E1C02 09/14/2010
13:35:18 Page 58
Fourier Cosine Series
If y(t) is even, its Fourier series will contain only cosine terms:
y t
ð Þ ¼
X 1
n¼1
A n cos
2pnt
T
¼
X 1
n¼1
A n cos nvt
ð2:22Þ
Fourier Sine Series
If y(t) is odd, its Fourier series will contain only sine terms
y t
ð Þ ¼
X 1
n¼1
B n sin
2pnt
T
¼
X 1
n¼1
B n sin nvt
ð2:23Þ
Note: Functions that are neither even nor odd result in Fourier series that contain both sine and cosine
terms.
Example 2.3
Determine the Fourier series that represents the function shown in Figure 2.14.
KNOWN T ¼ 10 (i.e., À5 to +5)
A 0 ¼ 0
FIND The Fourier coefficients A 1 , A 2 , . . . and B 1 , B 2 , . . .
SOLUTION Since the function shown in Figure 2.14 is odd, the Fourier series will contain only
sine terms (see Eq. 2.23):
y t
ð Þ ¼
X 1
n¼1
B n sin
2pnt
T
where
B n ¼
2
10
ð 0
À5
À1
ð Þsin
2pnt
10
dt þ
ð 5
0
1
ð Þsin
2pnt
10
dt
!
B n ¼
2
10
10
2np
cos
2pnt
10
! 0
À5
þ
À10
2np
cos
2pnt
10
! 5
0
(
)
B n ¼
2
10
10
2np
1 À cos Ànp
ð
ÞÀcos np
ð Þ þ 1
½
Š
&
'
t
5
–5
1
–1
0
y
Figure 2.14 Function represented by a Fourier
series in Example 2.3.
58 Chapter 2 Static and Dynamic Characteristics of Signals