Our objective in this chapter is to develop the basic properties associated with the Fourier transform and its inverse. This transform is particularly well-suited for determining the spectral character of energy signals such as those that arise in the analysis of linear shift-invariant systems. We also extend our analysis to the two-dimensional Fourier transform commonly used in optics applications, and to the related Hilbert transform, fractional Fourier transform, and wavelet transform.

9.1 Introduction

In Chapter 8 we found that periodic functions are classified as power signals that can be represented by a Fourier series. Nonperiodic functions that are also square-integrable are classified as energy signals, and they can be analyzed through application of the Fourier transform.

The use of Fourier transforms in mathematics, physics, and engineering applications dates back to the pioneering work of Joseph Fourier (see historical comments in Chapter 8). During the last few decades, however, there have been significant generalizations of the idea of integral transform, and many new uses of the transform method have evolved. Advances in computing power and software have also contributed to the increased feasibility of numerical evaluation of Fourier integrals and series.

Integral transforms of various types have become essential working tools of nearly every engineer and applied scientist. Of several varieties, the integral transforms of Fourier and Laplace are the most widely used in practice. In this chapter we consider the Fourier transform and briefly discuss some closely related transforms. The Fourier transform is basic to the spectral analysis of time-varying waveforms and linear systems. We introduce the Laplace transform in Chapter 10, which is basic to circuit analysis and control theory.

9.2 Fourier Integral Representations

In the last chapter we showed that a variety of periodic functions can be represented by a Fourier series. Here we wish to examine what happens when the period of a periodic function is allowed to become unbounded. The limiting result is clearly a function defined on the entire x-axis that is not periodic.

If f(x) is a piecewise continuous periodic function, with period T = 2p and a piecewise continuous derivative, it has the pointwise convergent Fourier series representation (see Section 8.2)