Biomedical Engineering Reference
In-Depth Information
1)
n
d
n
x
(
s
)
ds
n
ˆ
t
n
x
(
t
)
=
(
−
.
(5.119)
•
If a function
x
(
t
) has Laplace transform
x
(
s
), then the Laplace transform of
x
(
t
)
ˆ
/
t
,
assuming that lim
t
0
x
(
t
)
/
t
exists, is given by
→
∞
ˆ
(
x
(
t
)
/
t
)
=
x
(
a
)
da
.
(5.120)
s
The Fourier transform
x
∗
(
t
) of time function
x
(
t
) is given by
∞
x
∗
(
ω
)
=
x
(
t
)e
i
ω
t
dt
,
(5.121)
−∞
√
−
=
with
i
1. The Fourier transform has similar properties to the Laplace
transform. The most important properties of Fourier transforms are:
•
Fourier transform is a linear operation.
•
When
x
(
t
) is a continuous function, the Fourier transform of the time derivative
x
(
t
)of
˙
x
(
t
)isgivenby
x
∗
(
t
)
x
∗
(
˙
=
i
ω
ω
) .
(5.122)
•
Convolution in the time domain is equivalent to a product in the Fourier domain.
Using two time functions
x
(
t
)and
y
(
t
) with Fourier transforms
x
∗
(
)and
y
∗
(
ω
ω
), the
following convolution integral
I
(
t
) could be defined:
∞
I
(
t
)
=
x
(
τ
)
y
(
t
−
τ
)
d
τ
.
(5.123)
−∞
In that case the Fourier transform of this integral can be written as
I
∗
(
ω
)
=
x
∗
(
ω
)
y
∗
(
ω
) .
(5.124)
If a function
x
(
t
) has a Fourier transform
x
∗
(
ω
), then the Fourier transform of the
function
t
n
x
(
t
), with
n
•
=
1, 2, 3,
...
can be written as
(
i
)
n
d
n
x
∗
(
ω
)
d
t
n
x
(
t
)
∗
=
.
(5.125)
n
ω
If a function
x
(
t
) has Fourier transform
x
∗
(
•
ω
), then the Fourier transform of
x
(
t
)
/
t
is
given by
∞
)
∗
=
(
x
(
t
)
/
t
i
ω
x
(
a
)
da
.
(5.126)
ω
Finally, Table
5.1
gives some Laplace and Fourier transforms of often used func-
tions. In the table the step function H(
t
) (Heavyside function) as defined in Eq.
(
5.18
) is used, as well as the delta function
δ
(
t
), defined as
