A time-causal and time-recursive analogue of the Gabor transform

TL;DR

This paper introduces a time-causal and recursive analogue of the Gabor transform, enabling real-time, multi-scale time-frequency analysis suitable for systems where future data is inaccessible.

Contribution

It develops a theoretically grounded time-causal Gabor transform using a new kernel, ensuring scale covariance and recursive properties for real-time applications.

Findings

Provides a time-causal Gabor filter with scale covariance.

Ensures recursive computation over multiple scales.

Guarantees well-founded analysis in real-time and biological systems.

Abstract

This paper presents a time-causal analogue of the Gabor filter, as well as a both time-causal and time-recursive analogue of the Gabor transform, where the proposed time-causal representations obey both temporal scale covariance and a cascade property with a simplifying kernel over temporal scales. The motivation behind these constructions is to enable theoretically well-founded time-frequency analysis over multiple temporal scales for real-time situations, or for physical or biological modelling situations, when the future cannot be accessed, and the non-causal access to future in Gabor filtering is therefore not viable for a time-frequency analysis of the system. We develop the theory for these representations, obtained by replacing the Gaussian kernel in Gabor filtering with a time-causal kernel, referred to as the time-causal limit kernel, which guarantees simplification properties from finer to coarser levels of scales in a time-causal situation, similar as the Gaussian kernel can be shown to guarantee over a non-causal temporal domain. In these ways, the proposed time-frequency representations guarantee well-founded treatment over multiple scales, in situations when the characteristic scales in the signals, or physical or biological phenomena, to be analyzed may vary substantially, and additionally all steps in the time-frequency analysis have to be fully time-causal.