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Gabor滤波器的时间因果模拟，以及同时具备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变性和一种通过简化核在时间尺度上的级联性质。这些构造的动机是为实时场景或无法访问未来的物理/生物建模场景，在理论上建立多时间尺度上的时频分析方法——由于Gabor滤波需要非因果地访问未来信息，因此无法用于此类系统的时频分析。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时间因果核（称为时间因果极限核）来实现，该核在时间因果情况下保证了从精细到粗糙尺度层次的简化性质，类似于高斯核在非因果时间域中可证明的特性。通过上述方式，所提出的时频表示能够在信号特征尺度或待分析的物理/生物现象尺度可能存在显著变化，且时频分析的所有步骤必须完全时间因果的场景下，确保多尺度分析的严格理论基础。