Biomedical Engineering Reference
In-Depth Information
version obtained with the Fourier transform,
(o). The frequency domain expression there-
fore provides all of the necessary information for the signal and allows one to analyze and
manipulate biological signals from a different perspective.
X
EXAMPLE PROBLEM 11.9
Find the Fourier Transform (FT) of the rectangular pulse signal
x
ð
t
Þ¼
1,
jj<
a
0,
jj>
a
Solution
Equation (11.6) is used.
a
a
e
j
o
t
dt
¼
e
j
o
t
j
o
2 sin o
a
o
X
ðoÞ¼
¼
a
a
As for the Fourier series representation of a signal, the magnitude and the phase are important
attributes of the Fourier transform. As stated previously,
X
(o) is a complex valued function,
meaning that it has a real, Re{
X
(o)}, and imaginary, Im{
X
(o)}, component and can be expressed as
X
ð
o
Þ¼
Re
f
X
ð
o
Þg þ
j
Im
f
X
ð
o
Þg:
ð
11
:
8
Þ
As for the Fourier series, the magnitude determines the amplitude of each complex expo-
nential function (or equivalent cosine) required to reconstruct the desired signal,
x
(
t
), from
its Fourier transform
q
Re
2
2
j
X
ð
o
Þ
j ¼
f
X
ð
o
Þg
þ
Im
f
X
ð
o
Þg
ð
11
:
9
Þ
In contrast, the phase determines the time shift of each cosine signal relative to a reference
of time zero. It is determined as
Im
f
X
ð
o
Þg
tan
1
y
ð
o
Þ¼
:
ð
11
:
10
Þ
Re
f
X
ð
o
Þg
Note the close similarity for determining the magnitude and phase from the trigonometric
and compact forms of the Fourier series (Eqs. (11.4a, b, c)). The magnitude of the Fourier
transform,
j
X
ð
o
Þ
j
, is analogous to
A
m
, whereas
a
m
and
b
m
are analogous to Re{
X
(o)} and
Im{
X
(o)}, respectively. The equations are identical in all other respects.
EXAMPLE PROBLEM 11.10
Find the magnitude and phase of the signal with the Fourier transform
1
X
ð
o
Þ¼
1
þ
j
o
Continued
