Multi-Tone Sinusoidal Superpositions

📐 Definition

A multi-tone signal sums multiple sinusoids or complex exponentials, allowing control over amplitudes, phases, and frequency spacing.

x(t) = \sum_{k=1}^{m} A_k \sin(\omega_k t + \phi_k)

Domain and Codomain

Defined over time or space. Outputs remain bounded and inherit periodicity (if frequencies are rationally related) or quasi-periodicity (otherwise).

⚙️ Key Properties

Beating from Close Frequencies

Close frequencies produce beating envelopes at rates set by frequency differences.

Energy Additivity (Orthogonal Modes)

For orthogonal sinusoidal modes over a period, energies add:

E \propto \tfrac{1}{2}\sum_k A_k^2

🎯 Special Cases and Limits

Two-tone beating: sum-of-sines identities expose carrier/envelope decomposition.
Identical frequencies collapse into one tone with combined amplitude and phase.
Rationally related frequencies yield periodicity with a common fundamental period.

Fourier series represent the same structure with potentially infinite tones; standing waves and phase-shifted sinusoids are individual components; complex exponentials offer an equivalent representation with $e^{i\omega_k t}$ .

Usage in Oakfield

Oakfield provides multi-tone harmonic sums via its spectral stimulus operators:

stimulus_spectral_lines operator synthesizes sums of harmonics (multi-tone spectra) with configurable harmonic count and decay law.
stimulus_random_fourier generates randomized multi-tone structure (random Fourier features) with band limits and spectral slope controls.

Historical Foundations

📜 Fourier Decomposition

Multi-tone superpositions are the finite-dimensional analogue of Fourier series and underpin spectral representations of periodic and quasi-periodic signals.

🌍 Modern Perspective

They are standard for building test signals with prescribed spectral content and for modeling multi-frequency forcing in linear and weakly nonlinear systems.

📚 References

Stein & Shakarchi, Fourier Analysis
Oppenheim & Schafer, Discrete-Time Signal Processing