Square Waves

📐 Definition

A square wave alternates between two plateaus each half-period. A canonical $2\pi$ -periodic choice is

s(x) = \operatorname{sgn}(\sin x)

Domain and Codomain

Defined for periodic time or spatial axes. For the sign-based definition, outputs take values in $\{-1,1\}$ (or include $0$ at discontinuities depending on the sign convention).

⚙️ Key Properties

Fourier Series (50% Duty Cycle)

s(x) = \frac{4}{\pi}\sum_{n\ \text{odd}}^\infty \frac{1}{n}\sin(nx)

Odd harmonics dominate with amplitudes decaying as $1/n$ .

Zero Mean (Symmetric Case)

\int_0^{2\pi} s(x)\,dx = 0

🎯 Special Cases and Limits

Duty cycle $D$ generalizes to a rectangular pulse train of width $2\pi D$ .
Smooth approximation:

s_\beta(x) = \tanh(\beta \sin x) \to \operatorname{sgn}(\sin x) \quad (\beta \to \infty)

Fourier series connect square waves to sums of sinusoids. Thresholding and smooth saturation functions provide hard and soft versions of the plateau transitions.

Usage in Oakfield

Oakfield does not currently ship a dedicated “square wave” stimulus operator.

Closest equivalents in the current runtime:

Checkerboard stimulus provides a piecewise-constant periodic pattern over space (useful for Gibbs/aliasing stress tests).
User-defined Lua operators can implement time-domain on/off pulses (square waves in time) if needed.
Soft thresholding/saturation can approximate hard switching when differentiability is important.

Historical Foundations

📜 Harmonics and Fourier Series

Square waves are a canonical example in Fourier analysis: a simple discontinuous waveform with an explicitly computable harmonic series and a visible Gibbs phenomenon.

🌍 Modern Perspective

They are widely used as test signals and as idealized switching inputs, often paired with smoothing when differentiability is required.

📚 References

Stein & Shakarchi, Fourier Analysis
Bracewell, The Fourier Transform and Its Applications