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.

翻译：本文提出了一种时间因果的Gabor滤波器模拟，以及一种兼具时间因果性和时间递归性的Gabor变换模拟。所提出的时间因果表示在时间尺度上同时服从时间尺度协方差和级联性质（具有简化核）。构建这些表示的目的是在无法访问未来信息的实时场景、物理或生物建模场景中，实现跨多个时间尺度的具有理论基础的时频分析——由于Gabor滤波中面向未来的非因果访问在此类系统的时频分析中不可行。我们为这些表示发展了理论，通过将Gabor滤波中的高斯核替换为一种被称为时间因果极限核的时间因果核来实现。该核可保证在时间因果情况下，从精细到粗糙尺度级别的简化特性，类似于高斯核在非因果时域上所能保证的性质。通过上述方式，所提出的时频表示在待分析信号（或物理/生物现象）的特征尺度可能发生显著变化，且时频分析所有步骤必须完全满足时间因果性的场景下，能确保建立在多尺度处理的理论基础上。