12.1 the dirichlet conditions:
DESCRIPTION
Chapter 12 Fourier series. Advantages: describes functions that are not everywhere continuous and/or differentiable. represent the response of a system to a period input and depend on the frequency of the input - PowerPoint PPT PresentationTRANSCRIPT
12.1 The Dirichlet conditions:
Chapter 12 Fourier series
Advantages:(1) describes functions that are not everywhere continuous
and/or differentiable.(2) represent the response of a system to a period input and
depend on the frequency of the input (3) using in string vibration, light scattering, input signal
transmission in electronic circuit
(1) The function must be periodic.
(2) It must be single-valued and continuous, except possibly at a finite number of finite discontinuities.
(3) It must have only a finite number of maxima and minima within one period.
(4) The integral over one period of a function must converge.
all functions may be written as the sum of an odd and an even part
Chapter 12 Fourier series
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:)(
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oddeven
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Chapter 12 Fourier series
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Chapter 12 Fourier series
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(1) At a point of finite discontinuity, , the Fourier series converges to
(2) At a discontinuity, the Fourier series representation of the function will overshoot
its value. It never disappears even in the limit of an infinite number of terms.
This behavior is known as Gibb’s phenomenon.
Chapter 12 Fourier series
dx
)]()([lim2
10
dd xfxf
1 term 2 terms
3 terms
20 terms
overshooting
12.4 Discontinuous functions
Chapter 12 Fourier series
12.5 Non-periodic functions:
period=L, no particular symmetry
period=2L, antisymmetry; odd fun
period=2L, symmetry; even fun
Ex. : Find the Fourier series of 20 ,)( 2 x xxf
46
16
3
4116
3
42
20for )2
cos()1(
163
4
3
8
4
20
1for )1(16
cos16
)2
cos(8
|)2
cos(8
)2
sin(4
|)2
sin(2
)2
cos()4
2cos(
4
2
0 ,22 ,4L period function even an is )(
),()4( and )()(
2
21
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122
2
2
0
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2
20
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0
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2
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r
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r
r
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dxrx
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rxx
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dxrx
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rxxr
dxrx
xdxrx
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Chapter 12 Fourier series
(1) make the function periodic and symmetric
(2) make the function periodic and antisymmetric
Chapter 12 Fourier series
4not zero to converges seriesFourier the 2at
)2
sin()]1)1(()(
2)1(
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)1)1(()(
16)1(
8|)
2cos()
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8
])2
sin(2
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sin(2
[4
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)2
cos(22
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cos(2
)2
sin(
))(2
sin())((2
1)
2sin(
2
1
)4
2sin()(
4
2)
4
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4
2
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Integration and differentiation (1) The Fourier series of f(x) is integrated term by term then the resulting Fourier series converges to the integral of f(x). (2) f(x) is a continuous function of x and is periodic then the Fourier series that results from differentiating term by term converges to f(x).
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r
r
r
Chapter 12 Fourier series
Ex: Find the Fourier series of 20 ,)( 3 x xxf
Sol: from the previous example
integrate (1) term by term
)1()2
cos()1(
163
4
122
2
rx
rx
r
r
integrate (1) term by term
put (3) into (2)
)3()2
sin()1(
821
rx
rx
r
r
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)2
sin()1(
96)2
sin()1(
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331
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r
rx
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r
r
r
r
12.7 Complex Fourier series
rxirxirx sincos)exp(
* ,)(2
1 ,)(
2
1
is and to relations the has
for 0
for )2
exp()2
exp(
:ityorthogonal the using
)2
exp()(1
)2
exp()(
0
0
0
0
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x
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r
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dxL
irxxf
Lc
L
irxcxf
Complex Fourier series expansion is:
Chapter 12 Fourier series
Chapter 12 Fourier series
Ex: Find a complex Fourier series for in the range xxf )( 22 x
)2
exp()1(2
)1(2
sin2
cos2
|)2
exp(1
)]exp()[exp(1
)2
exp(2
1|)
2exp(
2)
2exp(
4
1
0 (2)
04
1
0 (1)
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22
2222
2
2
22
2
2
2
20
irx
r
ix
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ir
r
ir
r
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rr
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r
r
general proof:
Chapter 12 Fourier series
12.8 Parseval’s theorem:
)(2
1)
2
1(|||)(|
1 2
1
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Lx
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exp()()()(
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exp()( ),2
exp()(
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x
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r
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L
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L
irxxg
L
irxcxf
Ex: Using Parseval’s theorem and the Fourier series for
evaluate the sum
Chapter 12 Fourier series
22 ,)( 2 x-xxf
1
4
r
r
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9
16
5
16(
1
5
1616
2
1)
3
4()(
2
1)
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1(
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163
4 using and
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16
4
1
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