ABSTRACT
The idea of wavelets is to keep the wave idea, but drop the periodicity. So we may consider a wavelet to be a little part of a wave: a wave which is only non-zero in a small region. Figure 15.1 illustrates the idea. Figure 15.1(a) is just the graph of
y = sin(x)
for −10 ≤ x ≤ 10, and Figure 15.1(b) is the graph of a scaled version of
y = (1− x2)e−x2/2
over the same interval. This second function, in the more general form
y = 2√
3σpi1/4
( 1− x
σ2
) e−x
is known as the Mexican hat wavelet. Suppose we are given a wavelet. What can we do with it? Well, if
f = w(x)
is the function that defines our wavelet, we can:
Dilate it by applying a scaling factor to x: f(2x) would “squash” the wavelet; f(x/2) would expand it
Translate it by adding or subtracting an appropriate value from x: f(x − 2) would shift the wavelet 2 to the right; f(x+ 3) would shift the wavelet 3 to the left
Change its height by simply multiplying the function by a constant.
