...Considerations

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

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...Mann

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

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...Haykin

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

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...transform

When using a multidimensional parameter space, it is often impossible to establish frame boundsdaubechies: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.

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...``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.

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...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.

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...otherwise

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

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...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.

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...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.

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...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.

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...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.

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...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.

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...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.

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...sinusoidal

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

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...time

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

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