E1C02 09/14/2010
13:35:18 Page 57
A 0 ¼
1
T
ð T=2
ÀT=2
y t
ð Þdt
A n ¼
2
T
ð T=2
ÀT=2
y t
ð Þcos nvtdt
B n ¼
2
T
ð T=2
ÀT=2
y t
ð Þsin nvtdt
ð2:17Þ
where n ¼ 1, 2, 3, . . . , and T ¼ 2p/v is the period of y(t). The trigonometric series that results from
these coefficients is a Fourier series and may be written as
y t
ð Þ ¼ A 0 þ
X 1
n¼1
A n cos nvt þ B n sin nvt
ð
Þ
ð 2:18Þ
A series of sines and cosines may be written as a series of either sines or cosines through the
introduction of a phase angle, so that the Fourier series in Equation 2.18,
y t
ð Þ ¼ A 0 þ
X 1
n¼1
A n cos nvt þ B n sin nvt
ð
Þ
may be written as
y t
ð Þ ¼ A 0 þ
X 1
n¼1
C n cos nvt À f n
ð
Þ
ð 2:19Þ
or
y t
ð Þ ¼ A 0 þ
X 1
n¼1
C n sin nvt þ f
Ã
n
À
Á
ð2:20Þ
where
C n ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
A
2
n þ B
2
n
q
tan f n ¼
B n
A n
and tan f
Ã
n ¼
A n
B n
ð2:21Þ
Even and Odd Functions
A function g(t) is even if it is symmetric about the origin, which may be stated, for all t,
g Àt
ð Þ ¼ g t
ð Þ
A function h(t) is odd if, for all t,
h Àt
ð Þ ¼ Àh t
ð Þ
For example, cos nt is even, while sin nt is odd. A particular function or waveform may be even,
odd, or neither even nor odd.
2.4 Signal Amplitude And Frequency 57