Biomedical Engineering Reference
In-Depth Information
f
(
t
)
f
(
t
)
Sine
Triangle wave
t
t
-T/2
-T/2
T/2
T/2
(a) Sine wave
(b) Triangular wave
FIGURE 12.4
:
Odd functions. The sine wave (a) and the triangular wave are considered to be
odd functions since they satisfy the condition
f
(
t
)
=−
f
(
−
t
)
f
(
t
)
Half-wave
t
-T/2
T/2
FIGURE 12.5
:
Half-wave symmetric function. This triangular function is considered to have
half-wave symmetry since it satisfies the condition
f
(
t
)
T
=−
f
(
t
±
2
)
The relationship between the properties of symmetry and the Fourier Series are
summarized in Table 12.10 and as follows:
1.
Even functions contain only
a
0
and
a
n
cos
n
ω
0
t
terms.
2.
Odd functions contain only
b
n
sin
n
ω
0
t
terms.
3.
Half-wave symmetry functions contain only the odd complex Fourier coeffi-
cients,
a
n
and
b
n
terms, where
n
=
1
,
3
,
5
,
7,..., etc.
Example Problem:
What is the Fourier Series of the function,
v
(
t
), in Fig. 12.6?
The function
v
(
t
) is given below.
⎧
⎨
⎩
T
V
,
0
<
t
<
/
4
v
(
t
)
=
T
3
(12.9)
−
V
,
/
<
t
<
/
4
T
4
3
V
,
/
4
T
<
t
<
T
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