Gabor Function

📐 Definition

The Gabor function is a localized oscillatory function obtained by modulating a Gaussian envelope with a complex exponential.

In one dimension, a standard Gabor function is given by

g(x) = e^{-x^2/(2\sigma^2)}\,e^{ikx}

where $\sigma > 0$ controls spatial localization and $k \in \mathbb{R}$ is the carrier frequency.

The Gaussian envelope ensures simultaneous localization in both physical and frequency space.

Domain and Codomain

The Gabor function is defined for $x \in \mathbb{R}$ . It is complex-valued and belongs to $L^2(\mathbb{R})$ .

⚙️ Key Properties

Time–Frequency Localization

The Gabor function achieves minimal joint uncertainty in position and frequency:

\Delta x\,\Delta k = \frac{1}{2}

This property makes it optimal under the Heisenberg uncertainty principle.

Fourier Transform

The Fourier transform of a Gabor function is also a Gaussian, shifted in frequency:

\hat{g}(\xi) = \sqrt{2\pi}\,\sigma\,e^{-\tfrac{1}{2}\sigma^2(\xi-k)^2}

under the convention $\hat f(\xi)=\int_{-\infty}^{\infty} f(x)e^{-i\xi x}\,dx$ .

Translation and Modulation

Translations and modulations act naturally:

g_{x_0,k_0}(x) = e^{-(x-x_0)^2/(2\sigma^2)}\,e^{ik_0 x}

forming a family of localized basis functions.

Normalization

With appropriate normalization,

\int_{-\infty}^{\infty} |g(x)|^2\,dx = 1

allowing use as atoms in orthonormal or frame-based expansions.

🎯 Special Cases

Real-Valued Gabor Functions

Taking the real or imaginary part yields cosine- or sine-modulated Gaussians:

e^{-x^2/(2\sigma^2)}\cos(kx), \qquad e^{-x^2/(2\sigma^2)}\sin(kx)

These appear frequently in signal processing and vision models.

Zero Frequency

For $k = 0$ , the Gabor function reduces to a Gaussian window.

Gabor functions underlie the windowed Fourier transform and Gabor frames. They are closely related to Gaussian functions, complex exponentials, and coherent states in quantum mechanics.

Usage in Oakfield

Oakfield uses Gabor functions primarily as stimulus kernels (Gaussian-windowed sinusoids) rather than as an analysis transform:

stimulus_gabor operator writes a Gaussian envelope times an oscillatory carrier into a 1D field, with optional motion (velocity), carrier frequency (wavenumber, omega), and phase.
Complex outputs support an additional global phase rotation for writing complex-valued Gabors, useful for injecting analytic wavepackets.
Gabor-style packets are commonly used to seed or probe dynamics with localized, band-limited structure without introducing sharp spatial edges.

Historical Foundations

📜 Origin (20th Century)

Dennis Gabor introduced these functions while developing time–frequency representations for signal analysis, anticipating modern wavelet and frame theories.

🔬 Mathematical Development

Subsequent work placed Gabor functions within the theory of frames, harmonic analysis, and coherent states.

🌍 Modern Perspective

Gabor functions are now fundamental tools in signal processing, numerical analysis, and time–frequency methods.

📚 References

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