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滤波器类比，以及一种兼具时间因果性和时间递归性的Gabor变换类比。所提出的时间因果表征同时遵循时间尺度协变性和跨时间尺度的简化核级联特性。这些构建的动机在于：针对实时场景或物理/生物建模场景中无法获取未来信息的情形，当Gabor滤波所需的非因果未来信息访问不可行时，为系统时频分析提供理论完备的多时间尺度分析框架。我们通过将Gabor滤波中的高斯核替换为时间因果核（称为时间因果极限核）来发展这些表征的理论体系，该核能保证在时间因果场景中从细尺度到粗尺度的简化特性，类似于高斯核在非因果时间域中所证明的性质。通过这种方式，所提出的时频表征方法在被分析信号、物理或生物现象的特征尺度可能存在显著变化，且时频分析的所有步骤必须完全满足时间因果性的场景下，确保了多尺度处理的严谨性。