Improving Modeling of Thermal Noise Behavior at Antenna

Question

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. Namely, I'm asking separately about modeling the signal chain, and a more accurate theoretical model of the behavior of noise itself, and the resulting antenna response.

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). I'd like to begin by developing a more sophisticated theoretical model of the behavior of thermal noise, and how it is observed at the antenna, and that's what I'll discuss here.

Where My Understanding Lies

Much of this is admittedly motivated by my lack of a fully confident understanding of what, precisely, is actually measured by an antenna. Here's my understanding of the problem thus far:

In most texts, the Gaussian model is developed by approximating the electric field vectors due to thermal noise as complex phasors whose real and imaginary components are identically and independently Gaussian distributed (this coming, obviously, from the central limiting behavior of the contributions of a large number of e.g. atoms or molecules). When the phasor is observed at the antenna, the real part is what is observed and digitized (corresponding to the in-phase part of the electric field magnitude). The magnitude of the real part is distributed according to a folded Gaussian, while the magnitude of the phasor itself is Rayleigh distributed.

The actual electric field vectors, however, are \$3\$-dimensional, of course, and their magnitudes are distributed according to a "\$3\$D" Rayleigh distribution (this likely has a more precise name, however I'm unable to find one). Below, I've plotted the distribution of magnitude of the real part of a large number number of complex phasors (in blue), the distribution of the "full" magnitude of these phasors (in pink), and the distribution of the magnitude of a large number of \$3\$D vectors (in red):

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I had thought that what is measured at the antenna was the magnitude of this \$3D\$ electric field vector, which follows a very different distribution from the real part of the complex phasor.

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What I'd Like to Achieve

1. What is measured at the antenna, and how is it distributed? That is, when I look at the voltage measured due to thermal noise (or more generally), what can I expect to see? Is it the magnitude of the electric field vector, or the real part of the complex phasor? Why?

2. How is this effected by the properties of the antenna? Suppose I know the dimensions, material composition, polarization, response (as a function of frequency), and radiation pattern of the antenna. How does this effect the voltage that is observed due to thermal noise, if at all? I know for one that, in the simple Gaussian model, an ideal rectangular passband is assumed (hence, in a more sophisticated model, I might left $B$ be the integral under the true passband (modeled e.g. by a Butterworth filter)).

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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'd also simply like to achieve a deeper theoretical understanding. I'm open (and encourage) detailed and complex theoretical answers if necessary. I can trim those down to something I can apply for my practical use case.

Answer 1

Thermal noise is an Additive White Gaussian Noise process (AWGN). What is received at the antenna depends on the radiator, but if the radiator is a black-body thermal source only, then the signal would be thermal noise only and AWGN. If the receiver were to take the absolute value of the received signal (such as done with a simple diode detector), then the noise process would then be Rayleigh distributed.

What may help to understand this further is the relationship between real passband signals, and the equivalent baseband complex signal. The noise at the antenna is a real passband signal which can be equivalently modelled as a complex baseband signal. The easy way to relate the two is to consider the passband signal relative to the carrier, from which we can derive a magnitude and phase versus time.

For this reason it is often quite inefficient to model a passband signal directly, since the carrier itself adds no information. When a complex representation is used, the carrier frequency can be DC or 10 GHz with no change to the underlying magnitude and phase versus time characteristics.

I depict this with the graphic below showing on a complex (IQ) plane the magnitude and phase of a signal versus time. On the left is a passband signal centered at carrier frequency $F_o$. This is frequency translated to a complex "baseband" where the new carrier frequency is now 0 or DC. In both cases, the signal variation relative to the carrier has not changed. (If the input signal was real at an RF carrier at frequency $F_o$, and we multiplied it by $e^{j2\pi F_o t}$, we would also need to filter out the high frequency component so that we have the complex baseband signal alone). The main point here is the passband signal and complex baseband signal are equivalent for purposes of analysis of the RF signal.

The thermal noise floor is indeed well modelled as an Additive White Gaussian Noise process, in practical terms white means being of constant power spectral density over a finite bandwidth of interest. If we were to process the signal in the receiver to specifically take the magnitude of thermal noise, the resulting distribution would be Rayleigh, but in most receivers where we are processing a complex signal (often as "I" for in-phase and "Q" for quadrature), the resulting distribution of thermal noise is independent and identically distributed on I and Q as Gaussian noise. I detail the relationship between AWGN and Rayleigh distributions further in this post with additional references that may be helpful.

That said, what is measured at the antenna may be affected by any other sort of radiator and sources of interference that may not be AWGN. The properties of the antenna may affect this in terms of being more sensitive to radiation from particular directions. As far as thermal noise specifically, this is often dominated by a local contribution of the electronics in the receiver (often the "low noise amplifier" which is placed first), unless the antenna is pointed at a hot body radiator with power that is stronger than the noise generated locally (typically we get the sum of the two). The locally generated AWGN is what we would reliably measure if we were to disconnect the antenna and terminate the antenna input assuming we were not generating interference from other sources within the radio (such as local oscillator feedthrough, or spurious signals from other clocks, digital circuits etc).

If we can be assured the antenna is receiving energy from a radiator of thermal noise only, then the resulting signal will be well modelled as AWGN over a finite bandwidth of interest with the following considerations detailed with the diagrams below:

In this diagram above I am showing a lossless antenna with gain $G_a$ pointed at a black-body radiator with temperature in Kelvin of $T_a$. Under this condition the antenna would present AWGN with a total power of $kT_aBG_a$ to the receiver, where $k$ is Boltzmann's Constant, and $B$ is the receiver bandwidth (typically this would be the same bandwidth as the signal we would be interested in receiving). For terrestrial communications where the antenna is basically pointed at earth, $T_a=T_r$, but when the antenna is pointed upward at the sky (with narrow enough beamwidth), we can have the case of $T_a<<T_r$. $N_1$ refers to locally added noise by electronics in the receiver (resulting in a quality factor given as "Noise Figure), and $G_r$ is the gain of a perfect noise-less receiver. $R_L$ is the load resistance where the received signal power is dissipated, which itself generates thermal noise as indicated. In a well designed receiver, this local noise at the final termination is swamped out by large values for $G_r$ and can be inconsequential.

In comparison the diagram shown below depicts what would happen when the antenna is disconnected and the input is instead terminated:

Further Background in IQ Processing

This post Frequency Shifting of a Quadrature Mixed Signal further details complex signals and their related spectrums. A real signal, which we can receive with a single antenna or measure with a single scope probe has an equal (complex conjugate) positive and negative frequency spectrum. A simple and intuitive example is Euler's formula for a cosine:

$$2\cos(\omega t) = e^{j\omega t}+e^{-j\omega t}$$.

A complex signal can have a positive only or negative only frequency, but we need two scope probes to measure it's two real components: I and Q!

$$ e^{j\omega t} = \cos(\omega t) + j\sin(\omega t) + I +jQ$$

Note that we always need two real signals to express a complex signal, on paper or in implementation. (Example $Ke^{j\theta}$ or $K\angle{\theta}$ in polar coordinates, or $I+jQ$ in Cartesian coordinates.