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和时间递归特性的Gabor变换模拟，其中所提出的时间因果表示同时满足时间尺度协方差性和级联性质（伴随时间尺度上的简化核）。这些构造的动机是：在无法获取未来信息的实时场景，或物理/生物学建模场景中，为多时间尺度的时频分析提供理论基础——由于Gabor滤波中非因果性的未来信息访问不可行，故无法直接用于此类系统的时频分析。我们建立了这些表示的理论基础，其实现方式是用一种称为“时间因果极限核”的时间因果核替代Gabor滤波中的高斯核。该核保证在时间因果条件下，从精细到粗糙尺度层次的简化性质，其作用类似于高斯核在非因果时间域中的保证作用。通过上述方法，所提出的时频表示能够确保在多尺度情境下（当待分析信号或物理/生物学现象的特征尺度可能发生显著变化时）的理论严谨性，且时频分析中的所有步骤均需完全满足时间因果性。