Xwpl is a graphical tool to analyse one-dimensional signals using adapted waveform analysis, under the X Window System.
Wavelets, wavelet packets and local trigonometric waveforms are collections of short oscillatory waveforms each of which can be viewed as a ``musical note'' having a time duration, a pitch and an amplitude (level of loudness). The waveforms used here are synthesized mathematical notes, as synthesized by different mathematical instruments (corresponding to different processes used to generate the notes).
To pursue the musical analogy further, Xwpl displays the ``musical score'' for a signal using the Phase Cell representation, in which a ``note'' (a function in the wavelet or wavelet-packet basis) is represented by a box in the time-frequency space (time is horizontal, frequency is vertical). The boxes all have the same area as a result of Heisenberg's inequality on fourier transforms. See A.3.1 for more details on how this representation should be interpreted.
For instance, the signal in figure 1.1 is a chirp of the form
, which is difficult to analyse using classical fourier theory,
as it's instantaneous frquency varies with time (as a matter of fact, even the
graphical representation is deceived into showing a signal with a varying
envelope).
Figure 1.2: Best-basis representation of the chirp
Figure 1.2 shows a ``musical transcription'' of the same chirp using a ``C 12'' Quadrature Mirror Filter or QMF. This representation shows clearly that the local frequency of the signal increases linearly with time. The darkness of each box is proportional to the intensity of the corresponding wavelet-packet coefficient. Actually, the whole square is filled with boxes, but most of the coefficients are so small or even zero that thay do not appear on the graph.
There are three types of bases when using wavelet packets: best-basis, best-level and wavelet basis. They are described in more detail later but, in a nutshell, they differ in the constraints that are placed on the shape of the boxes. In this case, the best-level basis is the same as the best-basis, but this is not necessarily the case.
Figure 1.3: Wavelet basis representation
