How to determine noise figure: part 4

Two incompatible definitions of noise factor can lead to confusion, which you can alleviate by understanding where the differences lie.

In part 3 of this series, we discussed an equivalent noise temperature T_E and corresponding noise factor F_STD as defined in an IEEE standard. Then we described how to convert from one to another. Given T_E, where T₀ equals the reference temperature 290 K, find F_STD as follows:

And given F_STD, find T_E as follows:

So if we know either spec — we find it on a datasheet, for example — we can easily obtain the other.

We also described an alternative noise factor F_SNR that equals the input signal-to-noise ratio (SNR) divided by the output SNR. Can we convert between F_SNR and F_STD?
No. As we discussed last time, F_STD can apply to a single-port device such as a noise source or a signal generator. The latter will generate noise along with the desired signal at its output, and we can use a spectrum analyzer to measure the SNR at the output, but there is no input SNR with which to compare it.

Last time we described extending the noise-temperature concept from a one-port network to a two-port network. Can’t we do something similar here to try to relate F_STD and T_SNR?
Let’s take a look at Figure 1. We have a DUT with an equivalent noise temperature T_E and a signal generator with its own equivalent noise temperature — we’ll call it T_S, where the subscript stands for source.

We know from Equation 10 in part 2 of this series that the noise added by a component equals its equivalent noise temperature times Boltzmann’s constant (k = 1.38×10^-23 J/K) times the measurement bandwidth (B). Therefore, we can model our signal-generator output as a pure sine wave S_IN plus a noise power level N_IN = T_SkB. Similarly, we can model our DUT as an ideal amplifier with gain G and a noise level T_SkB added at the input, producing an output signal S_OUT and noise level N_OUT. We can now calculate SNR_IN, the SNR at the input to the DUT:

Noting that the DUT will amplify the input signal and noise plus its own input-referred noise by G, we can compute the output SNR:

Now we can write the equation for F_SNR in terms of noise temperatures:

This equation looks similar to Equation 1 above for FSTD, which repeats Equation 7 from part 3.
They do look similar but on closer inspection, it’s clear that they are only equivalent when T_E equals absolute zero or when T_S = T₀ = 290 K. Figure 2 shows how F_STD varies from F_SNR for various values of T_S not equaling 290 K. The blue line shows F_SNR. The upper and lower red lines show F_STD for T_S equaling 270 K and 310 K, respectively, and error rates are only about ±5%. However, the error reaches nearly 20% for T_S equaling 390 K and exceeds 40% for T_S equaling 190 K (orange lines).

So where does this leave us — which version of noise factor should we use?
Good question. The F_SNR version is intuitive and gives us a quantitative result, but it doesn’t apply to one-port networks. The article “Noise Figure One and Two, Friis and IEEE” provides an in depth analysis of your question. One possible takeaway? Use the SNR version for two-port devices and use the equivalent noise temperature for one-port devices, immediately converting T_STD to T_E, using Equation 1, whenever you come across it.

We haven’t talked about noise figure here in part 4.
Right. Just to review, given either version of noise factor F, we can calculate noise figure NF in dB as follows:

Where can I learn more about making noise-figure measurements?
See the “For further reading” list below as well as the related EE World content.

For further reading
Noise Figure Measurement Methods (Anritsu)
What is Noise Figure? (Copper Mountain Technologies)
The Essential Noise Figure Measurement Guide – Measuring Noise Figure with Signal Analyzers – Part 1 (Keysight)
The Y Factor Technique for Noise Figure Measurements (Rohde & Schwarz)
Noise Figure: Overview of Noise Measurement Methods (Tektronix)

Related EE World content
Using a scope to read signals obscured by noise
Understanding noise in electronic instrumentation
Key considerations for RF power measurement equipment
Should I use a spectrum, signal, or vector network analyzer? part 1
Overview of electrical measuring instruments, Part 3
The difference between arbitrary function generators and arbitrary waveform generators