# Improving Modeling of Thermal Noise in Radio Antennae

### Background

I'm attempting to write a physics simulation code, one portion of which involves the simulation of the voltage observed at a set of $$\N\$$ radio antennae immersed in some medium due to thermal (or "Johnson-Nyquist") noise, and which may expand to include other noise sources (such as triboelectric and anthropogenic) in the future.

For now, I do this by modeling thermal noise simplistically as Gaussian-distributed white noise centered on 0, and with $$\V_{RMS} = \sqrt{4k_BTBR}\$$ (see), where $$\B\$$ is the bandwidth of the antenna, $$\T\$$ is its temperature in Kelvin, and $$\R\$$ is its resistance. The voltage "waveforms" are produced by drawing $$\N\$$ sets of $$\\textrm{sampling rate} \times \textrm{duration}\$$ samples from the Gaussian distribution. This simplification is rooted in the assumption that the electric field vectors associated with each e.g. molecule in the antenna are 2 -dimensional random phasors, and in the limit of a large number of molecules follow Gaussian statistics (and involves other additional approximations, such as assuming a flat frequency response).

After some discussion with various faculty, I've decided that I'd like to improve upon this simplistic model. Unfortunately, however, I'm not sure where to start, and I can't seem to turn up any literature on this beyond simplistic models such as that linked previously (perhaps in large part due to simply not knowing what to search for).

### What I'd Like to Achieve

1. Firstly, I'd like to incorporate more sophisticated hardware modeling. In principle, I know the gain, response (as a function of frequency), dimensions, material composition, and relative location of each antenna. I'd also like to take into account the signal chain, which includes a low-noise amplifier whose noise figure and gain are known. It has been suggested that I approach this using methods Finite Element Analysis.

2. I'd like to more carefully consider the medium in which the antennae are immersed, to the extent to which it is relevant. In principle, I know the temperature and material composition of the medium, in addition to its attenuation properties, refractive index, etc.

3. Thirdly, discussion with faculty advisors and other knowledgeable people has yielded suggestions for various other considerations, most of which I'm not sure the relevance of nor how to implement. It has been suggested that I take into consideration radiative transport modeling, Einstein coefficients, and "modeling the noise beam-by-beam". I know what these are individually, but not how they should be taken into consideration here. I'm also not sure what other perhaps important considerations exist that I haven't considered, nor to what order these various considerations are important.

I understand that the simplistic Gaussian model comes from approximating the electric field vectors (phasors, if you like) due to thermal noise as $$\2\$$-dimensional, and considering the central limiting behavior of a large number of such electric field vectors (e.g. due to a large number of molecules). I'm told that the distribution is not truly Gaussian when you instead consider more realistic three-dimensional vectors. What, then, does the distribution look like? How does it change when, instead of assuming the antennae to be ideal resistors, I take into account both resistance and reactance? How would I do so?

How can I implement more sophisticated hardware modeling, taking into account the detailed properties of the antennae, signal chain, and media in which they are immersed? What is the relevance of radiative transport modeling and the Einstein coefficients? What other considerations should be involved so as to improve the model?

I understand that this is likely a rather involved task, and that what is sufficiently complex in any physics model is subjective. I'm looking for (ideally mathematically-motivated) suggestions for how to move forward, and resources for further reading (ideally accessible to a (perhaps slightly advanced) upper-level undergraduate). As of now, I simply don't know where to begin (nor what would be a good framework to begin with that would allow for additional layers of complexity to be easily added on).

I mentioned that we find the simplistic model to be insufficiently close to data. I've created a plot demonstrating this. The red and blue dotted lines are data (in red, the distribution of voltage values observed at the antennae, in blue, the distribution of their magnitude (i.e. absolute value)), and the green and pink lines are simulation (for each real data point, I generated a simulated data point according to the model and added it to the histogram).

These differences are not localized in time, and do not appear to be due to any equipment issues/calibration issues/etc.

It occurs to me that in some texts the simple Gaussian model is derived by approximating the electric field vectors due to thermal noise as $$\2\$$-dimensional, complex phasors whose components are identically and independently normally distributed. Their magnitude is then Rayleigh distributed, however when observed at an antenna only the real part is recorded, hence the voltage values are Gaussian distributed, and their magnitude is distributed according to a folded Gaussian.

In practice, of course, the electric field vectors are $3$-dimensional, and their magnitudes are distributed according to a "three-dimensional" Rayleigh distribution. In the $$\2\$$D bivariate normal model, the real part (corresponding to the magnitude of the in-phase part of the electric field vector) is what is recorded when the signal is digitized by the antenna, and it is Gaussian distributed. In the more accurate $$\3\$$D model, what is recorded, and how is it distributed? Below, I've plotted the magnitude distribution of a $$\3\$$D vector with IID Gaussian-distributed components in red, the magnitude of a $$\2\$$D complex phasor with IID Gaussian-distributed components in pink, and the distribution of the real parts of those complex phasors in blue (for 1E+9 randomly-generated phasors, and some arbitrary value of $$\\sigma\$$):

• It's also possible some signals aspects of this question may be askable (or even already answered) in Signal Processing SE. Search there for antenna for example. The only thing I know about thermal noise can be found in Why doesn't thermal radio emission from a DSN "hot dish" completely swamp the benefits of a cold LNA? Spoiler alert: emissivity!
– uhoh
Jun 18 at 23:50
• For section 2, is there any particular way you expect the medium to influence the noise level at the input to the LNA? It might be that it has a very low impact, and the thermal noise floor is dominating it, enough that nobody cares to model the weak effect. Of course you want to improve things, modeling the weak things too is kinda how you do that, but unless you have a really hostile environment in mind it might be better to ignore. Jun 22 at 23:07
• Also, have you got any hardware that you are modeling and measuring against? You might find your model is already good enough, or learn something by how close it is. Jun 22 at 23:09
• @RMcHenry I do not have a theoretical model for how I expect the medium to influence the noise level at the interface to the LNA, unfortunately. As for your second question, yes and no. I would like this model to eventually allow someone to input some arbitrary hardware configuration consisting of antennae, filters & amplifiers, such that it is applicable to current and future hardware. We do have a couple sets of hardware that I am measuring against, however... Jun 23 at 15:59
• @RMcHenry ...All of this hardware (current and future) will be operating at Antarctic temperatures, ranging from 184-213 Kelvin (but relatively constant in time). We find the model to be close but insufficiently so to measurement, however it is difficult using data to unravel what effects contribute to that deviation (and to what order each effect contributes). Jun 23 at 16:02

This is not an answer but an extended comment.

It occurs to me that in some texts the simple Gaussian model is derived by approximating the electric field vectors due to thermal noise as 2-dimensional, complex phasors whose components are identically and independently normally distributed. Their magnitude is then Rayleigh distributed, however when observed at an antenna only the real part is recorded, hence the voltage values are Gaussian distributed, and their magnitude is distributed according to a folded Gaussian.

In practice, of course, the electric field vectors are $3$-dimensional...

You seem to be confusing the 2 dimensional nature of a band-limited signal (typically represented as a 2D phasor giving the in-phase and quadrature channels) with the 3 dimensional nature of the EM fields.

Assuming we are talking about far-fields, if you have reception from one direction you have a further 2 dimensions due to polarization. If you want to account for reception from all directions, typically you can do some sort of modal expansion of the fields around the antenna and consider the coupling of the antenna to each mode. The response to each mode, for a band-limited signal, will be a 2D phasor. There is a lot in the literature about modal analysis of antennas.

In your first diagram, I don't understand how the blue trace, that peaks at zero, can be the magnitude of the red trace that peaks away from zero.

• Thanks for the input. Yes, I do seem to be a bit confused about the 2D complex phasor, 3D electric field, and their relationship (this is perhaps stretching the limit of my undergraduate level knowledge). I'll try reading the literature on modal analysis. Jun 26 at 0:52

The distribution of values shown in your plot has a significant mean and asymmetry which indicates that there is some spectral component to the delta between your model and measurements. This means that there is some source other than thermal noise visible in your data.

It could be a real incoming signal at the precise frequency of your receiver, some interference from itself, or some kind of subtle malfunction. Check if this difference is within the expected accuracy of your equipment. If you want to further work on eliminating or modeling it, look at the spectrum of a long streak of data and spectrograms showing the signal over time frequency.