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 pyramidalburt 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. 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.