- ...Considerations
- This work was
supported by the
Natural Sciences and
Engineering Research
Council of Canada
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...Mann
- Massachusetts Institute of Technology
Room E15-389, 20 Ames Street, Cambridge, MA 02139
e-mail
`steve@media.mit.edu`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...Haykin
- Communications Research Laboratory,
McMaster University
Hamilton, Ontario, Canada L8S 4K1.
e-mail
`haykin@synapse.crl.mcmaster.ca`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...transform
- When using a
multidimensional parameter space, it is often impossible
to establish
*frame bounds*daubechies:CBMS on the energy in the parameter space. With only one parameter, we cannot always reconstruct the signal. With two*effective*parameters, we can reconstruct the signal, and also bound the energy of the representation. With three or more parameters, the energy in the transform space will be infinite. To the extent that multi-dimensional parameter spaces are still useful, we will not let this infinite energy hinder our progress.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...``time-frequency-scale-chirprate
- Traditionally, the
term
*chirp-rate*(with a hyphen) is used, but in this paper, we use the single word ``chirprate'', to avoid confusion arising out of hyphens in compounded parameter lists.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...wavelet
- The term ``wavelet'' will appear in quotes
when it is used in this less restrictive sense.
In particular, a ``wavelet'' will be permitted to
have a non-zero DC component.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...otherwise
- In general,
*g*(*t*) is a complex-valued function of a real variable, and has finite energy: 6#6.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
*affine* - Segal [44]
and others sometimes refer to these coordinate
transformations as
*symplectomorphisms*. It is well-knownfolland,guilleminsymplectic that the actual geometry of phase space is symplectic geometry, and that it is a coincidence that 28#28 corresponds to area-preserving affine geometry. Therefore, we must keep in mind, that if we desire to extend our thinking to the analysis of signals of dimension*n*>1, then we must consider the symplectic geometry of 29#29.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...space
- Note that, if we were interested in exploiting the
phase of this representation, we would need to add a fourth parameter,
to account for the extent to which the operators
do not commute.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...rectangular
- The term ``rectangular''
is used here in the context of ``rectangular window'', meaning a 1-D function
that is unity in a certain frequency interval and zero outside that interval,
not to be confused with our later use of ``rectangular'' which will be more
consistent with its everyday usage to specify a 2-
*D*shape.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...elsewhere
- In actual fact, there is a small
amount of frequency smearing, but zero time smearing, as the energy is
entirely contained in the time interval under consideration.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...function
- We do not attempt to
address issues of discretization
in this paper, except to the extent to which
they have influenced the development of the
continuous chirplet transform.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...unit
- For this experiment, we positioned the radar horn
facing upward and held a volleyball two meters above
the horn, and released the ball after the recording began.
We recorded only the in-phase component, and ignored
the quadrature component of the radar.
The sampling rate was 8kHz.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...sinusoidal
- Here we are simplifying the description.
The dynamics of the sea are more fully described inpierson1.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...time
- Although there are devices, known as pitch transposers,
that attempt to perform such an operation in a highly nonlinear way.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Thu Jan 8 19:50:27 EST 1998