Harmonic Analysis, Basis Functions, Wavelets and Multi-Resolution Analysis February 21, 2006
The order in the title is significant. Say you have some signal, and want to analyze, to figure out what its components are. i.e., is this signal the sum of some set of basis functions ? For example, the Fourier transform of a signal gives that signal in terms of a set of sinusoids, the sinusoind being the Fourier basis function. A given sinusoid, sin(2πft) is at a fixed frequency f, over all time. Imagine another kind of basis function, which is local to some time range, but contains several “frequency components”. This is esssentially a wavelet. One method for determining the appropriate set of wavelets for a given signal, is by doing multi-resolution analysis, where you iteratively “zoom out” and look at the signal at coarser and coarser grain.
The wavelets have the property that:
i.e., the wavelet function has local support, or is localized in time. (Unlike, say,
sin(t)which is localized in frequency but not localized in time and has infinite support.) Also:
i.e., most of the energy is confined to a finite duration [Rao].
We can decompose and function into a smooth component, say,
fj(t) and a set of detail components, say,
The most concise and clear explanation of wavelets for beginners that I have seen is in Ra0 and Bopadikar, “Wavelet transforms, introduction to theory and applications” (Addisson-Wesley). The tutorial accompanying the Mathematica Wavelet Toolkit is also fairly helpful. I cursorily looked at Daubechies’s “Ten Lectures on Wavelets”(SIAM), and liked what I saw.