The method of multiple windows may be extended further to the chirplet framework.

This further extension
makes use of the same families of multiple
windows that are used in the Thomson method,
and that we first extended to the true rectangular tiling of the TF plane,
but instead they will now be used within the context of the operators of
Table 1.
In the same way that the Thomson method consists of
computing power spectra with a plurality of windows, and
averaging the power spectra together, we compute the power CCTs with a
plurality of windows, and average the results together.
To compute an appropriately-smoothed version the chirplet transform,
we compute a CCT (11) using each one
of the multiple windows as the
mother chirplet. We then average the squared-magnitudes of the resulting
CCTs together, weighting by the eigenvalues, just as with the Thomson
method.
This gives us the CCT at a particular value of *NW*.

Alternatively, we may consider a given point in the five-dimensional CCT parameter space, say, , is given by applying the operator to the set of multiple windows and then computing the sum of absolute squared energy:

S(t_c,f_c,(_t),c,d) = _i | C_t_c,f_c,(_t),c,d g_i(t) \: | s(t) |^2

We now refer to the multiple windows as
``multiple mother chirplets'' as they have collectively taken the
role of the single mother chirplet.
They act collectively to produce an
idealized parallelogram-shaped smoothing of the TF (Wigner) distribution
(Fig. 5), where the area of the parallelogram is *NW*.

For example, if we apply a frequency shear, with parameter *c*=0.85,
to each of the mother chirplets, the new set of functions will
collectively occupy the parallelogram-shaped region of the TF plane
indicated in the Fig 5, lower right.
This energy concentration represents a single point located at
coordinates (0, 0, 0, 0.85, 0) in the averaged squared-magnitude CCT.

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

Thu Jan 8 19:50:27 EST 1998