chap 5 fourier series
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中華大學 資訊工程系
Fall 2002
Chap 5 Fourier Series Chap 5 Fourier Series
Page 2
Fourier Analysis
FourierSeries
FourierSeries
FourierIntegral
FourierIntegral
DiscreteFourier
Transform
DiscreteFourier
Transform
FourierTransform
FourierTransform
FastFourier
Transform
FastFourier
Transform
Discrete Continuous
Page 3
Outline
Periodic Function
Fourier Cosine and Sine Series
Periodic Function with Period 2L
Odd and Even Functions
Half Range Fourier Cosine and Sine
Series
Complex Notation for Fourier Series
Page 4
Fourier, Joseph
Fourier, Joseph 1768-1830
Page 5
Fourier, Joseph
In 1807, Fourier submitted a paper to the Academy of Sciences of Paris. In it he derived the heat equation and proposed his separation of variables method of solution. The paper, evaluated by Laplace, Lagrange, and Lagendre, was rejected for lack rigor. However, the results were promising enough for the academy to include the problem of describing heat conduction in a prize competition in 1812. Fourier’s 1811 revision of his earlier paper won the prize, but suffered the same criticism as before. In 1822, Fourier finally published his classic Theorie analytique de la chaleur, laying the fundations not only for the separation of variables method and Fourier series, but for the Fourier integral and transform as well.
Page 6
Periodic Function
Definition: Periodic Function
A function f(x) is said to be periodic with
period T if for all x
)()( xfTxf
T
f(x)
x
Page 7
Periodic Function
f(x+p)=f(x), f(x+np)=f(x) If f(x) and g(x) have period p, the the
function H(x)=af(x)+bg(x) , also has the period p
If a period function of f(x) has a smallest period p (p >0), this is often called the fundamental period of f(x)
Page 8
Periodic Function
Example Cosine Functions: cosx, cos2x, cos3x, …
Sine Functions: sinx, sin2x, sin3x, …
eix, ei2x, ei3x, …
e-ix, e-i2x, e-i3x, …
Page 9
Fourier Cosine and Sine Series
Lemma: Trigonometric System is
Orthogonal
)( ,0coscos nmnxdxmx
)( ,0sinsin nmnxdxmx
),( ,0sincos nmanynxdxmx
Page 10
Fourier Cosine and Sine Series
A function f(x) is periodic with period 2
and
10
321
3210
sincos
3sin2sinsin
3cos2coscos)(
nnn nxbnxaa
xbxbxb
xaxaxaaxf
Page 11
Fourier Cosine and Sine Series(Euler formulas)
Then
dxxfa )(
2
10
nxdxxfan cos)(
1
nxdxxfbn sin)(
1
Page 12
Fourier Cosine and Sine Series
Proof:
n
n
n
n
nnn
a
a
xdxnnxdxnna
nxdxnxa
nxdxnxbnxaa
nxdxxf
22
1.
)cos(2
1)cos(
2
1
coscos1
cossincos1
cos)(1
10
Page 13
Fourier Cosine and Sine Series
Example 5-1: Find the Fourier coefficients
corresponding to the function
Sol:
0)(2
10
dxxfa
)()2(
0 ,
0 ,)(
xfxf
xk
xkxf
Page 14
Fourier Cosine and Sine Series
Sol:
0
sinsin
coscos)(1
cos)(1
0
0
0
0
n
nx
n
nxk
nxdxknxdxk
nxdxxfan
Page 15
Fourier Cosine and Sine Series
Sol:
2,4,6,n ,0
1,3,5,n ,2cos1
cos12
coscos
sinsin)(1
sin)(1
0
0
0
0
n
nn
k
n
nx
n
nxk
nxdxknxdxk
nxdxxfbn
Page 16
Periodic Function with Period 2L
A periodic function f(x) with period 2L
2L
f(x)
x
10 sincos)(
nnn L
xnb
L
xnaaxf
Page 17
Periodic Function with Period 2L
Then
L
Ldxxf
La )(
2
10
L
Ln dxL
xnxf
La
cos)(
1
L
Ln dxL
xnxf
Lb
sin)(
1
Page 18
Periodic Square Wave
2,2
21,0
11 ,12 ,0
)(
LLp
x
xkx
xf
Page 19
Odd and Even Functions
A function f(x) is said to be even if
A function f(x) is said to be odd if
)()( xfxf
)()( xfxf
Page 20
Odd and Even Functions
)()( xfxf )()( xfxf
f(x)
x
f(x)
x
Even Function Odd Function
Page 21
Odd and Even Functions
Property
The product of an even and an odd function is odd.
evenisxfifdxxfdxxfLL
L )( ,)(2)(
0
oddisxfifdxxfL
L )( ,0)(
Page 22
Odd and Even Functions
Fourier Cosine Series
Fourier Sine Series
1
0 )( ,cos)(n
n evenisxfifL
xnaaxf
1
)( ,sin)(n
n oddisxfifL
xnbxf
Page 23
Sun of Functions
The Fourier coefficients of a sum f1+f2 are the sum of the corresponding Fourier coefficients of f1 and f2.
The Fourier coefficients of cf are c times the corresponding Fourier coefficients of f.
Page 24
Examples
Rectangular Pulse The function f*(x) is the sum in Example 1
of Sec.10.2 and the constant k.
Sawtooth wave Find the Fourier series of the function
...)5sin5
13sin
3
1(sin
4kk )(* xxxxf
)()2(,
x )(
xfxf
xifxf
Page 25
Half-Range Expansions
A function f is given only on half the range, half the interval of periodicity of length 2L. even periodic extention f1 of f
odd period extention f2 of f
Page 26
Complex Notation for Fourier Series
inx
nnecxf
)(
dxexfc inx
n )(2
1
1
0 sincos)(n
nn nxbnxaaxf
Page 27
Complex Notation for Fourier Series
A periodic function f(x) with period 2L
L
xin
nnecxf
)(
L
L
L
xin
n dxexfL
c
)(2
1
Page 28
Complex Fourier Series
Find complex Fourier series
)()2(
)(
xfxf
xifexf x
Page 29
Exercise
Section 10-4 #1,
Section 10-2 #5, #11
Section 10-3 #5, #9
Section 10-4 #1, #15
Page 30
Fourier Cosine and Sine Integrals
Example 1
)()2(
1 ,0
11 ,1
1 ,0
)(
xfLxf
Lx
x
xL
xf
Page 31
Fourier Cosine and Sine Integrals
n
nn LLn
Ln
Ldx
L
xnxf
La
Ldxxf
La
sin2
/
)/sin(2cos)(
1
1)(
2
1
1
1
1
10
x
x
1
sin(x)
Page 32
Fourier Cosine and Sine Integrals
x
x
xsin
Page 33
Fourier Cosine and Sine Integrals
Page 34
Fourier Cosine and Sine Integrals
Page 35
Fourier Cosine and Sine Integrals
L
sin2
)( A
Page 36
Gibb’s Phenomenon
Sine Integral u
duSi0
sin)(
Page 37
Gibb’s Phenomenon
Gibb’s Phenomenon
Page 38
Gibb’s Phenomenon
))1(())1((1
sin1sin1
)sin(1)sin(1
)sin()sin(1
sincos2)(
)1(
0
)1(
0
00
0
0
xaSixaSi
dtt
tdt
t
t
dx
dx
dxx
dx
xf
axax
aa
a
a
Page 39
Fourier Cosine and Sine Integrals
Fourier Cosine Integral of f(t)
0
)cos()( )( dtAtf
0
)cos()(2
)( dtttfA
Page 40
Fourier Cosine and Sine Integrals
Fourier Sine Integral of f(t)
0
)sin()( )( dtBtf
0
)sin()(2
)( dtttfB
Page 41
Fourier Integrals
Fourier Integral of f(t)
dtttfB
dtttfA
)sin()(1
)(
)cos()(1
)(
0
)sin()()cos()( )( dtBtAtf
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