Signals and Systems - Electrical Engineering

CHAPTER 5 Frequency Analysis: The Fourier Transform...........................................

Imagination is the beginning of creation.

You imagine what you desire, you will what you imagine,

and at last you create what you will.

George Bernard Shaw (1856–1950)

Irish dramatist

5.1 Introduction

In this chapter we continue the frequency analysis of signals. In particular, we will concentrate in the
following issues:

n Generalization of the Fourier series—The frequency representation of signals as well as the frequency
response of systems are tools of great significance in signal processing, communications, and
control theory. In this chapter we will complete the Fourier representation of signals by extend-
ing it to aperiodic signals. By a limiting process the harmonic representation of periodic signals
is extended to the Fourier transform, a frequency-dense representation for nonperiodic signals.
The concept of spectrum introduced for periodic signals is generalized for both finite-power and
finite-energy signals. Thus, the Fourier transform measures the frequency content of a signal, and
unifies the representation of periodic and aperiodic signals.
n Laplace and Fourier transform—In this chapter the connection between the Laplace and the Fourier
transforms will be highlighted for computational and analytical reasons. The Fourier transform
turns out to be a very important case of the Laplace transform for signals of which the region of
convergence includes thejaxis. There are, however, signals where the Fourier transform cannot
be obtained from the Laplace transform; for those cases, properties of the Fourier transform will
be used. The duality of the direct and inverse transforms is of special interest in computing the
Fourier transform.
n Basics of filtering—Filtering is an important application of the Fourier transform. The Fourier rep-
resentation of signals and the eigenfunction property of LTI systems provide the tools to change

Signals and Systems Using MATLAB®. DOI: 10.1016/B978-0-12-374716-7.00008-9
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