Mathematical models for electromagnetic interference (EMI) noise spectra and clutter statistics derived from first principles in Mathematical Geoenergy (GEM), Chapter 21 (Electromagnetic Energy: Noise and Uncertainty).
Chapter 21 shows that the pervasive phenomena of 1/f noise (flicker noise) and electromagnetic clutter (Rayleigh/K-distributed fading) arise naturally from Maximum Entropy (MaxEnt) applied to disordered collections of EM processes. Three application domains are treated:
Information packets traversing a network experience variable round-trip times (RTTs) because the effective transfer rate $r$ varies hop-to-hop. Applying MaxEnt with only a known mean rate yields an exponential distribution for $r$. Combining this with the fixed-distance ($T$) constraint via the delta-function transport rule (identical to the porous-media rule, GEM Ch. 20, Eq. 20-16) gives an inverse-square dispersive latency PDF (Eq. 21-4):
\[p(t) = \frac{T}{t^2}\cdot e^{-T/t}\]with complementary CDF $P(t) = e^{-T/t}$ (Eq. 21-1). This fits measured SLAC network RTT histograms with a single parameter $T = 50\,\mu$s.
Any single-rate Markov switching process (Random Telegraph Noise, RTS) has a Lorentzian (Cauchy) power spectral density (PSD):
\[S(\omega) = \frac{2}{\pi(B^2 + \omega^2)}\]with switching rate $B$. Applying MaxEnt to $B$ itself — admitting that $B$ is uniformly distributed on $[0, R]$ (maximum ignorance within a bandwidth) — and integrating over all rates gives:
\[S(\omega) = \frac{2}{\pi\cdot\omega}\cdot\arctan\left(\frac{R}{\omega}\right) \\;\xrightarrow{R\to\infty}\\; \frac{1}{\omega} \propto \frac{1}{f}\]The 1/f spectrum emerges as the large-$R$ limit of superimposed Lorentzians — no fractal argument or self-organised criticality is required. An even simpler MaxEnt argument notes that uniform energy density across the spectrum requires the photon count density $p(f) \propto 1/f$.
In a wireless fading channel the received signal power (energy per unit time) is the only measurable. MaxEnt with known mean power $1/k$ gives an exponential power distribution (Eq. 21-12). Converting to amplitude via $E = r^2$ and applying the chain rule reproduces the Rayleigh distribution for signal amplitude (Eq. 21-11):
\[p(r) = 2k\cdot r\cdot e^{-k r^2}\]For heterogeneous (compound-heterogeneous) clutter — such as K-distributed sea/land radar returns — the local mean $\mu$ of the exponential intensity distribution itself varies. A second MaxEnt prior $p(\mu) = (1/\mu_0)\cdot e^{-\mu/\mu_0}$ and integration over $\mu$ yields the Bessel-$K_0$ compound clutter PDF:
\[p(I) = \frac{2}{\mu_0}\cdot K_0\left(2\sqrt{\frac{I}{\mu_0}}\right)\]This is the $\nu{=}1$ K-distribution — the same functional form as the terrain slope PDF (GEM Ch. 16, Eq. 16-49), demonstrating the universality of compound MaxEnt dispersion.
A GPS receiver must search a 2-D time-frequency grid; with FFT assistance the search time scales as $t^2$ for a given signal acquisition rate $R$. MaxEnt on $R$ (Eq. 21-15) and integration gives a hyperbolic/Lomax acquisition-time law (Eq. 21-17):
\[P(t) = \frac{k\cdot t^2}{k\cdot t^2 + c} = \frac{1}{1 + a^2/t^2}, \qquad a^2 = \frac{c}{k}\]This is the same $1/(1+kt)$ Lomax form as oil-reservoir production decline (GEM Ch. 14) and waste persistence (GEM Ch. 20), now appearing in the EM/communications domain.
Eq. 21-1 — Cumulative latency distribution
\[P(t) = e^{-T/t}\]The fraction of packets that have arrived by time $t$ when each packet travels distance $T$ at a MaxEnt-distributed rate.
Eq. 21-2 — MaxEnt rate distribution
\[p(r) = \frac{1}{T}\cdot e^{-r/T}, \quad r \geq 0\]Maximum Entropy with only the mean rate $T^{-1}$ known gives an exponential.
Eq. 21-3 — Delta-function transport rule
\[p(t) = \int_0^\infty r\cdot p(r)\cdot\delta(x - r\cdot t)\\,dr = \frac{x}{t^2}\cdot p\left(\frac{x}{t}\right)\]Identical in structure to the porous-media transport rule (GEM Ch. 20, Eq. 20-16): any velocity/rate PDF maps directly to a transit-time PDF.
Eq. 21-4 — Dispersive latency PDF
\[p(t) = \frac{T}{t^2}\cdot e^{-T/t}, \quad t > 0\]Result of applying Eq. 21-3 to the MaxEnt rate PDF (Eq. 21-2). The distribution is normalised: $\int_0^\infty p(t)\,dt = 1$.
Eq. 21-6 — Lorentzian (Cauchy) RTS PSD
\[S(\omega) = \frac{2}{\pi(B^2 + \omega^2)}\]Fourier transform of the two-sided MaxEnt autocorrelation $C(\tau) = e^{-B|\tau|}$ for a Markov two-state process with switching rate $B$. Peak at $\omega = 0$: $S(0) = 2/(\pi B^2)$.
Eq. 21-7 — Superposition integral
\[S(\omega) = \int_0^R S(\omega, B)\\,dB = \int_0^R \frac{2}{\pi(B^2 + \omega^2)}\\,dB\]Average over a uniform distribution of switching rates $B \in [0, R]$ — the maximum-entropy assignment when only the range $[0, R]$ is known.
Eq. 21-8 — Analytical 1/f superposition result
\[S(\omega) = \frac{2}{\pi\cdot\omega}\cdot\arctan\left(\frac{R}{\omega}\right)\]Closed-form result of Eq. 21-7. Transitions from $1/\omega^2$ (single Lorentzian) above $\omega \approx R$ to $1/\omega$ below it.
Eq. 21-9 — 1/f limit ($R \to \infty$)
\[S(\omega) \\;\to\\; \frac{1}{\omega} \propto \frac{1}{f}\]As $R \to \infty$, $\arctan(R/\omega) \to \pi/2$, so $S(\omega) \to (2/\pi)\cdot(\pi/2)/\omega = 1/\omega$.
Eq. 21-10 — MaxEnt photon flatness condition
\[E(f)\cdot p(E(f)) = h f \cdot \frac{1}{f} = h = \text{constant}\]If EM radiation carries uniform energy density per unit frequency band (MaxEnt), photon count density must be $p(f) \propto 1/f$.
Eq. 21-12 — MaxEnt signal power distribution
\[p(E) = k\cdot e^{-k E}, \quad E \geq 0\]MaxEnt with only the mean signal power $1/k$ known.
Eq. 21-11 / 21-13 — Rayleigh amplitude PDF (MaxEnt chain rule)
\[p(r) = 2k\cdot r\cdot e^{-k r^2}, \quad r \geq 0\]Derived from $p(E)$ (Eq. 21-12) via $E = r^2$ and $p(r) = p(E)\cdot|dE/dr| = k\cdot e^{-kr^2}\cdot 2r$. Mean power $\langle r^2\rangle = 1/k$.
K-distribution compound clutter PDF
\[p(I) = \frac{2}{\mu_0}\cdot K_0\left(2\sqrt{\frac{I}{\mu_0}}\right), \quad I \geq 0\]Super-statistical (compound MaxEnt) result: exponential conditional $p(I|\mu)$ integrated over exponential prior $p(\mu)$. Identical functional form to the terrain slope PDF (GEM Ch. 16, Eq. 16-49) and wind super-statistics (GEM Ch. 11).
Complementary CDF:
\[P(I > i) = 2\sqrt{\frac{i}{\mu_0}}\cdot K_1\left(2\sqrt{\frac{i}{\mu_0}}\right)\]Eq. 21-14 — Conditional GPS acquisition CDF
\[P(t \mid R) = e^{-c R / t^2}\]With $N^2$ search steps reduced to $N$ by FFT parallel processing, the acquisition probability for signal rate $R$ scales as $e^{-cR/t^2}$.
Eq. 21-15 — MaxEnt signal rate distribution
\[p(R) = k\cdot e^{-k R}, \quad R \geq 0\]MaxEnt with only the mean acquisition rate $1/k$ known.
Eq. 21-16 — Marginal acquisition probability
\[P(t) = \int_0^\infty P(t \mid R)\cdot p(R)\\,dR = \int_0^\infty e^{-cR/t^2}\cdot k\cdot e^{-kR}\\,dR\]Eq. 21-17 — Hyperbolic (Lomax) GPS acquisition law
\[P(t) = \frac{k\cdot t^2}{k\cdot t^2 + c} = \frac{1}{1 + a^2/t^2}, \qquad a = \sqrt{c/k}\]Closed-form result of Eq. 21-16. As $t\to\infty$, $P(t)\to 1$. For small $t$: $P(t)\approx t^2/a^2$ — quadratic growth reflecting the $t^2$ search-space scaling. Fitted parameter $a \approx 62\,\text{s}$ in GPS cold-start data (GEM Ch. 21, Fig. 21-10).
| File | Purpose |
|---|---|
EMI_symbolic.py |
Symbolic derivation of all Chapter 21 equations using SymPy |
EMI_numerical.py |
Numerical implementation, validation, and composite figure |
EMI_model_output.png |
Output figure (6 panels) generated by EMI_numerical.py |
Install dependencies (from models/requirements.txt):
pip install -r ../requirements.txt
Run symbolic derivation (all assertions print ✓):
python EMI_symbolic.py
Run numerical model and generate figure:
MPLBACKEND=Agg python EMI_numerical.py
1/f noise needs no mystery: superimposing Lorentzian spectra from a MaxEnt (uniform) distribution of switching rates $B \in [0, R]$ gives $S(\omega) \propto 1/\omega$ for $\omega \ll R$ (Eq. 21-8, 21-9). No fractal geometry, self-organised criticality, or long-range correlations are required — uniform ignorance about the rate is sufficient.
Latency and contaminant transport are structurally identical: the dispersive latency PDF (Eq. 21-4) and the porous-media breakthrough curve (GEM Ch. 20, Eq. 20-11) both follow from the same MaxEnt delta-function transport rule, $n(x,t) = (1/t)\cdot p(x/t)$. The physics (network packets vs. solute parcels) differs; the mathematics is the same.
Rayleigh is a MaxEnt amplitude distribution: the Rayleigh distribution for fading-channel amplitude arises from MaxEnt applied to signal power (Eq. 21-12) combined with the $E = r^2$ energy-amplitude relationship (Eq. 21-13). No random-phase interference argument is needed — energy MaxEnt alone suffices.
K-distribution clutter = terrain Bessel-K₀: applying MaxEnt twice (first on clutter intensity conditional on local mean, then on the local mean itself) gives the Bessel-$K_0$ K-distribution (compound clutter PDF). This is exactly the same functional form as the terrain slope distribution (GEM Ch. 16, Eq. 16-49) and the wind super-statistics (GEM Ch. 11), proving the universality of the compound MaxEnt argument across geophysics and electromagnetics.
GPS acquisition obeys the Lomax / hyperbolic law: the same $P(t) = 1/(1+a/t^\alpha)$ form that describes oil-reservoir production decline (GEM Ch. 14) and waste persistence (GEM Ch. 20) governs GPS cold-start times. Disorder in the signal acquisition rate + the quadratic search-space scaling combine to produce this law.
Single-parameter fits throughout: each model requires at most two free parameters (a mean and a scale). The observed fat-tailed distributions in network latency, 1/f noise, GPS acquisition, and radar clutter all follow from elementary MaxEnt assignments — complex multi-parameter heuristics are unnecessary.