Understanding Wavelets: Definition and Functions Explained

[dimensionless]

Homework Statement

What is a wavelet?

Homework Equations

Here is the Haar wavelet mother waveley function:
<tex>\{ t \mapsto \psi(2^n t-k) ; n \in \N, 0 \leq k < 2^n\}</tex>

The Attempt at a Solution

Wavelets are used for analysis. There is a wavelet function and a scaling function. There are also mother and daughter wavelets.

Everything I see talks about what a wavelet does. I can't seem to find very much written about what a wavelet is. I'm not concerned about the application, I'm just looking for a clear definition.

 

[AEM]

Here is a quote from Percival and Waldren, "Wavelet Methods for Time Series Analysis" :

"What is a wavelet? As the name suggests, a wavelet is a 'small wave'. A small wave grows and decays essentially in a limited time period. The contrasting notion is obviously a 'big wave'. An example of a big wave is the sine function, which keeps on oscillating up and down on a plot of sin(u) vs $$u \in (-\infty, \infty)$$. "

Also from Percival and Waldren:

A wavelet defined over the real axis $$( -\infty, \infty)$$ has two basic properties:

(1) The integral of $$\psi ( \cdot )$$ is zero. and,

(2) The square of $$\psi ( \cdot )$$ is unity.

Those integrals are from $$( - \infty, \infty )$$. The first property means there is as much of the wavelet below the axis as above, while the second means that the nonzero portion of the wavelet is limited in length. For example the sine function fails the second property.

There are pictures of three wavelets on page 3 of Percival and Waldren that illustrate three different wavelets.

A book on wavelets for the nontechnical reader is Barbra Hubbard's "The World According to Wavelets"