2
Fourier Transform
2.1 INTRODUCTION AND OVERVIEW
Among all transforms used in signal and image processing, Fourier transform (FT)
is probably the most commonly used transform. In this chapter, we first describe the
definition as well as the concepts of FT and then discuss some of the properties of FT
that are commonly used in signal processing. The emphasis of this chapter is on the
conceptual interpretations as well as the applications of one-dimensional (1-D) and twodimensional (2-D) continuous and discrete FT as opposed to mathematical formulation.
2.2 ONE-DIMENSIONAL CONTINUOUS FOURIER TRANSFORM
As mentioned in Chapter 1, a signal can be expressed in many different domains
among which time is probably the most intuitive domain. Time signals can answer
questions regarding “when” events happen, whereas FT domain addresses questions
starting with “how often” (this is why FT domain is also called frequency domain).
As an example, assume that you are to study the shopping habits of members in
a community by preparing a questionnaire. You will obtain some useful information when you ask questions such as “What days do you normally go shopping?” or
“What time of the day you never go shopping?” This information helps you understand and visualize the “time” elements of people’s shopping habits. Also, if you
prepare a time signal that shows the number of people shopping at every instance of
time (i.e., a graph of number of people shopping vs. time), you can acquire answers to
all the aforementioned questions. Now, consider a different set of questions such as
“How often do you go shopping?” or “What percentages of people go shopping twice
a week?” Answers to these questions form the frequency domain, which in signal
processing is formed by FT. Let us remind ourselves that the information in time
and frequency are exactly the same, i.e., neither of the domains are more informative
than the other and one can acquire all information on one domain from the other.
However, considering the computation size and the visibility of certain information
to humans, one domain can be preferred over the other, as discussed in Chapter 1.
Now, we give a formal definition for 1-D continuous FT. Consider g(t) as a continuous signal in time. The FT of this signal, shown as G(f), is defined as follows:
+∞
G( )
f = FT {g t
( )} =
−∞
∫
g t
( ) e
− j f
2p t dt
(2.1)
where
f is the frequency variable (which is often expressed in units such as Hz, kHz,
MHz, and so on)
j is the imaginary number (i.e., j 2 = −1)
15
Fourier Transform
2.1 INTRODUCTION AND OVERVIEW
Among all transforms used in signal and image processing, Fourier transform (FT)
is probably the most commonly used transform. In this chapter, we first describe the
definition as well as the concepts of FT and then discuss some of the properties of FT
that are commonly used in signal processing. The emphasis of this chapter is on the
conceptual interpretations as well as the applications of one-dimensional (1-D) and twodimensional (2-D) continuous and discrete FT as opposed to mathematical formulation.
2.2 ONE-DIMENSIONAL CONTINUOUS FOURIER TRANSFORM
As mentioned in Chapter 1, a signal can be expressed in many different domains
among which time is probably the most intuitive domain. Time signals can answer
questions regarding “when” events happen, whereas FT domain addresses questions
starting with “how often” (this is why FT domain is also called frequency domain).
As an example, assume that you are to study the shopping habits of members in
a community by preparing a questionnaire. You will obtain some useful information when you ask questions such as “What days do you normally go shopping?” or
“What time of the day you never go shopping?” This information helps you understand and visualize the “time” elements of people’s shopping habits. Also, if you
prepare a time signal that shows the number of people shopping at every instance of
time (i.e., a graph of number of people shopping vs. time), you can acquire answers to
all the aforementioned questions. Now, consider a different set of questions such as
“How often do you go shopping?” or “What percentages of people go shopping twice
a week?” Answers to these questions form the frequency domain, which in signal
processing is formed by FT. Let us remind ourselves that the information in time
and frequency are exactly the same, i.e., neither of the domains are more informative
than the other and one can acquire all information on one domain from the other.
However, considering the computation size and the visibility of certain information
to humans, one domain can be preferred over the other, as discussed in Chapter 1.
Now, we give a formal definition for 1-D continuous FT. Consider g(t) as a continuous signal in time. The FT of this signal, shown as G(f), is defined as follows:
+∞
G( )
f = FT {g t
( )} =
−∞
∫
g t
( ) e
− j f
2p t dt
(2.1)
where
f is the frequency variable (which is often expressed in units such as Hz, kHz,
MHz, and so on)
j is the imaginary number (i.e., j 2 = −1)
15