Biomedical Engineering Reference
In-Depth Information
Redefining coefficients:
a
n
−
jb
n
a
n
+
jb
n
c
n
=
,
c
−
n
=
,
and
c
0
=
a
0
2
2
Then, the Fourier expression for a function may be written as (12.12):
∞
c
n
e
jn
ω
0
t
c
−
n
e
−
jn
ω
0
t
c
0
f
(
t
)
=
+
+
(12.12)
n
=
1
Letting
n
range from 1 to
∞
is equivalent to letting
n
range from
−∞
to
+∞
, including 0.
∞
c
n
e
jn
ω
0
t
Then
f
(
t
)
=
n
=−∞
The coefficients
c
n
can be evaluated in terms of
a
n
and
b
n
as shown in (12.13).
a
n
+
1
2
1
2
c
n
b
n
a
n
tan
−1
c
n
|
| =
b
n
=
and
φ
=
(12.13)
n
e
j
θ
n
c
n
= |
c
n
|
c
n
= |
e
−
j
θ
n
c
n
=
c
n
|
Note that in (12.13), the coefficients of the exponential Fourier Series are
1
/
2
the mag-
nitude of the geometric Fourier Series coefficients.
12.6 LIMITATIONS
Overall, the Fourier Series method has several limitations in analyzing linear systems for
the following reasons:
1. The Fourier Series can be used only for inputs that are periodic. Most inputs in
practice are nonperiodic.
2. The method applies only to systems that are stable (systems whose natural
response decays in time).
The first limitation can be overcome if we can represent the nonperiodic input,
f
(
t
)
,
in terms of exponential components, which can be done by either the Laplace or
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