The area occupied by a particular family of DPSS is
just the *time-bandwidth product* and is denoted by the
letters *NW* (the notation used by Thomson and others). The quantity
*N* denotes the number of samples (duration) of a window,
and *W* denotes the bandwidth collectively defined by a plurality of
such windows of equal length.
We can compute the TF plane, of a particular signal, at any desired
value of *NW*, by using the
discrete prolate sheroidal sequences (subject to the constraints
that *NW* can only be adjusted in integer increments,
and that it also has a lower bound dictated by
the uncertainty relationslepcom). If we compute the
TF plane at each possible value of *NW*, and stack these one above the
other (Fig. 4), we obtain a 3-parameter space,
where the axes are time, frequency, and resolution (1/*NW*).

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

*Hierarchical* or *pyramidal*burt representations
have been previously formulated, in the context of
image processing, using multiple scales in the physical domain
(e.g. spatial scale).
The proposed multi-resolution TF representation, however, is new.
In particular, here the scale axis is
*NW* - the area of the rectangular tiles at each level of the pyramid.
Here the scale is in the time-frequency plane, not the physical (time or space)
domain.

At this point, a reasonable question to ask might be: Why vary the area; do we not always desire maximum resolution or maximum concentration in the TF plane? The same answer we gave earlier, regarding smoothed spectral estimates, applies here.

*Smoothing* is well-known in time-frequency analysis, particularly
with the Wigner distribution where we wish to reduce or eliminate
cross terms. Many smoothing kernels have been
proposedcohen1[54].
Each of these smoothing kernels has a particular shape and many of
these are optimum in one sense or another.
The use of the DPSSs, however,
has been shown to be equivalent to a rectangular smoothing of
the Wigner distributionshenoyweyl, and therefore deserves
special attention, particularly when we wish to describe a tiling
of the TF plane in a very simple way.

We may use the result of Shenoy and Parksshenoyweyl to generalize the pyramidal true-rectangular TF tiling further, by smoothing the TF distribution with a continuously variable rectangle size. When uncountably many of these rectangularly smoothed TF planes are stacked, one above the other, a continuous volumetric parameter space results, having parameters time, frequency, and resolution.

Thu Jan 8 19:50:27 EST 1998