Biomedical Engineering Reference
In-Depth Information
Convolution Theorem
The convolution between two signals,
x
1
(
) and
x
2
(
), in the time domain is defined as
t
t
1
c
ð
t
Þ¼
x
ð
t
Þ
x
ð
t
t
Þ
d
t
¼
x
ð
t
Þ
x
ð
t
Þ
ð
11
:
14
Þ
*
1
2
1
2
1
where * is shorthand for the convolution operator. The convolution has an equivalent
expression in the frequency domain
ðÞ¼
Fc
ð
fg
¼
Fx
o
ðÞ
ðÞ
g
¼
X
1
o
ðÞ
X
2
o
ðÞ:
ð
11
:
15
Þ
C
f
x
*
1
2
Convolution in the time domain, which is relatively difficult to compute, is a straightfor-
ward multiplication in the frequency domain.
Next, consider the convolution of two signals,
X
2
(o), in the frequency domain.
The convolution integral in the frequency domain is expressed as
X
1
(o) and
1
X
ð
o
Þ¼
X
1
ð
n
Þ
X
2
ð
o
n
Þ
d
n
¼
X
1
ð
o
Þ
*
X
2
ð
o
Þ
ð
11
:
16
Þ
1
It can be shown that the inverse Fourier transform (IFT) of
X
(o)is
x
ðÞ¼
F
1
g ¼
F
1
f
X
ðÞ
o
f
X
1
o
ðÞ
*
X
2
o
ðÞ
g ¼
2p
x
ðÞ
x
ðÞ
ð
11
:
17
Þ
1
2
Consequently, the convolution of two signals in the frequency domain is 2
times the prod-
uct of the two signals in the time domain. As we will see subsequently for linear systems,
convolution is an important mathematical operator that fully describes the relationship
between the input and output of a linear system.
p
EXAMPLE PROBLEM 11.11
What is the FT of 3 sin (25
)
þ
4 cos (50
)? Express your answer only in a symbolic equation.
t
t
Do not evaluate the result.
Solution
F
f
3 sin
ð
25
t
Þþ
4 cos
ð
50
t
Þg ¼
3
F
f
sin
ð
25
t
Þg þ
4
F
f
cos
ð
50
t
Þg
11.5.6 Discrete Fourier Transform
In digital signal applications, continuous biological signals are first sampled by an
analog-to-digital converter (see Figure 11.4) and then transferred to a computer, where they
can be further analyzed and processed. Since the Fourier transform applies only to contin-
uous signals of time, analyzing discrete signals in the frequency domain requires that we
first modify the Fourier transform equations so they are structurally compatible with the
digital samples of a continuous signal.
