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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., [1–3], for more theoretical details on this trans-
form.
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