chapter 3: fourier series
DESCRIPTION
Chapter 3: Fourier Series. Introduction In 1808, Fourier wrote the first version of his celebrated memoir on the theory of heat “Theorie Analytique de la Chaleur”. He made a detailed study of trigonometric series, which he used to solve a variety of heat conduction problems. - PowerPoint PPT PresentationTRANSCRIPT
Da-Chuan Cheng, PhD 1112/04/20 Da-Chuan Cheng, PhD 1
Chapter 3: Fourier Series
• Introduction– In 1808, Fourier wrote the first version of his
celebrated memoir on the theory of heat “Theorie Analytique de la Chaleur”. He made a detailed study of trigonometric series, which he used to solve a variety of heat conduction problems.
– Nearly two centuries after Fourier’s work, the series that bears his name is still important, practically and theoretically, and still a topic of current research.
Da-Chuan Cheng, PhD 2Da-Chuan Cheng, PhD 2
Computation of Fourier series : Real form
1 10 )sin()cos()(
:seriesFourier in the , and ts,coefficienFourier thecompute we interval On the
k kkk
kk
kxbkxaaxf
baπ,x-π
Important properties of Fourier series:
otherwise0
0 if2
1 if1
)cos()cos(1
kn
kn
dxkxnx(1)
otherwise0
1 if1)sin()sin(
1 kndxkxnx(2)
.,integer allfor 0)sin()cos(
1kndxkxnx
integers. are ,kn
(3)
,2,1k
Da-Chuan Cheng, PhD 3112/04/20 Da-Chuan Cheng, PhD 3
Computation of Fourier series : Real form
,
)2sin(,
)sin(,
2
1,
)cos(,
)2cos( ,
xxxx
An equivalent way of starting this theorem is that the collection
is an orthonormal set of functions in ]),([2 L
Da-Chuan Cheng, PhD 4
Matlab program
Da-Chuan Cheng, PhD 4
% Confirm page 3.L=37; % L can be any positive integer.t=[-pi:(2*pi)/L:pi]; % range is from -pi to (pi- dt)t=t(1:end-1);y=cos(t)/sqrt(L/2); % equation in page 3. y=cos(t).disp(['The length of vector y=cos(t) is : ']);sum(y.^2) % confirm the length=1.figure(1); plot(t,y); hold on;
長度不是 π﹐而是 L/2
The length of vector y=cos(t) is :
ans =
1.0000
Da-Chuan Cheng, PhD 5Da-Chuan Cheng, PhD 5
Da-Chuan Cheng, PhD 6Da-Chuan Cheng, PhD 6
z1=sin(t)/sqrt(L/2); % equation in page 3. y=sin(t).disp(['The length of vector y=sin(t) is : ']);sum(z1.^2) % confirm the length=1.figure(1); plot(t,z1); hold on;
The length of vector y=sin(t) is :
ans =
1.0000
Da-Chuan Cheng, PhD 7Da-Chuan Cheng, PhD 7
y1=cos(2*t)/sqrt(L/2); % y=cos(2t).disp(['The length of vector y1=cos(2t) is : ']);sum(y1.^2) % confirm the length=1.figure(1); plot(t,y1,'y');
The length of vector y1=cos(2t) is :
ans =
1
Da-Chuan Cheng, PhD 8Da-Chuan Cheng, PhD 8
% Confirm cos(t)sin(t)=0z = cos(t).*sin(t);sum(z)
ans =
4.9960e-016
% confirm cos(t)*sin(2t)=0z = cos(t).*sin(2*t);sum(z)
ans =
-1.1102e-015
Da-Chuan Cheng, PhD 9
Computation of Fourier series : Real form
Da-Chuan Cheng, PhD 9
Proof of the properties on page 2:
The derivations of the first two equalities use the following identities:
.sinsincoscos))cos((sinsincoscos))cos((kxnxkxnxxknkxnxkxnxxkn
) if(0)sin()sin(
2
1
)))cos(())(cos((2
1coscos
knkn
xkn
kn
xkn
dxxknxknkxdxnx
Da-Chuan Cheng, PhD 10112/04/20 Da-Chuan Cheng, PhD 10
Computation of Fourier series : Real form
dxnxnxdxkn )2cos1)(2/1(cos then,1 If 2
proof.) the(Complete.21)/1( then ,0 If
dxkn
Equation (2) and (3) can be proved in a similar way.
Fourier coefficients computation:
1 1
0 )sin()cos()(k k
kk kxbkxaaxfAssume
nxdxkxbkxaanxdxxfa
k kkn cos)sin()cos(1
cos)(1
: find To10
Note that k starts from 1, why? (Hint: see page 2, Eq.(1))
Da-Chuan Cheng, PhD 11112/04/20 Da-Chuan Cheng, PhD 11
Computation of Fourier series : Real form
nn
kk
kk
kk
kk
k kk
aa
dxnxkxb
dxnxkxanxn
a
dxnxkxb
dxnxkxadxnxa
dxnxkxbkxaa
00
)cos()sin(1
)cos()cos(1
|)sin(11
)cos()sin(1
)cos()cos(1
)cos(1
)cos()sin()cos(1
1
10
1
10
10
Da-Chuan Cheng, PhD 12112/04/20 Da-Chuan Cheng, PhD 12
Computation of Fourier series : Real form
1)cos()(1
nadxnxxf n
1)sin()(1
Similarly,
nbdxnxxf n
000 2
1)(
2
1: find To adxadxxfa
實際上就是訊號 f(x)的平均值
Da-Chuan Cheng, PhD 13112/04/20 Da-Chuan Cheng, PhD 13
Computation of Fourier series : Real form
Theorem:
1 1
0 )sin()cos()(k k
kk kxbkxaaxfIf
then,
dxxfa )(
2
10
1)cos()(1
ndxnxxfan
1)sin()(1
ndxnxxfbn
The Fourier coefficients for a given function are unique.
Da-Chuan Cheng, PhD 14
Matlab implementation
Da-Chuan Cheng, PhD 14
% Page 13.L=64; % L can be any positive integer.t=[-pi:(2*pi)/L:pi]; % range is from -pi to (pi- dt)t=t(1:end-1); f=rand(1,L).*sin(rand(1,L)*2); figure; plot(t,f); a0=mean(f) a1=sum(f.*cos(t))/L/2b1=sum(f.*sin(t))/L/2
Da-Chuan Cheng, PhD 15
Change the representation form
)]cos(,),[cos(),cos(; kkkxx see p.13
)]cos(,),[cos(),cos(;
kka
xkaxa
)]cos(,),[cos(),2
2cos(;
kk
a
xkaxa
or
Da-Chuan Cheng, PhD 16
Computation of Fourier series : Real form
Theorem:
1 10 )/sin()/cos()(
k kkk axkbaxkaaxfIf
then, a
adxxf
aa )(
2
10
1)2
2cos()(
1
ndx
a
nxxf
aa
a
an
1)2
2sin()(
1
ndx
a
nxxf
ab
a
an
Da-Chuan Cheng, PhD 17112/04/20 Da-Chuan Cheng, PhD 17
Computation of Fourier series : Real form
Even and odd functions
).()( if odd is );()( ifeven is
function; a be :Let :Definition
xfxffxfxffRRf
The following properties follow from the definition.
Even X Even = EvenEven X Odd = OddOdd X Odd = Even
If F is an even function, then
aa
adxxFdxxF
0)(2)(
If F is an odd function, then 0)( a
adxxF
Da-Chuan Cheng, PhD 18112/04/20 Da-Chuan Cheng, PhD 18
Computation of Fourier series : Real form
Theorem:
a
k
a
kk
dxaxkxfa
a
dxxfa
a
axkaaxf
0
00
10
)/cos()(2
)(1
with)/cos()(
cosines. haveonly will][ interval on the seriesFourier its then function,even an is )( If
-a,axf
sines. haveonly will][ interval on the seriesFourier its then function, oddan is )( If
-a,axf
a
k
kk
dxaxkxfa
b
axkbxf
0
1
)/sin()(2
with)/sin()(
Da-Chuan Cheng, PhD 19112/04/20 Da-Chuan Cheng, PhD 19
Computation of Fourier series : Real form
Gibbs phenomenon
Approximation of square wave in 5 steps
The height of the blip is approximately the same no matter how many terms are considered in the partial sum.
Da-Chuan Cheng, PhD 20112/04/20 Da-Chuan Cheng, PhD 20
Computation of Fourier series (1) : Complex form
Complex form of Fourier Series
Often, it is more convenient to express Fourier series in complex form using the complex exponentials due to the simple computational properties of these functions.
Definition:
1 where)sin()cos(
:is lexponentiacmplex thenumber t, realany For
ititeit
This definition is motivated by substituting x=it into the Taylor series for xe
)sin()cos()!5!3
()!4!2
1(
!4
)(
!3
)(
!2
)()(1
with !4!3!2
1
5342
432
432
tittt
titt
itititite
itxxxx
xe
it
x
Da-Chuan Cheng, PhD 21112/04/20 Da-Chuan Cheng, PhD 21
Computation of Fourier series (2) : Complex form
Lemma:
itit
stiisit
stiisit
itit
it
itti
ieedt
deeeeee
eee
ee
)(
/
1||
)(
)(
)2(
Theorem:
]).,([Lin lorthonorma is ,2,1,0,1,2, ,2
functions ofset The 2
ne int
Proof:
mnmn
mni
edtedteeee
tmnitmniimtimt
if2 if0
)(,
)()(intint
Da-Chuan Cheng, PhD 22112/04/20 Da-Chuan Cheng, PhD 22
Computation of Fourier series (3) : Complex form
Theorem:
dtetfa
teatf
n
nn
int
int
)(2
1then
, interval on the )( If
Combine with the previous theorem in Chap2 p.7, we get the following theorem:
Proof: To find na we simply substitute f(t) into:
dtntmtntmtintmtintmta
dtntintmtimtadteea
nn
nn
tin
n
imtn
))sin()sin()sin()cos()cos()sin()cos()(cos(2
1
))sin()))(cos(sin()(cos(2
1
2
1
Use the properties in p.2.
Da-Chuan Cheng, PhD 23112/04/20 Da-Chuan Cheng, PhD 23
.,integer allfor 0)sin()cos(
1mndxmxnx
dtntmtntmtintmtintmtann
))sin()sin()sin()cos()cos()sin()cos()(cos(2
1
Property (3)
0 0
(1) n≠m
otherwise0
0 if2
1 if1
)cos()cos(1
mn
mn
dxmxnx
otherwise0
1 if1)sin()sin(
1 mndxmxnx
=0
Da-Chuan Cheng, PhD 24112/04/20 Da-Chuan Cheng, PhD 24
dtntmtntmtann
))sin()sin()cos()(cos(2
1
(2) n=m=0 000 22
1)0cos(
2
1aadta
(3) n=m 1≧ nnn adtadtntnta
2
1))(sin)((cos
2
1 22
Da-Chuan Cheng, PhD 25112/04/20 Da-Chuan Cheng, PhD 25
Computation of Fourier series (4) : Complex form
So the complex Fourier series of f is
k
tki
n
tinn e
k
iea )12(
)12(
2
Example:
0 if1
0 if1)(
t
ttf
The n-th complex Fourier coefficients is:
even isn if0
odd isn if2]1)[cos(
))]cos(1(1)[cos(2
]||[)(
1
2
1
2
1
2
1)(
2
1 00
0
0
n
i
n
ninn
n
i
eein
dtedtedtetfa tintintintinntin
Da-Chuan Cheng, PhD 26112/04/20 Da-Chuan Cheng, PhD 26
Computation of Fourier series (5) : Complex form
t=[-pi:2*pi/1024:pi];t=t(1:end-1);fs=zeros(1,length(t));N=100;K=[-N:N];for k=K; fs=fs+2/((2*k+1)*pi)*sin((2*k+1)*t);end; figure(1); subplot(121); plot(t,fs); fc=zeros(1,length(t));for k=K, fc=fc-2/((2*k+1)*pi)*cos((2*k+1)*t);end;figure(1); subplot(122); plot(t,fc);
Matlab implementation
k
tkiek
i )12(
)12(
2
0 if1
0 if1)(
t
ttf
Da-Chuan Cheng, PhD 27112/04/20 Da-Chuan Cheng, PhD 27
Computation of Fourier series (6) : Complex form
Result: k=-10:10
real part imaginary part
Da-Chuan Cheng, PhD 28112/04/20 Da-Chuan Cheng, PhD 28
Computation of Fourier series (7) : Complex form
Result: k=-50:50
real part imaginary part
Da-Chuan Cheng, PhD 29112/04/20 Da-Chuan Cheng, PhD 29
Computation of Fourier series (8) : Complex form
Result: k=-100:100
real part imaginary part
Da-Chuan Cheng, PhD 30112/04/20 Da-Chuan Cheng, PhD 30
Computation of Fourier series (9) : Complex form
Result: k=-10000:10000
real part imaginary part
Da-Chuan Cheng, PhD 31112/04/20 Da-Chuan Cheng, PhD 31
Computation of Fourier series (10) : Complex form
Theorem: The set of functions
]).,([Lfor basis lorthonormaan is ,2,1,0,1,2, ,2
2 aana
e a
t/in
a
a
atinn
n
atinn dtetf
aαetf // )(
2
1 then ,)( If
Example:
01 if1
10 if1)(
t
ttf
n
ni
ninninn
i
eein
dtedtedtetfa
ndst
tintintintintnin
]1)[cos(
]))sin()(cos()01()01())sin()(cos([2
]||[)(
1
2
1
2
1
2
1)(
2
1
term2 term1
01
10
0
1
1
0
1
1
Da-Chuan Cheng, PhD 32112/04/20 Da-Chuan Cheng, PhD 32
Computation of Fourier series (11) : Complex form
Relation between the real and complex Fourier series
If f is a real valued function, the real form of its Fourier series can be derived from its complex form and vice versa. For simplicity, we discuss this derivation on the interval -π t π, but this discussion also holds for other ≦ ≦intervals as well.
.)(2
1 where)(
termpositive
10
termnegative
1
dtetfeetf tin
nn
tinn
n
tinn
If f is real valued, then because nn
ntintin
n dtetfdtetf
)(2
1)(
2
1
n≠0
Da-Chuan Cheng, PhD 33112/04/20 Da-Chuan Cheng, PhD 33
Computation of Fourier series (12) : Complex form
termnegative
1
termpositive
10)(
n
tinn
n
tinn eetf
10 Re2)(
}Re{2
n
tinnetf
zzz
Da-Chuan Cheng, PhD 34112/04/20 Da-Chuan Cheng, PhD 34
Computation of Fourier series (13) : Complex form
00 )(2
1adttf
)(2
1
))sin())(cos((2
1
1for )(2
1
nn
tinn
iba
dtntinttf
ndtetf
similar to p. 13
(see p. 13)
Da-Chuan Cheng, PhD 35112/04/20 Da-Chuan Cheng, PhD 35
Computation of Fourier series (14) : Complex form
10
10
10
)sin()cos(
))sin())(cos((Re
Re2)( Recall
nnn
nnn
n
tinn
ntbnta
ntintiba
etf
Exactly the same to page 13.
Da-Chuan Cheng, PhD 36
Homework
• 請自行找出一個長 32的信號﹐寫 Matlab程式將其 Fourier series的係數 (a0, an, bn)找出。
• Issue date: 5/5
• Due date: 19/5