To each wavelet packet or local trigonometric function we can associate a time
t and a frequency f. These will be uncertain by amounts 
and
, respectively. The result may be interpreted as a rectangular patch
of dimensions 
by 
, located around 
. We shall call
the patch a phase cell, or Heisenberg box, in honor of the uncertainty
principle, which limits how small the area of the patch may be. The cells may
be colored in proportion to the amplitude of the corresponding wavelet packet
component.
Figure A.4: Cells in the Phase Plane.
An orthonormal basis corresponds to a disjoint cover of the phase plane by phase cells (Heisenberg boxes). Certain bases have characterizations in terms of the shapes of the boxes present in the cover. For example, the standard basis consists of the cover by the tallest, thinnest patches allowed by the sampling interval. The Fourier transform may be regarded as the transpose of the standard basis, in the sense that the cells are transposed by interchanging time and frequency (cf. figure A.5). The standard basis has optimal time localization and no frequency localization, while the Fourier basis has optimal frequency localization, but no time localization.
Figure A.5: Phase Plane Decomposition by the Standard and Fourier Bases.
Windowed Fourier or cosine transforms with a fixed window size correspond to
covers with congruent cells whose width 
is the window
width. The ratio of frequency uncertainty to time uncertainty is the aspect
ratio of the cells.
Figure A.6: Phase Plane Decomposition by Windowed Cosine Transforms.
The wavelet basis is an octave-band decomposition of the phase plane, as in figure A.7.
Figure A.7: Phase Plane Decomposition by Wavelet Transform.
The best-basis of wavelet packets fits a cover to the signal so as to
minimize the amount of dark phase cell boxes. The compressibility of a
sampled signal is easily seen to be the ratio of the total area of the phase
plane (
for a signal sampled at N points) divided by the total
area of the dark cells (each of area N). This method allows
rectangles of all aspect ratios. The best-level or adapted subband basis
fits a cover of equal aspect ratio rectangles to the signal, so as to
minimize the amount of dark.
We may automatically analyze signals by expanding them in the best basis,
then drawing the corresponding phase plane representation. As is clear, the
negligible components will not be drawn, as it is not relevant which
particular basis is chosen for a subspace containing negligible energy.