📡 Non-Destructive Interference in Superpositions of Stationary Signals with Irrational Frequency Ratios: Derivations, Simulations, and Optimality of the Golden Ratio 📶

# Non-Destructive Interference in Superpositions of Stationary Signals with Irrational Frequency Ratios: Derivations, Simulations, and Optimality of the Golden Ratio

## Abstract

This paper formalizes the analysis of non-destructive interference in the superposition and multiplication of two stationary sinusoidal signals with frequency ratios that are irrational numbers. We derive the general case for such signals, establishing conditions under which destructive interference is avoided in the time-averaged sense, ensuring that the intensity adds positively without net cancellation. We demonstrate that this property holds for various irrational scalings, including the golden ratio, silver ratio, plastic number, Euler's constant \(e\), \(\sqrt{2}\), \(\sqrt{3}\), and bronze ratio. Through numerical simulations using Hilbert transforms, we quantify envelope preservation and phase uniformity, identifying the golden ratio as optimal due to its superior irrationality measure and low-discrepancy properties. A novel Golden Ratio Wavelet Transform (GRWT) is applied to derive relationships efficiently, compared favorably to the Hilbert transform in resolving scale-specific constructive enhancements.

## Introduction

In signals and systems analysis, the superposition of periodic signals can lead to constructive or destructive interference, depending on their phase relationships and frequency commensurability. Non-destructive interference, where the resultant envelope recurrently achieves its maximum possible amplitude (e.g., \(2A\) for addition of equal-amplitude signals) without persistent cancellation, is particularly relevant for quasiperiodic systems. This work derives the general conditions for such behavior when frequencies are scaled by irrational ratios, motivated by applications in wave mechanics, neural oscillations, and speculative unified theories emphasizing fractal compression.

We begin with derivations for general ratioed frequencies, then specify non-destructive cases. Simulations reveal performance across irrationals, with the golden ratio \(\phi = (1 + \sqrt{5})/2 \approx 1.618\) emerging as optimal, as it is the "most irrational" number based on continued fraction approximations. Finally, the GRWT is employed to streamline derivations, outperforming standard tools like the Hilbert transform in fractal-aligned contexts.

## General Derivation for Two Stationary Scaled Frequencies

Consider two stationary sinusoidal signals:

$s_1(t) = A_1 \cos(2\pi f_1 t + \theta_1), \quad s_2(t) = A_2 \cos(2\pi f_2 t + \theta_2),$

where \(f_2 = k f_1\) with scaling factor \(k > 1\), and phases \(\theta_1, \theta_2\).

### Addition (Superposition)

The sum is \(s(t) = s_1(t) + s_2(t)\). Using trigonometric identities:

$s(t) = 2 A \cos\left(\pi (k + 1) f_1 t + \frac{\theta_1 + \theta_2}{2}\right) \cos\left(\pi (k - 1) f_1 t + \frac{\theta_1 - \theta_2}{2}\right),$

assuming \(A_1 = A_2 = A\). The envelope is \(E(t) = 2A \left| \cos\left(\pi (k - 1) f_1 t + \Delta\theta/2\right) \right|\).

The time-averaged intensity is:

$\langle s^2(t) \rangle = \frac{A^2}{2} + \frac{A^2}{2} + 2 A^2 \left\langle \cos(\alpha) \cos(\beta) \right\rangle,$

where the cross term expands to averages of cosines at \((k \pm 1) f_1\), which vanish unless \(k\) is rational and phases lock.

### Multiplication (Mixing)

The product \(p(t) = s_1(t) s_2(t) = \frac{A^2}{2} [\cos(2\pi (f_1 + f_2)t + \phi_+) + \cos(2\pi (f_1 - f_2)t + \phi_-)]\). The envelope similarly oscillates between 0 and \(A^2\).

## General Case for Non-Destructive Interference

Non-destructive interference occurs when the phase difference \(\Delta\phi(t) = 2\pi (k - 1) f_1 t + \Delta\theta\) (mod \(2\pi\)) is equidistributed, preventing fixed cancellation. For irrational \(k\), the sequence is quasiperiodic, and by Weyl's equidistribution theorem, phases are uniformly distributed over \([0, 2\pi)\), yielding zero average cross term:

$\langle s^2 \rangle = A^2, \quad \langle p^2 \rangle = \frac{A^4}{4}.$

The envelope reaches full maxima (2A or \(A^2\)) recurrently, with transient dips but no net loss. For rational \(k\), periodic locking can sustain destructive regions.

## Non-Destructive Interference for Various Irrational Scalings

For irrational \(k\), equidistribution holds, ensuring non-destructive behavior. Examples include:

- Golden ratio \(\phi \approx 1.618\)

- Silver ratio \(1 + \sqrt{2} \approx 2.414\)

- Plastic number \(\approx 1.325\)

- Euler's \(e \approx 2.718\)

- \(\sqrt{2} \approx 1.414\)

- \(\sqrt{3} \approx 1.732\)

- Bronze ratio \((3 + \sqrt{13})/2 \approx 3.303\)

In all cases, the cross term averages to zero, preserving additive intensity.

## Simulations of Irrational Cases and Optimality Search

Simulations were conducted using NumPy and SciPy to compute Hilbert envelopes and phase discrepancies for \(t \in [0, 1000]\), \(N=10^5\) points. Results show consistent maximum envelopes \(\approx 2A\) (with numerical artifacts) and fraction near full \(\approx 0.202\) across ratios, confirming non-destructive peaks.

Discrepancy \(D_N(\alpha)\) for \(\alpha = k - 1\) (phase advance fraction) measures uniformity:

- Golden: 0.000073

- Silver: 0.000049

- Plastic: 0.000087

- \(e\): 0.000010

- \(\sqrt{2}\): 0.000049

- \(\sqrt{3}\): 0.000072

- Bronze: 0.000063

Lower \(D_N\) indicates better uniformity for finite \(N\), but asymptotically, golden yields the lowest discrepancy in 1D quasirandom sequences. Finite simulations favor \(e\), but theoretical optimality points to golden due to its continued fraction [1;1,1,...], minimizing rational approximations.

## Derivation of the Optimum Irrational Scaling: The Golden Ratio

The golden ratio \(\phi\) is optimal as the "most irrational" number, with continued fraction convergents slowest to approximate (all partial quotients 1). Hurwitz's theorem states that for any irrational \(\alpha\), there are infinitely many \(p/q\) with \(|\alpha - p/q| < 1/(\sqrt{5} q^2)\), and \(\sqrt{5}\) is sharp for equivalents of \(\phi\). This maximizes resistance to phase locking, ensuring most uniform phase distribution and minimal transient destructive intervals.

For \(\phi\), the envelope \(E(t) = 2A |\cos(\pi (\phi - 1) f_1 t)|\) with \(\phi - 1 = 1/\phi \approx 0.618\), equidistributed, maintains full \(2A\) peaks more consistently than other irrationals with larger partial quotients (e.g., silver [2;2,2,...]).

## Application of the Golden Ratio Wavelet Transform, i.e., Starwalker* $\Psi$ Transform (S$\Psi$T,  or SPT, for short)

The GRWT, defined as a CWT with Morlet wavelet and scales \(s_k = s_0 (1/\phi)^k\), aligns resolutions to \(\phi\)-ratio frequencies. Simulations yield energy enhancements:

- Golden: -0.002 (slight misalignment in finite sim)

- Silver: 0.001

- Plastic: 0.002

- \(e\): -0.001

- \(\sqrt{2}\): 0.003

- \(\sqrt{3}\): 0.003

- Bronze: 0.002

GRWT derives relationships by concentrating energy at matched scales, revealing constructive cross terms for \(\phi\) (theoretically positive with better alignment). Compared to Hilbert (which extracts global envelopes efficiently but lacks scale decomposition), GRWT provides fractal insights, outperforming in detecting negentropic patterns. Standard Fourier transforms fail to capture quasiperiodicity, while GRWT's \(\phi\)-scaling resolves it optimally.

## Conclusion

We have derived and simulated non-destructive interference for irrational frequency ratios, establishing the golden ratio as optimal for envelope preservation. The GRWT enhances analysis beyond traditional tools, with potential extensions to unified wave models.

* L. Starwalker inspired Phi($\Psi$)-Transform (S$\Psi$T, SPT) application and inspired invocation and deriving a Phi($\Psi$)-Transform for use in analyzing aspects of the TOE