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

TL;DR

This work constructs a time-causal, time-recursive analogue of the Gabor transform by replacing the non-causal Gaussian window with a time-causal limit kernel, enabling real-time, multi-scale time-frequency analysis. It derives the complex-valued time-causal kernel, proves temporal scale covariance and cascade smoothing, and implements a discrete, strictly time-recursive cascade of first-order filters for efficient real-time computation. Theoretical analyses and experiments quantify temporal delays, frequency selectivity, and robustness to noise, showing the approach yields accurate local frequency estimates with modest spectral broadening compared to the classical Gabor transform. The framework supports principled time-frequency analysis for real-time signal processing and biological/physical modelling where access to the future is not possible, with practical discrete implementations and extensive appendices detailing theory and inverse transforms.

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.

Figures (7)

• Figure 1: (top row) Gaussian kernels at temporal scale levels $\sigma = \sqrt{\tau} = 1$ and $2$. (middle row) The time-causal limit kernel at temporal scale levels $\sigma = \sqrt{\tau} = 1$ and $2$ for $c = \sqrt{2}$. (bottom row) The time-causal limit kernel at temporal scale levels $\sigma = \sqrt{\tau} = 1$ and $2$ for $c = 2$.
• Figure 2: (top two rows) The real and imaginary parts of Gabor kernels at temporal scale levels $\sigma = \sqrt{\tau} = 1$ and $2$ for $\omega = 10$. (middle two rows) The real and imaginary parts of the complex-valued extension of the time-causal limit kernel at temporal scale levels $\sigma = \sqrt{\tau} = 1$ and $2$ for $\omega = 10$ and $c = \sqrt{2}$. (bottom two rows) The real and imaginary parts of the complex-valued extension of the time-causal limit kernel at temporal scale levels $\sigma = \sqrt{\tau} = 1$ and $2$ for $\omega = 10$ and $c = 2$.
• Figure 3: Commutative diagram for the regular Gabor transform under temporal scaling transformations, for temporal scaling factors $S > 0$. (This commutative diagram should be read from the lower left corner to the upper right corner, and means that irrespective of whether we first rescale the input signal $f(t)$ to a rescaled signal $f'(t')$ and then compute the Gabor transform, or first compute the Gabor transform and then rescale it, we get the same result, provided that the values of the angular frequency parameter and the temporal scale parameter are matched according to $\omega' = \omega/S$ and $\tau' = S^2 \tau$.)
• Figure 4: Commutative diagram for the time-causal analogue of the Gabor transform under temporal scaling transformations for scaling factors $S$ that are integer powers of the distribution parameter $c$ of the time-causal limit kernel, i.e., $S = c^j$ for integer $j$ for some $c > 1$. (This commutative diagram should be read from the lower left corner to the upper right corner, and means that irrespective of whether we first rescale the input signal $f(t)$ to a rescaled signal $f'(t')$ and then compute the time-causal analogue of the Gabor transform, or first compute the time-causal analogue of the Gabor transform and then rescale it, we get the same result, provided that the values of the angular frequency parameter and the temporal scale parameter are matched according to $\omega' = \omega/S$ and $\tau' = S^2 \tau$.)
• Figure 5: Frequency selectivity properties of the Gabor transform and the time-causal analogue of the Gabor transform, based on the entity $R(\omega) = |\hat{h}(\omega - \omega_0;\; \tau(\omega))|$ in (\ref{['eq-freq-sel-entity']}), for different values of the wavelength proportionality factor $N$ in (\ref{['eq-def-tau-from-omega-N']}) and different values of the distribution parameter $c$ in the time-causal limit kernel. As can be seen from these graphs, the frequency selectivity becomes more narrow for larger values of $N$, which in turn correspond to temporal window functions at coarser temporal scales. The frequency selectivity of the time-causal analogue of the Gabor transform also becomes more narrow when the distribution parameter $c$ is decreased, however, then at the cost of longer temporal delays. Choosing appropriate parameter settings does in this respect correspond to a trade-off, which should be balanced for any given application. (Horizontal axis: angular frequency in the time-frequency transform, on a logarithmic scale, and relative to a reference angular frequency of $\omega_0 = 1$. Vertical axis: Value of $R(\omega;\ \tau) = |\hat{h}(\omega - \omega_0;\; \tau(\omega))|$ in dB.)