Gaussian Window

📐 Definition

The Gaussian window is a smooth, localized weighting function defined by a Gaussian profile. In one dimension, it is given by

w_\sigma(x) = (\pi\sigma^2)^{-1/4}\,e^{-x^2/(2\sigma^2)}

where $\sigma > 0$ controls the width of the window. The Gaussian window is commonly applied multiplicatively to a signal or function to enforce localization.

Domain and Codomain

The Gaussian window is defined for $x \in \mathbb{R}$ . It is real-valued, strictly positive, and belongs to $L^2(\mathbb{R})$ and $L^\infty(\mathbb{R})$ .

⚙️ Key Properties

Smooth Localization

The Gaussian window is infinitely differentiable and rapidly decaying:

\lim_{|x|\to\infty} x^n w_\sigma(x) = 0

for all integers $n \ge 0$ . This ensures smooth tapering without sharp cutoffs.

Optimal Time–Frequency Concentration

Among all windows, the Gaussian minimizes the joint uncertainty in position and frequency:

\Delta x\,\Delta \xi \ge \frac{1}{2}

Equality holds for Gaussian windows (with $\Delta$ denoting standard deviations).

Fourier Transform

The Fourier transform of a Gaussian window is also Gaussian:

\widehat{w_\sigma}(\xi) = \sqrt{2}\,\pi^{1/4}\,\sigma^{1/2}\,e^{-\tfrac{1}{2}\sigma^2 \xi^2}

Thus, localization in physical space corresponds to localization in frequency space.

Scaling Behavior

Scaling the window width rescales its frequency spread:

w_{\sigma_1}(x)\,w_{\sigma_2}(x) \propto w_{\sigma}(x),\qquad \frac{1}{\sigma^2}=\frac{1}{\sigma_1^2}+\frac{1}{\sigma_2^2}

reflecting additivity of precisions (inverse variances) under multiplication.

🎯 Special Cases

Unit-Width Window

For $\sigma = 1$ ,

w_1(x) = \pi^{-1/4}\,e^{-x^2/2}

is often used as a canonical Gaussian window.

Limiting Behavior

As $\sigma \to 0$ , the window concentrates at the origin. As $\sigma \to \infty$ , the window approaches a constant function on bounded intervals.

The Gaussian window underlies:

Gabor functions and Gabor frames,
the windowed Fourier transform,
Gaussian convolution kernels.

It provides a canonical example of smooth localization.

Usage in Oakfield

Oakfield uses Gaussian “window” ideas mostly as localized tapers for operator kernels and stimuli:

Stimulus shaping: the Gaussian envelope used by stimulus_gaussian and stimulus_gabor is a window/taper over the 1D field domain.
Smoothing kernels: sieve generates a normalized discrete Gaussian window (over taps) to implement low-pass filtering.

Oakfield does not currently implement a full windowed Fourier transform/Gabor transform pipeline; Gaussian windows appear mainly as the building blocks inside operators.

Historical Foundations

📜 Origin

Gaussian profiles arose naturally in probability theory and error analysis in the work of Gauss.

🔬 Time–Frequency Analysis

Dennis Gabor identified the Gaussian window as optimal for joint time–frequency localization, forming the basis of modern Gabor analysis.

🌍 Modern Perspective

Gaussian windows are standard tools in harmonic analysis, signal processing, and numerical spectral methods.

📚 References

D. Gabor, Theory of Communication (1946)
Gröchenig, Foundations of Time-Frequency Analysis
NIST Digital Library of Mathematical Functions (DLMF), §1.11