2. Introduction
Jean Baptiste Joseph Fourier
(Mar21st 1768 –May16th 1830)
French mathematician, physicist
Main Work:
(The Analytic Theory of Heat)
•Any function of a variable, whether continuous or
discontinuous, can be expanded in a series of sines of
multiples of the variable (Incorrect)
•The concept of dimensional homogeneity in
equations
•Proposal of his partial differential equation for
conductive diffusion of heat
Discovery of the "greenhouse effect“
•Fourier series is very useful in solving ordinary and
partial differential equation.
http://en.wikipedia.org/wiki/Joseph_Fourier
3. Even, Odd, and Periodic Functions
Even, Odd, and Periodic Functions
0
T
f
4. Fourier Series of a Periodic Function
Fourier Series of a Periodic Function
Definition : A Fourier series may be defined as an expansion of a
function in a series of sines and cosines such as ,
The coefficients are related to the periodic function f(x)
by definite integrals:
Henceforth we assume f satisfies the following (Dirichlet)
conditions:
(1) f(x) is a periodic function;
(2) f(x) has only a finite number of finite discontinuities;
(3) f(x) has only a finite number of extrem values, maxima and
minima in the interval [0,2p].
5. EULER’S FORMULA
EULER’S FORMULA
The formula for a Fourier series is: N
We have formulae for the coefficients (for the derivations see the
course notes):
One very important property of sines and cosines is their orthogonality,
expressed by:
n
n
n
n
T
x
n
b
T
x
n
a
a
x
f
1
0
2
sin
2
cos
)
(
2
2
0 )
(
1
T
T
dx
x
f
T
a
2
2
2
cos
)
(
2
T
T
n dx
T
x
n
x
f
T
a
2
2
2
sin
)
(
2
T
T
n dx
T
x
n
x
f
T
b
m
n
T
m
n
dx
T
x
m
T
x
n
T
T
2
0
2
sin
2
sin
2
2
7. Find ,
0
a
2
2
0 )
(
1
T
T
dx
x
f
T
a
dx
x
f
a )
(
2
1
0
0
0
1
xdx
a
2
0
a
f (x) is an even function so:
dx
x
f
a )
(
2
1
0
0
0 )
(
1
dx
x
f
a
0
2
0
2
1
x
a
8. Find n
a
2
2
2
cos
)
(
2
T
T
n dx
T
x
n
x
f
T
a
dx
x
n
x
f
an
2
2
cos
)
(
1
Since both functions are even their product is even:
dx
nx
x
f
an cos
)
(
1
0
cos
2
dx
nx
x
an
n
b
2
2
2
sin
)
(
2
T
T
n dx
T
x
n
x
f
T
b
dx
x
n
x
f
bn
2
2
sin
)
(
1
dx
nx
x
f
bn sin
)
(
1
0
n
b
9. So we can put the coefficients back into the Fourier series formula:
n
n
n
n
T
x
n
b
T
x
n
a
a
x
f
1
0
2
sin
2
cos
)
(
n
n
n
nx
n
x
f
1
2
cos
1
1
2
2
)
(
x
x
x
f 3
cos
9
4
0
cos
4
2
)
(
10. Summary of finding coefficients
Summary of finding coefficients
function
even
function
odd
function
neither
0
a
n
a
n
b
0
)
(
1 2
2
0
T
T
dx
x
f
T
a
2
2
2
cos
)
(
2
T
T
n dx
T
x
n
x
f
T
a
Though maybe easy to find
using geometry
2
2
2
sin
)
(
2
T
T
n dx
T
x
n
x
f
T
b
2
2
2
sin
)
(
2
T
T
n dx
T
x
n
x
f
T
b
0
0
0
)
(
1 2
2
0
T
T
dx
x
f
T
a
2
2
2
cos
)
(
2
T
T
n dx
T
x
n
x
f
T
a
Though maybe easy to find
using geometry
0
11. Half range Expansions
Half range Expansions
It often happens in applications, especially when we solve partial
differential equations by the method of separation of variables, that we
need to expand a given function f in a Fourier series, where f is
defined only on a finite interval.
We define an “extended function”, say fext, so that fext is periodic in the
domain of -∞< x < ∞, and fext=f(x) on the original interval 0<x<L.
There can be infinite number of such extensions.
Four extensions: half- and quarter- range cosine and sine extensions,
which are based on symmetry or antisymmetry about the endpoints
x=0 and x=L.
12. HRC (half range cosines)
HRC (half range cosines)
fext is symmetric about x=0 and
also about x=L. Because of its
symmetry about x=0, fext is an
even function, and its Fourier
series will contain only cosines, no sines. Further, its
period is 2L, so L is half the period.
18. PARSEVAL’S FORMULA
PARSEVAL’S FORMULA
(
1
)
If a function has a Fourier series given by
then Bessel's inequality becomes an equality known as Parseval's
theorem. From (1),
Integrating
so
19. Math for CS Lecture 11 19
Fourier Integral
Fourier Integral
If f(x) and f’(x) are piecewise continuous in every finite interval, and f(x) is
absolutely integrable on R, i.e.
converges, then
Remark: the above conditions are sufficient, but not necessary.
dw
dt
t
f
e
e
x
f
x
f iwt
iwx
)
(
2
1
)]
(
)
(
[
2
1
20. DIFFERENT FORMS OF FOURIER
DIFFERENT FORMS OF FOURIER
INTEGRAL THEOREM
INTEGRAL THEOREM
24. Math for CS Lecture 11 24
Properties of Fourier transform
Properties of Fourier transform
1 Linearity:
For any constants a, b the following equality holds:
2 Scaling:
For any constant c, the following equality holds:
)
(
)
(
)}
(
{
)}
(
{
)}
(
)
(
{ w
bG
w
aF
t
g
bF
t
f
aF
t
bg
t
af
F
)
(
|
|
1
)}
(
{
c
w
F
c
ct
f
F
25. Math for CS Lecture 11 25
3. Time shifting:
proof:
4. Frequency shifting:
Proof:
)
(
)}
(
{ 0
0 w
F
e
t
t
f
F iwt
du
e
u
f
e
dt
e
t
t
f
t
t
f
F iwu
iwt
iwt
)
(
)
(
)}
(
{ 0
0
0
)
(
)}
(
{ 0
0
w
w
F
t
f
e
F iwt
)
(
)
(
)}
(
{ 0
0
0
w
w
F
dt
e
t
f
e
t
f
e
F iwt
iwt
t
iw
)
(
2
)}
(
{ w
f
t
F
F
dw
e
w
F
w
f
F
t
f iwt
)
(
2
1
)}
(
{
)
( 1
)}
(
{
)
(
2
1
)
(
2 t
F
F
dt
e
t
F
w
f itw
26. Math for CS Lecture 11 26
6. Modulation:
Proof:
Using Euler formula, properties 1 (linearity) and 4 (frequency shifting):
)]
(
)
(
[
2
1
)}
sin(
)
(
{
)]
(
)
(
[
2
1
)}
cos(
)
(
{
0
0
0
0
0
0
w
w
F
w
w
F
t
w
t
f
F
w
w
F
w
w
F
t
w
t
f
F
)]
(
)
(
[
2
1
)}]
(
{
)}
(
{
[
2
1
)}
cos(
)
(
{
0
0
0
0
0
w
w
F
w
w
F
t
f
e
F
t
f
e
F
t
w
t
f
F t
iw
t
iw
27. Periodically forced oscillation: mass-spring
Periodically forced oscillation: mass-spring
system
system
m = mass
c = damping factor
k = spring constant
F(t) = 2L- periodic forcing function
mx’’(t) + cx’(t) + k x(t) = F(t)
http://www.jirka.org/diffyqs/
Differential Equations for Engineers
m F(t)
k
28. The particular solution xp of the above equation is periodic with
the same period as F(t) .
The coefficients are k=2, and m=1 and c=0 (for simplicity). The
units are the mks units (meters-kilograms- seconds). There is a
jetpack strapped to the mass, which fires with a force of 1
newton for 1 second and then is off for 1 second, and so on.
We want to find the steady periodic solution.
The equation is:
x’’ + 2x = F(t)
Where F(t) => 0 if -1<t<0
1 if 0<t<1
![Fourier Series of a Periodic Function
Fourier Series of a Periodic Function
Definition : A Fourier series may be defined as an expansion of a
function in a series of sines and cosines such as ,
The coefficients are related to the periodic function f(x)
by definite integrals:
Henceforth we assume f satisfies the following (Dirichlet)
conditions:
(1) f(x) is a periodic function;
(2) f(x) has only a finite number of finite discontinuities;
(3) f(x) has only a finite number of extrem values, maxima and
minima in the interval [0,2p].](https://image.slidesharecdn.com/fourier-series-250409170905-302e657a/85/Fourier-Series-SAMPLE-PRESENTATION-FOR-LEARNING-4-320.jpg)
![Math for CS Lecture 11 19
Fourier Integral
Fourier Integral
If f(x) and f’(x) are piecewise continuous in every finite interval, and f(x) is
absolutely integrable on R, i.e.
converges, then
Remark: the above conditions are sufficient, but not necessary.
dw
dt
t
f
e
e
x
f
x
f iwt
iwx
)
(
2
1
)]
(
)
(
[
2
1
](https://image.slidesharecdn.com/fourier-series-250409170905-302e657a/85/Fourier-Series-SAMPLE-PRESENTATION-FOR-LEARNING-19-320.jpg)
![Math for CS Lecture 11 26
6. Modulation:
Proof:
Using Euler formula, properties 1 (linearity) and 4 (frequency shifting):
)]
(
)
(
[
2
1
)}
sin(
)
(
{
)]
(
)
(
[
2
1
)}
cos(
)
(
{
0
0
0
0
0
0
w
w
F
w
w
F
t
w
t
f
F
w
w
F
w
w
F
t
w
t
f
F
)]
(
)
(
[
2
1
)}]
(
{
)}
(
{
[
2
1
)}
cos(
)
(
{
0
0
0
0
0
w
w
F
w
w
F
t
f
e
F
t
f
e
F
t
w
t
f
F t
iw
t
iw
](https://image.slidesharecdn.com/fourier-series-250409170905-302e657a/85/Fourier-Series-SAMPLE-PRESENTATION-FOR-LEARNING-26-320.jpg)
