Following are certain canonical signals and their automatic analyses by Xwpl.
We first analyze a relatively smooth transient, spread over 9 samples in a 256 sample signal (figure A.8).
Figure A.8: Representing a Fast Transient
Notice that the wavelet analysis at the right correctly localizes the peak in the high-frequency components, but is forced to include poorly localized low-frequency elements as well. The best-basis analysis finds the optimal representation within the library, which in this case is almost a single wavelet packet.
The second signal is taken from a recording (at 8012 samples per second) of a person whistling (figure A.9), using WPLab, the NeXTstep equivalent of Xwpl.
Figure A.9: Representing a Whistle
Here the wavelet basis is only able to localize the frequency within an octave, even though the best-basis analysis shows that it falls in a much narrower band. The vertical stripes among the wavelet Heisenberg boxes may be used to further localize the frequencies, but the best-basis decomposition performs this analysis automatically.
Let us now combine the transient and periodic parts in different ways. For example, we may take a critically damped oscillator which receives an impulse, and decompose the resulting solution in the wavelet and best-level bases, as in figures A.10 and A.11. The wavelet decomposition locates the discontinuity at the impulse, while the best-level analysis finds the resonant frequency of the oscillator more precisely.
Figure A.10: Critically Damped Harmonic Oscillator (Wavelet Basis)
Figure A.11: Critically Damped Harmonic Oscillator (Best Level)
Figure A.12: Critically Damped Harmonic Oscillator (Best Basis)
The exponential decay of the amplitude is visible in both analyses.
A chirp is an oscillatory signal with increasing modulation. Take for example
the functions 
and 
on the interval
0<x<1024, sampled 1024 times (figure A.13). The modulation increases
linearly and quadratically, respectively. The Heisenberg boxes form a line and
a parabolic arc, respectively. In the best-level analyses, all the Heisenberg
boxes have the same aspect ratio, which is appropriate for a line. In the
best-basis analysis, the Heisenberg boxes near the zero-slope portion have
smaller aspect ratio than those near the large-slope portion.
Figure A.13: Linear and Quadratic Chirps
Such a time-frequency analysis can separate superposed chirps. In figure
A.14 are pairs of linear chirps, differing either by modulation law
or phase. Both are functions on the interval 0<t<1024, sampled 1024 times. On
the left is the function 
analyzed in the
best wavelet packet basis. Note that the milder slope chirp is represented by
Heisenberg boxes of lower aspect ratio. On the right is 
, analyzed by best-level wavelet packets. The
downward-sloping line comes from the aliasing of negative frequencies.
Figure A.14: Superposed Chirps