Digital Signal Processing Reference
In-Depth Information
5
THE FOURIER TRANSFORM
The Fourier transform is a method of representing mathematical models of signals
and systems in the frequency domain. We begin to get a hint of this process as we rep-
resent periodic time-domain signals in terms of their harmonic frequency, components,
using the Fourier series. The Fourier transform is an extension of this concept.
Engineers use the Fourier transform to simplify the mathematical analysis of
signals and systems and for explaining physical phenomena mathematically. It is
widely used in the field of electrical engineering, especially in the study of electron-
ic communication signals and systems. For this reason, every student of electrical
engineering should become familiar with the Fourier transform and its applications.
In this chapter, the Fourier transform is introduced in a way that will give each
student an understanding of its mathematical basis and a glimpse at its utility in the
analysis and design of linear signals and systems. The relationship between the
Fourier transform and the Fourier series is presented with the intent to give
the reader an intuitive feeling for the Fourier transform. Mathematical properties
of the Fourier transform are presented with the emphasis on application of the
properties rather than formal, mathematical proof.
5.1
DEFINITION OF THE FOURIER TRANSFORM
We approach the definition of the Fourier transform by first considering the Fouri-
er series, which is described in Chapter 4; there the Fourier series is defined, in the
exponential form
, as
f(t) =
q
k=-
q
C
k
e
jkv
0
t
,
[eq(4.11)]
where
1
T
0
L
T
0
f(t)e
-jkv
0
t
dt.
[eq(4.23)]
C
k
=
197
