  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.
