fourier series - national tsing hua...
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© 2012, Ching-Han Hsu, Ph.D.
Fourier Series
Ching-Han Hsu
© 2012, Ching-Han Hsu, Ph.D.
Periodic Function
• A function f(x) is called periodic, if it is
defined for every x in the domain of f and if
there is some positive number p such that
f(x+p)= f(x)
• The number p is called a period of f(x).
• If a periodic function f has a smallest
period p, this is often called the
fundamental period of f(x).
© 2012, Ching-Han Hsu, Ph.D.
Periodic Function
p
f(x)
x
© 2012, Ching-Han Hsu, Ph.D.
Periodic Function
• For any integer n,
f(x+np) = f(x), for all x
• If f(x) and g(x) have period p, then so does
h(x) = af(x) + bg(x),
a and b are constants.
© 2012, Ching-Han Hsu, Ph.D.
Trigonometric Functions
• Consider the following periodic functions
with period 2π: 1, sin x, cos x, sin2x,
cos2x, …, sin nx, cos nx, …
© 2012, Ching-Han Hsu, Ph.D.
Trigonometric Series• The trigonometric series is of the form
a0, a1, a2, a3, …, b1, b2, b3, … are real
constants and are called the coefficients of
the series.
• If the series converges, its sum will be a
function of period 2π.
1
0
22110
sincos1
2sin2cossincos1
n
nn nxbnxaa
xbxaxbxaa
© 2012, Ching-Han Hsu, Ph.D.
Fourier Series
• Assume f(x) is a periodic function of
period 2π that can be represented by the
trigonometric series:
That is, we assume that the series
converges and has f(x) as its sum.
• Question: how to compute the coefficients?
1
0 sincos1n
nn nxbnxaaxf
© 2012, Ching-Han Hsu, Ph.D.
Properties of Trigonometric
Functions
• Recall some basic properties of
trigonometric functions
0sin1sin
0cos1cos
dxnxnxdx
dxnxnxdx
0sin2
1sin
2
1
sincos
xdxmnxdxmn
nxdxmx
© 2012, Ching-Han Hsu, Ph.D.
Properties of Trigonometric
Functions
mn
mnxdxmnxdxmn
nxdxmx
0cos
2
1cos
2
1
sinsin
mn
mnxdxmnxdxmn
nxdxmx
0cos
2
1cos
2
1
coscos
© 2012, Ching-Han Hsu, Ph.D.
Determination of the Constant
Term a0
• Integrating both sides from – to :
0
1
0
2
sincos
a
dxnxbnxaadxxfn
nn
dxxfa
2
10
© 2012, Ching-Han Hsu, Ph.D.
Determination of the Coefficients an
of the Cosine Term
• Multiply by cosmx with fixed positive integer
and then integrate both sides from – to :
m
n
nn
a
mxdxnxbnxaa
mxdxxf
cossincos
cos
1
0
mxdxxfam cos
1
© 2012, Ching-Han Hsu, Ph.D.
Determination of the Coefficients bn
of the Sine Term
• Multiply by sinmx with fixed positive integer
and then integrate both sides from – to :
m
n
nn
b
mxdxnxbnxaa
mxdxxf
sinsincos
sin
1
0
mxdxxfbm sin
1
© 2012, Ching-Han Hsu, Ph.D.
Summary
1
0 sincosn
nn nxbnxaaxf
dxxfa
2
10
nxdxxfbn sin
1
nxdxxfan cos
1
© 2012, Ching-Han Hsu, Ph.D.
Fourier Series: Square Wave
• Ex. Find the Fourier coefficients of the
periodic function f(x)
xfxfxk
xkxf
2,
0
0
© 2012, Ching-Han Hsu, Ph.D.
Fourier Series: Square Wave
02
1
2
1
2
1
0
0
0
kk
kdxkdxdxxfa
0sinsin1
coscos1
cos1
0
0
0
0
nxn
knx
n
k
nxdxknxdxk
nxdxxfan
Q: Is f(x) even or odd?!
© 2012, Ching-Han Hsu, Ph.D.
nn
k
nn
kn
n
k
nxn
knx
n
k
nxdxknxdxk
nxdxxfbn
cos12
1coscos11
coscos1
sinsin1
sin1
0
0
0
0
nn coscos
2,4,0
,3,12cos1
n
nn
© 2012, Ching-Han Hsu, Ph.D.
Fourier Series: Square Wave
,5
4,0,
3
4,0,
454321
kbb
kbb
kb
xxxk
nxnn
k
nxbnxaaxf
n
n
nn
5sin5
13sin
3
1sin
4
sincos12
sincos)(
1
1
0
© 2012, Ching-Han Hsu, Ph.D.
Fourier Series: Square Wave
• Consider the series expansion of finite
sine terms:
xxxk
S
xxk
S
xk
S
5sin5
13sin
3
1sin
4
3sin3
1sin
4
sin4
3
2
1
© 2012, Ching-Han Hsu, Ph.D.
© 2012, Ching-Han Hsu, Ph.D.
Representation by Fourier Series
• Thm. If a periodic function f(x) with period
2 is peicewise continuous in the interval
-≦x≦ and has a left-hand derivative and
right-hand derivative at each point of that
interval, then Fourier series of f(x) is
convergent. Its sum is f(x), except at a
point of x0 at which f(x) is discontinuous
and the sum of the series is the average of
the left- and right-hand limits of f(x) at x0.
© 2012, Ching-Han Hsu, Ph.D.
Representation by Fourier Series
01012
1 ff
© 2012, Ching-Han Hsu, Ph.D.
Functions of Any Period p=2L
• A function f(x) of period p=2L has a Fourier
series
1
0
1
0
sincos
2
2sin
2
2cos
n
nn
n
nn
xL
nbx
L
naa
nxL
bnxL
aaxf
L
Ldxxf
La
2
10
L
Ln dx
L
xnxf
La
cos
1
L
Ln dx
L
xnxf
Lb
sin
1
,3,2,1n
© 2012, Ching-Han Hsu, Ph.D.
Fourier Series: Periodic Square Wave
• Ex. Find the Fourier series of the function
2,42,
210
11
120
LLp
x
xk
x
xf
© 2012, Ching-Han Hsu, Ph.D.
Fourier Series: Periodic Square Wave
24
1
4
1 1
1
2
20
kkdxdxxfa
2sin
2
2sin
2cos
2
1
2cos
2
1
1
1
1
1
2
2
n
n
kx
n
n
k
dxxn
kdxxn
xfan
,11,7,3,2
,9,5,1,2
nn
kan
n
ka nn
© 2012, Ching-Han Hsu, Ph.D.
Fourier Series: Periodic Square Wave
xxx
kkxf
2
5cos
5
1
2
3cos
3
1
2cos
2
2)(
02
cos2
cos
2cos
2cos
2cos
2sin
2
1
2sin
2
1
1
1
1
1
2
2
n
n
kn
n
k
n
n
kn
n
kx
n
n
k
dxxn
kdxxn
xfbn
© 2012, Ching-Han Hsu, Ph.D.
Fourier Series: Periodic Square Wave
k=5
© 2012, Ching-Han Hsu, Ph.D.
Fourier Series: Half-wave Rectifier
• Ex. A sinusoidal voltage Esinwt, is passed
through a half-wave rectifier that clips the
negative portion of the wave. Find the
Fourier series of the resulting periodic
function:
w
w
w
LLp
LttE
tLtu ,
22,
0sin
00
© 2012, Ching-Han Hsu, Ph.D.
Fourier Series: Half-wave Rectifier
,4,3,20
12
1
1sin
1
1sin
2
1
1sin
1
1sin
2
1cos1cos2
sinsinsin
0
0
0
n
nE
n
n
n
nE
n
tn
n
tnE
dttntnE
tdtntEtdtntubn
w
w
w
w
w
ww
w
ww
ww
w
w
w
w
w
w
nndn
d
n
n
n
nn
1cos1sin
1
1coslim
1
1sinlim
11
© 2012, Ching-Han Hsu, Ph.D.
Fourier Series: Half-wave Rectifier
w
w
w
w
w
w
w
w
EEt
E
tdtEdttua
0coscos2
cos2
sin22
0
00
w
w
w
w
w
w
w
w
w
w
ww
w
ww
ww
w
0
0
0
1
1cos
1
1cos
2
1sin1sin2
cossincos
n
tn
n
tnE
dttntnE
tdtntEtdtntuan
© 2012, Ching-Han Hsu, Ph.D.
Fourier Series: Half-wave Rectifier
,6,4,211
2,5,3,10
,6,4,21
2
1
2
2
,5,3,10
1
1cos1
1
1cos1
2
nnn
En
nnn
E
n
n
n
n
nEan
tt
Et
EEtu ww
w
4cos
53
12cos
31
12sin
2
© 2012, Ching-Han Hsu, Ph.D.
Fourier Series: Half-wave Rectifier
k=5 =1
© 2012, Ching-Han Hsu, Ph.D.
Even and Odd Functions
• A function e(x) is even, if e(-x)=e(x).
• A function o(x) is odd, if o(-x)=-o(x).
xe xo
© 2012, Ching-Han Hsu, Ph.D.
Even and Odd Functions: Facts
• If e(x) is an even function, then
• If o(x) is an odd function, then
• The product of an even and an odd
function is odd
LL
Ldxxedxxe
02
0L
Ldxxo
xoxexoxe
© 2012, Ching-Han Hsu, Ph.D.
Fourier Cosine Series
• Thm. The Fourier series of an even
function of period 2L is a Fourier cosine
series
with coefficients
1
0 cosn
n xL
naaxf
LL
Ldxxf
Ldxxf
La
00
1
2
1
LL
Ln dx
L
xnxf
Ldx
L
xnxf
La
0cos
2cos
1
© 2012, Ching-Han Hsu, Ph.D.
Fourier Sine Series
• Thm. The Fourier series of an odd
function of period 2L is a Fourier sine
series
with coefficients
1
sinn
n xL
nbxf
LL
Ln dx
L
xnxf
Ldx
L
xnxf
Lb
0sin
2sin
1
© 2012, Ching-Han Hsu, Ph.D.
Fourier Series: Sum of Functions
• Thm. The Fourier coefficients of a sum
f1+f2 are the sums of the corresponding
Fourier coefficients of f1 and f2.
© 2012, Ching-Han Hsu, Ph.D.
xxxk
kxf
xxxk
xf
5sin5
13sin
3
1sin
4)(
5sin5
13sin
3
1sin
4)(
*
© 2012, Ching-Han Hsu, Ph.D.
Sawtooth Wave
• Ex. Find the Fourier series of the function
xfxfxxxf 2,,
xfxxffff 2121 ,
© 2012, Ching-Han Hsu, Ph.D.
Sawtooth Wave
• Since f1 is odd, an=0 for n=1,2,3,… and
nn
nxn
nn
dxnxn
nxxn
nxxdn
nxdxx
dxL
xnxf
Lb
L
n
cos2
sin2
cos2
cos2
cos2
cos2
sin2
sin2
0
2
00
00
01
11
1 sincos2
sinnn
n nxnn
nxbxf
© 2012, Ching-Han Hsu, Ph.D.
Sawtooth Wave
xfxf
xxxxf
21
3sin3
12sin
2
1sin2
© 2012, Ching-Han Hsu, Ph.D.
Half-Range Expansion• In some applications, we often need to
employ a Fourier series for a function f(x)
that is given only on some interval, say
0≦x≦L.
Suppose that we have a choice!! We can
use even or odd periodic expansion.
© 2012, Ching-Han Hsu, Ph.D.
Half-Range Expansion
• If we apply even periodic extension, we
obtain a Fourier cosine series:
• If we apply odd periodic extension, we
obtain a Fourier sine series:
© 2012, Ching-Han Hsu, Ph.D.
Triangle Function: Half-Range Expansion
• Ex. Find the two half-range expansion of
the triangle function
LxL
xLL
k
Lxx
L
k
xf
2
22
02
© 2012, Ching-Han Hsu, Ph.D.
Triangle Function: Half-Range Expansion
Even Expansion
Odd Expansion
© 2012, Ching-Han Hsu, Ph.D.
Triangle Function: Even Periodic Expansion
LxL
xLL
k
Lxx
L
k
xf
2
22
02
222
1
221
2
2
2
1
22
2/
2/
0
00
kL
L
kL
L
k
L
dxxLL
kxdx
L
k
L
dxxeL
dxxeL
a
L
L
L
L
f
L
Lf
L
L
L
L
f
L
Lfn
dxL
xnxL
L
kdx
L
xnx
L
k
L
dxL
xnxe
Ldx
L
xnxe
La
2/
2/
0
0
cos2
cos22
cos2
cos1
© 2012, Ching-Han Hsu, Ph.D.
Triangle Function: Even Periodic Expansion
12
cos2
sin2
sinsincos
22
22
2/
0
2/
0
2/
0
n
n
Ln
n
L
dxL
xn
n
L
L
xnx
n
Ldx
L
xnx
LL
L
2coscos
2sin
2
sinsin
cos
22
22
2/2/
2/
nn
n
Ln
n
L
dxL
xn
n
L
L
xnxL
n
L
dxL
xnxL
L
L
L
L
L
L
© 2012, Ching-Han Hsu, Ph.D.
Triangle Function: Even Periodic Expansion
,14,10,6,2,0
,10
16,
6
16
0,0,0,2
16,0
1cos2
cos24
2210226
5432221
22
na
ka
ka
aaak
aa
nn
n
ka
n
n
x
Lx
Lx
L
kkxf
10cos
10
16cos
6
12cos
2
116
2 2222
© 2012, Ching-Han Hsu, Ph.D.
Triangle Function: Even Periodic Expansion
© 2012, Ching-Han Hsu, Ph.D.
Triangle Function: Odd Periodic Expansion
x
Lx
Lx
L
kxf
5sin
5
13sin
3
1sin
1
182222
2sin
8
sin2
sin1
22
0
n
n
k
dxL
xnxo
Ldx
L
xnxo
Lb
L
f
L
Lfn
© 2012, Ching-Han Hsu, Ph.D.
Triangle Function: Odd Periodic Expansion
© 2012, Ching-Han Hsu, Ph.D.
Forced Oscillations
• Consider the forced oscillation of a body of mass m on spring of modulus are governed by the equation
where y(t) is the displacement from rest, c the damping constant, and r(t) the external force depending on time t.
trtkytyctym
© 2012, Ching-Han Hsu, Ph.D.
Forced Oscillations
• The electric analog to the spring is RLC-
circuit: tEtI
CtIRtIL
1
© 2012, Ching-Han Hsu, Ph.D.
Forced Oscillations• Ex. Let m=1(g), c=0.02(g/s), k=25(g/s2), so
that
where r(t) is measured in g‧㎝/s2. Let
Find the steady-state solution y(t).
trtytyty 2502.0
trtr
tt
tttr
2,
02
02
0
5sin5cos 01.001.0
tyt
tBetAety
h
tt
h
© 2012, Ching-Han Hsu, Ph.D.
Forced OscillationsThe Fourier series of r(t) is
Now we the following linear differential
equation
ttttr 5cos
5
13cos
3
1cos
1
14222
tkk
tr
tytyty
k
12cos12
4
2502.0
12
© 2012, Ching-Han Hsu, Ph.D.
Forced OscillationsThe steady-state (particular) solution yn(t)
corresponding to the sinusoidal input
cos(2k-1)t=cosnt can be computed using
the method of undetermined coefficients
tytytyty
nnD
DnB
Dn
nA
ntBntAty
p
nn
nnn
321
222
2
2
02.025
08.0,
254
sincos
© 2012, Ching-Han Hsu, Ph.D.
Forced OscillationsLet us find the amplitude
We have
22
nnn BAC
0003.0
0011.0
5100.0
0088.0
0530.0
9
7
5
3
1
C
C
C
C
C
© 2012, Ching-Han Hsu, Ph.D.
Parseval’s Theorem
© 2012, Ching-Han Hsu, Ph.D.
Fourier Series Formulae
1
0 sincosn
nn xL
nbx
L
naaxf
L
Ldxxf
La
2
10
L
Ln dx
L
xnxf
La
cos
1
L
Ln dx
L
xnxf
Lb
sin
1
,
© 2012, Ching-Han Hsu, Ph.D.
Parseval’s Theorem
• Show that:
1
222
0
22
1
n
nn
L
Lbaadxxf
L
1
0
1
0
2
sincos
sincos
n
nn
m
mm
xL
nbx
L
naa
xL
mbx
L
maaxf
© 2012, Ching-Han Hsu, Ph.D.
nm
dxxL
mnx
L
mnba
dxxL
nx
L
mba
L
Lnm
L
Lnm
,,0
sinsin2
1
sincos
2
000 2LadxaaL
L
© 2012, Ching-Han Hsu, Ph.D.
nmLa
dxxL
nadxx
L
na
dxxL
nx
L
naa
nmdxxL
nx
L
maa
n
L
Ln
L
Ln
L
Lnn
L
Lnm
,
2cos1
2
1cos
coscos
,0coscos
2
222
1,1 nm
© 2012, Ching-Han Hsu, Ph.D.
nmLb
dxxL
nbdxx
L
nb
dxxL
nx
L
nbb
nmdxxL
nx
L
mbb
n
L
Ln
L
Ln
L
Lnn
L
Lnm
,
2cos1
2
1sin
sinsin
,0sinsin
2
222
1,1 nm
© 2012, Ching-Han Hsu, Ph.D.
1,1,
1,1,
1
22222
0
1
0
1
0
2
sinsin
coscos
sincos2
sincos
sincos
sincos
nmnm
nm
nmnm
nm
nm
nm
n
nn
n
nn
n
mm
xL
nbx
L
mb
xL
nax
L
ma
xL
nbx
L
ma
xL
nbx
L
naa
xL
nbx
L
naa
xL
mbx
L
maaxf
© 2012, Ching-Han Hsu, Ph.D.
1
222
0
2
1
222
0
1
22222
0
2
21
2
sincos
n
nn
L
L
n
nn
L
Ln
nn
L
L
baadxxfL
baaL
dxxL
nbx
L
naadxxf
1
222
0
2
2
1
2
1
n
nn
L
Lbaadxxf
L
© 2012, Ching-Han Hsu, Ph.D.
Complex Fourier Series
© 2012, Ching-Han Hsu, Ph.D.
Functions of Any Period p=2L
• A function f(x) of period p=2L has a Fourier
series
1
0 sincosn
nn xL
nbx
L
naaxf
L
Ldxxf
La
2
10
L
Ln dx
L
xnxf
La
cos
1
L
Ln dx
L
xnxf
Lb
sin
1
,3,2,1n
© 2012, Ching-Han Hsu, Ph.D.
Euler Formulae
iiii
i
eei
ee
ie
2
1sin,
2
1cos
sincos
L
xni
nnL
xni
nn
L
xni
L
xni
nL
xni
L
xni
n
nn
eibaeiba
eebi
eea
L
xnb
L
xna
2
1
2
1
)(2
1)(
2
1
sincos
© 2012, Ching-Han Hsu, Ph.D.
Complex Fourier Series
0
)0(
0
)(
2
1
2
1
sincos2
1
2
1
2
1
sincos2
1
2
1
cdxexfL
a
cdxexfL
dxL
xni
L
xnxf
Liba
cdxexfL
dxL
xni
L
xnxf
Liba
L
L
L
xi
n
L
L
L
xni
L
Lnn
n
L
L
L
xni
L
Lnn
© 2012, Ching-Han Hsu, Ph.D.
Complex Fourier Series
n
L
xni
n
n
L
xni
n
n
L
xni
n
n
L
xni
n
n
L
xni
n
n
L
xni
n
n
L
xni
n
n
L
xni
nn
n
L
xni
nn
n
nn
ec
ececc
ececcececc
eibaeibaa
L
xnb
L
xnaaxf
11
0
1
)(
1
0
11
0
11
0
1
0
2
1
2
1
sincos
L
L
L
xni
n dxexfL
c
2
1
© 2012, Ching-Han Hsu, Ph.D.
Let Us Play Fourier Series!!
© 2012, Ching-Han Hsu, Ph.D.
Functions of Any Period p=2L
• A function f(x) of period p=2L has a Fourier
series
1
0 sincosn
nn xL
nbx
L
naaxf
L
Ldxxf
La
2
10
L
Ln dx
L
xnxf
La
cos
1
L
Ln dx
L
xnxf
Lb
sin
1
,3,2,1n
© 2012, Ching-Han Hsu, Ph.D.
Periodic Square Wave (I)
• Ex. Find the Fourier series of the function
Since the function f(x) is odd, there are no
constant and cosine terms in the Fourier's
series, i.e., a0=a1=a2=…=0!!
01
01
xL
Lxxf
© 2012, Ching-Han Hsu, Ph.D.
Periodic Square Wave (I)
nn
L
xn
Ln
Ldx
L
xn
L
dxL
xn
Ldx
L
xn
L
dxL
xnxf
Lb
LL
L
L
L
Ln
cos12
cos2
sin12
sin11
sin11
sin1
00
0
0
1
sincos12
n
xL
n
n
nxf
© 2012, Ching-Han Hsu, Ph.D.
Periodic Square Wave (II)
• Ex. Find the Fourier series of the function
00
01
xL
Lxxg
1
sincos1
2
11
2
1
n
xL
n
n
nxfxg
© 2012, Ching-Han Hsu, Ph.D.
Periodic Square Wave (III)
• Ex. Find the Fourier series of the function
2/0
2/2/1
2/0
LxL
LxL
LxL
xh
© 2012, Ching-Han Hsu, Ph.D.
Periodic Square Wave (III)
1
1
1
cos2
sincos1
2
1
2sincos
2cossin
cos1
2
1
2sin
cos1
2
1
2
n
n
n
xL
nn
n
n
L
Lnx
L
n
L
Lnx
L
n
n
n
Lx
L
n
n
n
Lxgxh
© 2012, Ching-Han Hsu, Ph.D.
Triangle Wave (I)
• Ex. Find the Fourier series of the function
Since the function f(x) is odd, there are no
constant and cosine terms in the Fourier's
series, i.e., a0=a1=a2=…=0!!
LxLxxf ,
© 2012, Ching-Han Hsu, Ph.D.
nn
L
L
xn
n
L
nn
n
L
dxL
xn
nL
xnx
n
L
xnxd
n
L
Ldx
L
xnx
L
dxL
xnx
Ldx
L
xnxf
Lb
L
LL
LL
L
L
L
Ln
cos2
sin2
cos2
cos2
cos2
cos2
sin2
sin1
sin1
0
00
00
1
sincos2
n
xL
nn
n
Lxf
© 2012, Ching-Han Hsu, Ph.D.
Triangle Wave (I)
© 2012, Ching-Han Hsu, Ph.D.
Triangle Wave (II)
• Ex. Find the Fourier series of the function
Since the function f(x) is odd, there are no
constant and cosine terms in the Fourier's
series, i.e., a0=a1=a2=…=0!!
LxLxxfxg ,
1
sincos2
n
xL
nn
n
Lxg
© 2012, Ching-Han Hsu, Ph.D.
© 2012, Ching-Han Hsu, Ph.D.
Triangle Wave (II)
© 2012, Ching-Han Hsu, Ph.D.
Triangle Wave (III)
• Ex. Find the Fourier series of the function
LxLLxLxfxh ,
1
sincos2
n
xL
nn
n
LLxh
© 2012, Ching-Han Hsu, Ph.D.
Triangle Wave (III)
© 2012, Ching-Han Hsu, Ph.D.
Triangle Wave (IV)
• Ex. Find the Fourier series of the function
LxLLxL
xLxhxp
20
,
© 2012, Ching-Han Hsu, Ph.D.
1
sincos2
n
LxL
nn
n
LL
Lxhxp
L
xnn
L
xn
L
Ln
L
Ln
L
xn
LxL
n
sincos
cossincossin
sin
1
sin2
n
xL
n
n
LLxp
© 2012, Ching-Han Hsu, Ph.D.
Triangle Wave (IV)