Scale and Resolution

Chapter
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)

Abstract

Suppose that w = w(t) is a function of one real variable. For a > 0 and any real b, the new function wab defined by
$$w{}_{ab}(t)^{\underline{\underline {\rm {def}}} } \frac{1}{{\sqrt a }}w(\frac{{t - b}}{a}),$$
(5.1)
is a shifted and stretched copy of w. For example, if w = 1 is the indicator function of the interval [0, 1], then wab is the indicator function of the interval [b, b + a],√a divided by

Now imagine that some fixed w is a waveform that is present in a signal, centered at an unknown location t = b and scaled to an unknown width a. The collection {wab : a > 0, bε R} consists of all shifted and stretched versions of w, and can be matched with the signal to determine the best values for a and b. In this context, w is called a mother function for the collection.

Keywords

Discrete Wavelet Transform Discrete Wavelet Unitary Transformation Scaling Function Haar Wavelet
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.