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Gabor滤波器的时间因果类比，以及一种兼具时间因果和时间递归特性的Gabor变换类比。所提出的时间因果表示同时满足时域尺度协方差性与级联特性（通过随时间尺度简化的核函数实现）。构建这些表示旨在为实时场景或物理/生物建模场景（其中无法访问未来信息，因此Gabor滤波中非因果的未来访问不适用于系统时频分析）提供具有理论基础的跨多个时间尺度时频分析方法。我们发展了这些表示的理论，通过将Gabor滤波中的高斯核替换为一种称为时间因果极限核的核函数，该核函数保证在时间因果条件下从细尺度到粗尺度的简化特性——类似于高斯核在非因果时域上保证的特性。通过上述方式，所提出的时频表示确保了当待分析信号、物理或生物现象的特征尺度存在显著变化，且时频分析所有步骤必须完全时间因果时，仍能提供多尺度分析的理论基础。