# Improving Modeling of Thermal Noise Propagation Through a Signal Chain

### Background

I recently asked this question over on Electrical Engineering Stack Exchange. On the advice of some commenters there, I've broken off those pieces which are appropriate for asking as smaller questions here.

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 (e.g. triboelectric, anthropogenic) in the future.

For now, I do so by modeling thermal noise simplistically as Gaussian-distributed white noise centered on \$0\$ and with \$V_{RMS} = \sqrt{4k_BTBR}\$. The voltage "waveforms" are produced by drawing \$N\$ sets of \$\textrm{sampling rate} \times \textrm{duration}\$ samples from the Gaussian distribution.

After some discussion with faculty advisors, I've decided that I'd like to improve upon this simplistic model (largely because we find it insufficiently close to data). One consideration I'd like to model is the signal chain, and that's what I'll discuss here.

### What I'd Like to Achieve

While I do have some hardware that I'm modeling against, I'd like to generalize this to both other current and future sets of hardware. In general, the hardware consists of:

1. The antennae. For each antenna, I know the polarization (vertical or horizontal, with respect to our coordinate system), dimensions, material composition, response as a function of frequency, and may also know the radiation pattern. I'd like to incorporate these where relevant.

2. A cable of known length and resistance, for each antenna.

3. Filters. A Butterworth band-pass filter, and in some cases a notch filter (so, we effectively have two Butterworth band-pass filters), for each antenna. I know that I can let $$B$$ in the $$V_{RMS}$$ calculation above be the integral under the filter transfer function across the pass-band (as-is, it assumes a perfect, rectangular pass-band).

4. LNA. A low-noise amplifier with known noise figure, for each antenna.

How can I implement more sophisticated hardware modeling, taking into account the detailed properties of the antennae and signal chain?

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).

The hardware modelling for the propagation of thermal noise in a receiver is well established as the process of determining the cascaded noise figure. Noise figure establishes the receiver sensitivity, and similarly cascaded analysis is done for linearity metrics such as 1 dB compression and 2 tone third order intercept, which help to establish the strongest signal before non-linearities disrupt reception. A complete analysis also includes the effect of phase noise, and cross modulation effects in frequency translation stages.

Here are other existing posts where cascaded noise figure and thermal noise propagation in a receiver is detailed further:

Detection Bandwidth for Noise Power Calculation

Is noise figure dependent on input noise power?

noise floor of attenuator

Detection Bandwidth for Noise Power Calculation

How to calculate a mixer noise?