Moving Gaussian Envelope

📐 Definition

For a center trajectory $c(t)$ , a moving Gaussian envelope is a translated Gaussian of fixed width $\sigma>0$ :

g(x,t) = \exp\!\left(-\frac{(x - c(t))^2}{2\sigma^2}\right)

often with $c(t) = x_0 + vt$ for constant velocity $v$ .

Domain and Codomain

Defined for $(x,t)$ on continuous or discretized domains. For each fixed $t$ , the envelope maps $x$ to a strictly positive real value and decays smoothly away from the moving center $x=c(t)$ .

⚙️ Key Properties

Translation Form

g(x,t) = g(x-c(t),0)

Peak and Width

\sup_x g(x,t) = 1 \quad \text{attained at } x=c(t)

Derivatives

\partial_x g(x,t) = -\frac{x-c(t)}{\sigma^2}\,g(x,t)

\partial_t g(x,t) = \frac{(x-c(t))\,c'(t)}{\sigma^2}\,g(x,t)

Transport Equation

The envelope is advected by the trajectory velocity:

\partial_t g + c'(t)\,\partial_x g = 0

🎯 Special Cases and Limits

Stationary case: $c(t)\equiv x_0$ gives a fixed Gaussian envelope.
Sharp limit: as $\sigma\to 0^+$ , $g(\cdot,t)$ concentrates near $x=c(t)$ .
Broad limit: as $\sigma\to\infty$ , $g(\cdot,t)$ approaches a near-uniform gate on bounded domains.

The base Gaussian envelope $e^{-x^2/(2\sigma^2)}$ is recovered when $c(t)$ is constant; modulation by $e^{i\omega t}$ yields a Gabor packet; Gaussian windows apply similar tapering without translation.

Usage in Oakfield

Oakfield implements moving Gaussian envelopes directly in its stimulus operators:

stimulus_gaussian supports a velocity term so the envelope center advances as center + velocity * t, with optional fixed_clock and nominal_dt to decouple stimulus time from adaptive stepping.
stimulus_gabor uses the same moving-center envelope idea while also applying a carrier oscillation.
stimulus_gaussian_travel (sinusoidal family) applies a traveling Gaussian envelope as part of a modulated wave stimulus.

Historical Foundations

📜 Gaussian Envelopes

Gaussian envelopes arise from the central role of Gaussians in localization and smoothing. Translating the envelope along a trajectory is a natural extension used in wavepacket and windowing constructions.

🌍 Modern Perspective

Moving Gaussian windows are a standard tool for localized analysis in time-dependent simulations, supporting tracking, gating, and adaptive focusing of computations.

📚 References

Gröchenig, Foundations of Time-Frequency Analysis
NIST Digital Library of Mathematical Functions (DLMF), §7