Phase-Shifted Sinusoids

📐 Definition

A phase-shifted sinusoid applies a constant phase offset to a base sinusoidal mode, adjusting zero crossings and peak alignment.

x(t) = \sin(\omega t + \phi)

Domain and Codomain

Defined over time or space for oscillatory fields. Outputs are bounded real-valued sinusoids (or complex-valued phasors if represented as $e^{i(\omega t + \phi)}$ ).

⚙️ Key Properties

Decomposition into Sine/Cosine Components

\sin(\omega t + \phi) = \sin(\omega t)\cos\phi + \cos(\omega t)\sin\phi

Time-Shift Interpretation

\sin(\omega t + \phi) = \sin\!\left(\omega\left(t + \tfrac{\phi}{\omega}\right)\right), \qquad \omega \ne 0

Phase offsets translate the waveform while preserving amplitude and frequency. Relative phases control constructive/destructive interference in superpositions.

🎯 Special Cases and Limits

$\phi=0$ yields the base sinusoid.
$\phi=\tfrac{\pi}{2}$ produces a cosine.
$\phi=\pi$ flips sign.
Small-phase approximation:

\sin(\omega t + \phi) \approx \sin(\omega t) + \phi \cos(\omega t) \quad (|\phi|\ll 1)

Sine and cosine correspond to specific phase choices. Complex exponentials encode phase via $e^{i\phi}$ , and standing waves combine counterpropagating phase-related modes.

Usage in Oakfield

Oakfield uses explicit phase offsets throughout its stimulus and mixing operators:

Sinusoidal stimuli accept a phase parameter (and related offsets) to control phase-shifted oscillatory forcing.
Complex stimuli can apply additional phase rotation when writing complex outputs, implemented as multiplication by $(\cos\theta + i\sin\theta)$ .
Mixer PM/FM modes also implement phase shifts/modulation in complex fields (e.g. $L\mapsto L\cdot e^{iR}$ ).

Historical Foundations

📜 Phasor View

Representing a phase shift as an additive constant in the argument (or as a multiplicative factor $e^{i\phi}$ ) is central to phasor methods in waves and signal processing.

🌍 Modern Perspective

Phase-shifted sinusoids are the basic unit of interference modeling and spectral composition in linear oscillatory systems.

📚 References

Bracewell, The Fourier Transform and Its Applications
Oppenheim & Schafer, Discrete-Time Signal Processing