Scale and resolution

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 5.1 (Formula Presented) 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.