2.1 Introduction

In this chapter we present the basic tools and techniques for carrying out Fourier transforms of suitable functions without using integration. In the rest of the book the definitions and results given here will be used to obtain useful results relatively quickly and easily. Some of these results are well established, but these derivations will serve as valuable illustrations of the method, indicating how similar or related problems may be tackled.

The method has already been outlined in Chapter 1. First, the function to be transformed is described formally in a suitable and precise notation. This defines the function in terms of some very basic, or elementary, functions, such as rectangular pulses or ?-functions, which are combined in various ways, such as by addition, multiplication, or convolution. Each of these elementary functions has a Fourier transform, the function and its transform forming a transform pair. Next, the transform is carried out by using the known set of pairs to replace each elementary waveform with its transform, and also by using a set of established rules that relates the way the transforms are combined to the way the input functions were combined. For example, addition, multiplication, and convolution of functions transform to addition, convolution, and multiplication of transforms, respectively. Finally, the transform expression needs interpretation, possibly after rearrangement. Diagrams of the functions and transforms can be helpful and are widely used here.

We begin by defining the notation used. Some of these terms, such as rect and sinc,...