Biomedical Engineering Reference
In-Depth Information
domain and multiplication in the frequency domain. Another is the use of an impulse
response function and a transfer function for describing system and filter responses in the
time and frequency, respectively.
An impulse function is a generalized function that has the unusual property that it sam-
ples the integrand:
1
t t
Þ g ð t Þ dt ¼ g ð t
Þ
ð
16
:
1
Þ
0
0
1
When the transform of the impulse function is taken, the result is an exponential
1
Þ e i o t dt ¼ e i o t 0
t t
ð
16
:
2
Þ
H ðoÞ¼
0
1
which shows that a delay in time is equivalent to a multiplicative exponential delay
factor in the frequency domain. When the impulse has no delay or
t 0 ¼
0,
H
(
o
)
¼
1.0,
aconstant.
The preceding relation can be generalized to the form where
represents the Fourier
I
transform operation,
` ½ g ð t b Þ ¼ e i 2p bf G ðoÞ
ð
16
:
3
Þ
A scaling factor can be added to this time shifting/delay theorem to make it even more useful,
` ½ g ð a ð t b ÞÞ ¼ e i o b
j a j G ðo= a Þ
ð
16
:
4a
Þ
` i ,
Note that a similar relation exists for the inverse transform denoted by
½ G ð a ðo b ÞÞ ¼ e ibt
j a j
` 1
g ð t = a Þ
ð
16
:
4b
Þ
Finally, an important unique property of the impulse function is
1
j a j t Þ
at Þ¼
ð
16
:
5
Þ
EXAMPLE PROBLEM 16.1
Find the inverse Fourier transform of
R
(
o
)
¼
sin[3(
o
-
o 0 )
t 1 ].
Solution
Recognize the basic function
in
(
o
)as
G
R
o t 1 ¼ e i o t 1
e i o t 1
2
G ðoÞ¼
sin
i
Continued
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