Underlying a great deal of
traditional signal processing theory is the notion of
a *sinusoidal wave*.
With the advent of modern computing, and
the Fast Fourier transform,
the use of and interest in frequency-domain signal processing has increased
dramatically. More recently,
however, researchers are becoming aware of the limitations of frequency-domain
methods. Although the Fourier transform yields perfect reconstruction
of a broad class of signals,
it does not necessarily provide a meaningful interpretation
when the signals lack global stationarity.
For example, consider the time series
formed by a typical passage of music.
An estimate of its power spectrum
tells us which musical notes are
present (how much energy there is around
each of the frequencies), but fails to tell us when each
of those notes was sounded.

Much of the recent focus of signal processing is on the
so-called *time-frequency* (TF) methods, which
allow us to observe how a spectral estimate evolves over time.
One of these TF methods,
the short-time Fourier transform
(STFT), has been used extensively for analyzing speech,
music, and other non-stationary signals.
Suppose we want to perform a STFT analysis, but
are uncertain what the window size should be.
We could
perform the STFT of a signal, *s*(*t*),
using a window of relatively short duration,
then stretch the window out a small amount and compute another STFT,
and so on, gradually increasing the window size and computing another STFT
for each value of window size.
Stacking uncountably many of these STFTs on top of one
another results in a continuous volumetric
representation of *s* that is a function of time, frequency,
and the size of the window (Fig. 1(a)).

**Figure 16:** FIGURE GOES SOMEWHERE IN THIS GENERAL VICINITY

We will refer to this
volumetric representation as the *time-frequency-scale* (TFS)
transform.

Another time-frequency representation (which might more appropriately be called
a time-scale representation) is the well-known wavelet
transformheil-walnut,mallat:theory,daubechies:transform,strangwavelets.
The wavelet transform can be expressed as an inner product
of the signal under analysis with a family of
translates and dilates of one
basic primitive. This primitive is known as the *mother wavelet*.
A member of the wavelet family is produced by a particular
one-dimensional affine
coordinate transformation acting on the time axis of the mother wavelet;
this geometric transformation is parameterized
by two numbers (corresponding to the amounts of translation and
dilation).
The continuous wavelet transform is formed by
taking inner products of the signal
with the uncountably many members of the two-parameter wavelet family.
The continuous wavelet transform is,
with an appropriate choice of window/mother-wavelet,
simply the time-scale (TS) plane of the TFS
volume (Fig. 1(a)).

We begin to see that, even if it is not practical from a computational
or data-storage point of view, the
*time-frequency-scale* space is useful from a conceptual point
of view. In particular, if we only desire the magnitude TFS volume,
we can easily extract this information from the
Wigner distribution, by the appropriate coordinate transformations
and uniform smoothing of the coordinate-transformed
Wigner distributions.
A continuous transition from the magnitude TF plane (spectrogram) to
the magnitude TS plane (scalogram) is possible
through appropriate smoothing of the Wigner
distributionrioulgeneralclass.

Now suppose we were to
multiply the signal, *s*(*t*), by a linear FM (chirp) signal
and then
compute its STFT.
If we vary the __hirp rate, c, continuously,
and repeat the process uncountably many times,
stacking the resulting STFTs one above
the other, we obtain a different three-dimensional
volume (Fig. 1(b)).
This time
we have a function of time, frequency and
chirprate.
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Of course there is no reason to limit ourselves to a choice between
these two parameter spaces;
to motivate what follows, it will prove helpful to keep in mind
a continuous
four-dimensional ``time-frequency-scale-chirprate''
(TFSC) parameter space.
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Thu Jan 8 19:50:27 EST 1998