Version 3 of wavelet

Updated 2005-04-21 10:37:49 by YAW

The Wikipedia's description [L1 ], while typically accurate, is too abstract for CL's taste. Here's the idea: 'know Laplace or Fourier? 'Seen "sonograms" that resolve, say, a voice signal into its constituent frequencies? Perhaps you have some notion of the ubiquity and potency of such techniques. They rely on the duality of a signal (typically conceived along a time dimension) and its frequency analysis (the signal decomposed into the "harmonics" that make it up).

The signal and its constituent frequencies are ideally "infinite", that is, defined over the entire real (or complex, ...) line. Instead of those pure tones, consider instead signals limited in time--a note that starts and stops. Doesn't that hint at a better match for physical reality? And what if the calculus of signal decomposition into band- and time-limited resolvents were tweaked so that its calculations matched up particularly well with computer capabilities, by, for example, admitting natural expressions in binary? At that point, you'd have powerful, very high-performance techniques for analysis of signals in terms of physically-meaningful fundamental bases.

You'd have wavelets.

As wavelet theory remains young, there's a lot of exploration and interfacing still to do. Tcl, of course, makes a great partner.

aricb: I'm familiar with spectrograms, which are commonly used in phonetics to analyze speech signals (and which the Snack extension can produce). My understanding of spectrograms is that they are derived from Fourier analysis of a signal. Can somebody explain how a wavelet differs from a spectrogram?

Serge Kazantzev Time-scale (wavelet transforms) analysis is often presented as an alternative method to time-frequency (fourier transforms) analysis, and strong links exist between the two. On-line tutorials such as [L2 ] offer an exploration the differences and similarities between the two.

Wavelet spectogram called scalogram, refers to a time-scale energy distribution defined as the square modulus of the wavelet coefficients. Scalogram tastes like spectogram, however there is no trivial transition between scalogram and spectrogram.

Category Glossary