TABLE OF CONTENTS In this chapter, both the Fourier transform and Fourier series will be discussed. The properties of the Fourier transform will be presented and the concept of impulse function will be introduced. The definition of convolution and its relation with Fourier transform will be presented. Examples of some commonly used Fourier transforms are given and the results are presented in a table for quick reference. There are many good books on this subject. A few of the books are listed at the end of this chapter. Readers with a background in this area can skip this chapter. However, Examples 3.10 and 3.11 may be of interest, because the former one is related to the radar pulse train and the latter will be used in the Hilbert transform.
Fourier series was introduced in 1807 by a French engineer, Jean Baptiste Joseph de Fourier (1768 1830). He suggested that any arbitrary function defined over a finite interval by any piecewise graph, continuous or discontinuous, could be represented as an infinite sum of continuous functions such as sine and cosine. Although almost all the members of the French Academy questioned its validity, it turned out to be one of the most powerful tools in signal processing.
The basic concept of the Fourier transform is that any function in the time domain can be represented by an infinite number of sinusoidal functions. The Fourier transform is defined as follows. A function x( t) in the time domain t
