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
dð
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
dð
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
dð
t
Þ
dð
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