. We will start with the mismatch terms: The numerator is already expressed in terms of magnitudes, to express this in terms of  magnitude and phase, we will let and and defining results in We can now express the uncertainty in this quantity as follows: where    It should be noted here that in the above equations all angles and angle uncertainties are expressed in radians.  The uncertainty terms that we do know above can be determined from network analyzer specifications. These terms are We are almost done with the mismatch term, the only quantities we don’t know at this point are the two uncertainties These can be determined from network analyzer specifications if the reflection coefficient is measured directly through the switch. If not, the magnitude and phase must be written in terms of the four S-parameter magnitudes and phases, and the magnitude and phase of the noise standard reflection coefficient, a total of 10 terms.

Starting with the expression for the magnitude one obtains where    It is convenient to write the uncertainty in the magnitude in the following form To facilitate the writing of the 10 coefficients we make the following definitions   it then follows that         This completes the derivation of the uncertainty in the magnitude of , what remains is to determine the uncertainty in the phase to complete the uncertainty analysis of the mismatch terms. Again the phase is a function of 10 terms.

One can write for the phase where  and    The uncertainty in the phase can no be written where  and    This completes the analysis for the mismatch terms, all that remains is the uncertainty analysis of the noise efficiency.

### Analysis of the Noise Efficiency Terms This expression has to be rewritten in terms of magnitude and phase before the appropriate partial derivatives can be taken to complete the uncertainty analysis.

We are going to define then so that Now we know and that solves the problem of the uncertainty analysis of this hypothetical noise system provided all the above partial derivatives are calculated. That will be left as an exercise, after all this is a training site. There should be enough examples here to allow one to do this. The procedure is to write the quantity G as a function of the magnitude and phase of all its components, and then take all the partial derivatives. It should only take a few pages of complex algebra.

## Conclusion

Except for the last part that is left as an exercise, the complete uncertainty analysis of the hypothetical noise system has been presented. One could implement a real system with the information provided here.