## TR#250: The Chirplet Transform: Physical Considerations

Available in
Trans. IEEE Sig. Proc.,

Sept. 1995.

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 ``chiplet 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.