## 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].

**Multiresolution Analysis:**

We can decompose and function into a smooth component, say, `fj(t) `

and a set of detail components, say, `dk(t) `

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.

## One Response to “Harmonic Analysis, Basis Functions, Wavelets and Multi-Resolution Analysis”

abbMarch 13th, 2006 at 1:42 pm | Permalinktechnically i’m as hard as steel