Steve Mann
-
Simon Haykin
Corresponding author: Steve Mann, currently with University of Toronto, Department of Electrical Engineering, Computer Group, 10 King's College Road, Sandford Fleming Building, Room 2001, (416)946-3387, mann@eecg.toronto.edu
We consider a multidimensional parameter space formed by inner products of a parameterizable family of chirp functions with a signal under analysis. We propose the use of quadratic chirp functions (which we will call q-chirps for short), giving rise to a parameter space that includes both the time-frequency plane and the time-scale plane as two-dimensional subspaces. The parameter space contains a ``time-frequency-scale volume'', and thus encompasses both the short-time Fourier transform (as a slice along the time and frequency axes), and the wavelet transform (as a slice along the time and scale axes).
In addition to time, frequency, and scale, there are two other coordinate axes within this transform space: shear-in-time (obtained through convolution with a q-chirp) and shear-in-frequency (obtained through multiplication by a q-chirp). Signals in this multidimensional space can be obtained by a new transform which we call the ``q-chirplet transform'', or simply the ``chirplet transform''.
The proposed chirplets are generalizations of wavelets, related to each other by two-dimensional affine coordinate transformations (translations, dilations, rotations, and shears) in the time-frequency plane, as opposed to wavelets which are related to each other by one-dimensional affine coordinate transformations (translations and dilations) in the time-domain only.