Biomedical Engineering Reference
In-Depth Information
Solution
The signal has to be put in a recognizable form similar to Eq. (11.8). To achieve this,
1
1
j o
1
j o
1
o
X ðoÞ¼
o
o ¼
o 2 ¼
o 2 j
o 2 :
1
þ j
1
j
1
þ
1
þ
1
þ
Therefore
1
o
Re
f
X ðoÞ
g ¼
and Im
f X ðoÞg ¼
þ o 2 :
1
þ o 2
1
Using Eqs. (11.9) and (11.10), the magnitude is
1
j
X ð
o
Þ
o 2
1
þ
and the phase
tan 1
tan 1
y
ð
o
Þ¼
ð
o
Þ¼
ð
o
Þ:
11.5.5 Properties of the Fourier Transform
In practice, computing Fourier transforms for complex signals may be somewhat tedious
and time consuming. When working with real-world problems, it is therefore useful to have
tools available that help simplify calculations. The FT has several properties that help
simplify frequency domain transformations. Some of these are summarized following.
Let
x 1 (
t
) and
x 2 (
t
) be two signals in the time domain. The FTs of
x 1 (
t
)and
x 2 (
t
) are
represented as
X 1 (o)
¼ F
{
x 1 (
t
)} and
X 2 (o)
¼ F
{
x 2 (
t
)}.
Linearity
The Fourier transform is a linear operator. Therefore, for any constants
a 1 and
a
2 ,
F f a 1 x 1 ð t Þþ a 2 x 2 ð t Þg ¼ a 1 X 1 ð
o
Þþ a 2 X 2 ð
o
Þ
ð
11
:
11
Þ
This result demonstrates that the scaling and superposition properties defined for a liner
system also hold for the Fourier transform.
Time Shifting/Delay
If
t t 0 ) is a signal in the time domain that is shifted in time, the Fourier transform can
be represented as
x 1 (
e j o t 0
F f x 1 ð t t 0 Þg ¼ X ð
o
Þ
ð
11
:
12
Þ
In other words, shifting a signal in time corresponds to multiplying its Fourier transform by
a phase factor, e jot 0 .
Frequency Shifting
If
o 0 ) is the Fourier transform of a signal, shifted in frequency, the inverse Fourier
transform is
X 1 (o
F 1
e j o 0 t
f
X
1 o
ð
o 0
Þ
g ¼ x ðÞ
ð
11
:
13
Þ
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