10 - Applications of the Fourier integral

Summary

INTRODUCTION

The Fourier transform is one of the most important tools in the study of the transfer of signals in control and communication systems. In chapter 1 we have already discussed signals and systems in general terms. Now that we have the Fourier integral available, and are familiar with the delta function and other distributions, we are able to get a better understanding of the transfer of signals in linear time-invariant systems. The Fourier integral plays an important role in continuous-time systems which, moreover, are linear and time-invariant. These have been introduced in chapter 1 and will be denoted here by LTC-systems for short, just as in chapter 5.

Systems can be described by giving the relation between the input u(t) and the corresponding output or response y(t). This can be done in several ways. For example, by a description in the time domain (in such a description the variable t occurs), or by a description in the frequency domain. The latter means that a relation is given between the spectra (the Fourier transforms) U(ω) and Y(ω) of, respectively, the input u(t) and the response y(t).

In section 10.1 we will see that for LTC-systems the relation between u(t) and y(t) can be expressed in the time domain by means of a convolution product. Here the response h(t) to the unit pulse, or delta function, δ(t) plays a central role.