E1C02 09/14/2010
13:35:19 Page 62
that the rectified sine wave is an even function. The coefficient A 0 is determined from
A 0 ¼ 2
1
T
ð T=2
0
y t
ð Þdt
"
#
¼
2
1=60
ð 1=120
0
120 sin 120ptdt
with the result that
A 0 ¼
2 Â 60 Â 2
p
¼ 76:4
The remaining coefficients in the Fourier series may be expressed as
A n ¼
4
T
ð T=2
0
y t
ð Þcos
2npt
T
dt
¼
4
1=60
ð 1=120
0
120 sin 120 pt cos nptdt
For values of n that are odd, the coefficient A n is identically zero. For values of n that are even, the
result is
A n ¼
120
p
À2
n À 1
þ
2
n þ 1
ð2:24Þ
The Fourier series for the function 120 sin 120ptj
j
is
76:4 À 50:93 cos 240pt À 10:10 cos 480pt À 4:37 cos 720pt . . .
Figure 2.17 shows the amplitude versus frequency content of the rectified signal, based on a Fourier
series expansion.
COMMENT The frequency content of a signal is determined by examining the amplitude of
the various frequency components that are present in the signal. For a periodic mathematical
function, expanding the function in a Fourier series and plotting the amplitudes of the contributing
sine and cosine terms can illustrate these frequency contributions.
0
0
40
60
100
120
80
20
0.005
0.01
Time (s)
Voltage
0.015
0.02
Figure 2.16 Rectified sine wave.
62 Chapter 2 Static and Dynamic Characteristics of Signals