169
C H A P T E R 6
Continuous-Time Fourier
Transform
In this chapter, the continuous-time Fourier transform (CTFT), often referred to as Fourier
transform, is computed numerically. is transform is then used to solve linear systems. Also,
noise cancellation and amplitude modulation are examined as applications of Fourier transform.
6.1 CTFT AND ITS PROPERTIES
e CTFT (commonly known as Fourier transform) of a signal x.t/ is expressed as
X.!/ D
1
Z
1
x.t/e
j!t
dt: (6.1)
e signal x.t/ can be recovered from X.!/ via this inverse transform equation
x.t/ D
1
2
Z
1
1
X.!/e
j!t
d!: (6.2)
Some of the properties of CTFT are listed in Table 6.1.
Table 6.1: CTFT properties
Properties Time Domain Frequency Domain
Time Shift x(t t
0
) X(ɷ)e
-jɷt
0
Time Scaling x(at)
X
ɷ
|a| a
Linearity a
1
x
1
(t) + a
2
x
2
(t) a
1
X
1
(ɷ) + a
2
X
2
(ɷ)
Time Convolution x(t) * h(t) X(ɷ)H(ɷ)
Frequency Convolution x(t)h(t) X(ɷ) * H(ɷ)
1
Refer to the signals and systems textbooks, e.g., [13], for more theoretical details on this trans-
form.

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