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1
Fourier Series
Any periodic function X
P
(t) can be represented by an infinite
sum of properly weighted sine and cosine functions of the
proper frequencies
X
P
(t) = a
0
+
∑
a
k
cos (2
π
k f
0
t )
+
b
k
sin (2
π
k f
0
t)
∞
k =1
Where
f
0
is the fundamental frequency
= 1 / T
and T
is the period of
x(t)
i.e.
x (t) = x (t + T)
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Fourier Series Illustration
Let us show how we can approximate a periodic Square wave of amplitude 1 and frequency
of f
0
=1 Hz by adding sinusoids of varying frequencies and amplitudes
b
k
sin (2
π
k
f
0
t
)
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1
0.5
0
0.5
1
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1
0.5
0
0.5
1
(
1/1
)
sin( 2
π
*
1
*1*
t
)
Approximated
square wave
base on
1
sinusoid
b
k
k
f
0
3
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1
0.5
0
0.5
1
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1
0.5
0
0.5
1
(1/1)
sin(2
π
*
1
*1*
t
)
+
(1/3)
sin(2
π
*
3
*1*
t
)
Approximated
square wave
base on
2
sinusoid
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0
0.2
0.4
0.6
0.8
1
1.2
1.4
1
0.5
0
0.5
1
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1
0.5
0
0.5
1
(1/1)
sin(2
π
*
1
*1*
t
)
+
(1/3)
sin(2
π
*
3
*1*
t
)
+
(1/5)
sin(2
π
*
5
*1*
t
)
Approximated
square wave
base on
3
sinusoid
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This note was uploaded on 10/06/2010 for the course EE EE 228 taught by Professor Saghri during the Spring '10 term at Cal Poly.
 Spring '10
 SAGHRI
 Frequency

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