1.1 Aim of the Work

The Fourier transform is a valuable theoretical technique, used widely in fields such as applied mathematics, statistics, physics, and engineering. However, the relationship between a function and its transform is given by an integral, and a certain amount of tedious integration may be required to obtain the transform in a given application. In general, the user of this mathematical tool is interested in the functions and their transforms and not in the process of obtaining one from the other, which, even if not difficult, can be complicated and require care to avoid any small slip that might lead to an error in the result. If the transform function could be obtained without integration, this would be welcomed by most users. In fact, anyone performing many transforms in a particular field, such as radar, where the spectra corresponding to various, perhaps rather similar, waveforms are required, would notice that certain waveforms have certain transforms and that certain relationships between waveforms lead to corresponding relationships between spectra. With the knowledge of a relatively small number of waveform-transform pairs and the rules for combining and scaling transforms, a very substantial amount of Fourier transform analysis can be carried out without any explicit integration at all-the integrations are prepackaged within the set of rules and pairs.

The aim of this book is to represent the rules-and-pairs approach to Fourier transforms, first defined systematically by Woodward [1], and to illustrate its use. The rules and pairs are given in Chapter...