Embed Size (px)
Citation preview
1
CHAPTER 5 : FOURIER SERIES
Introduction Periodic Functions Fourier Series of a function Dirichlet Conditions Odd and Even Functions Relationship Between Even and Odd
functions to Fourier series
2
5.1 : Introduction
Fourier Series ?
A Fourier series is a representation of a function as a series of constants times sine and/or cosine functions of different frequencies.
3
5.2 : Periodic Functions
A function f(x) is said to be periodic if its function values repeat at regular intervals of the independent variables.
For the following example, a function f(x) has the period p.
p
x
y
x1 x1 + p x1 + 2p x1 + 3p
4
In general, a function f(x) is called periodic if there is some positive number p such that ;
f(x) = f(x + np)
for any integer n.This number p is called a period of f(x).
5
5.3 : Fourier Series of a Function
If f(x) is defined within the interval c < x < c+2L. The Fourier Series corresponding to f(x) is given by
where
1
0 sincos2 n
nn xL
nbx
L
na
axf
Lc
c
n
Lc
c
n
Lc
c
xdxL
nxf
Lb
xdxL
nxf
La
dxxfL
a
2
2
2
0
sin1
cos1
1
6
5.4 : Dirichlet Conditions
If a function f(x) defined within the interval c < x < c+2L, the following conditions must be satisfied;
1. f(x) is defined and single-valued.
2. f(x) is continuous or finite discontinuity in the corresponding periodic interval.
3. f(x) and f ’(x) are piecewise continuous .
7
Example 5.1:
Determine whether the Dirichlet conditions are satisfied in the following cases :
i.Yes.
ii.No, because there is infinite discontinuity
at
xx
xf ; 1
xxxf ; 2
.0x
8
iii.
No, the function is not defined for x < 0.
iv.
Yes.
xxxf ; ln
11 ; 5
1
x
xxf
9
Example 5.2:
1. Find the Fourier series for the function defined within the interval
Answer :
2
22
2
,0
,4
,0
x
x
x
xf
xxxxxxf 9cos
9
17cos
7
15cos
5
13cos
3
1cos
82
. x
10
Solution
The Fourier series is
where
1
0 sincos2 n
nn nxbnxaa
xf
.sin)(1
cos)(1
)(1
0
dxnxxfb
dxnxxfa
dxxfa
n
n
. Therefore, .2 and get We LLcc
11
.4
)4(1
]4[1
0401
)(1
2
2
2
2
2
2
0
x
dxdxdx
dxxfa
12
.2
sin8
sin14
cos41
,cos)(1
2
2
2
2
n
n
nxn
dxnx
dxnxxfan
.0
cos14
sin41
sin)(1
2
2
2
2
nxn
dxnx
dxnxxfbn
13
.9cos9
17cos
7
15cos
5
13cos
3
1cos
82
9cos9
87cos
7
85cos
5
83cos
3
8cos
82
cos2
sin8
2
4
)sincos(2
is seriesFourier theTherefore,
1
1
0
xxxxx
xxxxx
nxn
n
nxbnxaa
xf
n
nnn
14
Graph of a Function
2
22
2
,0
,4
,0
x
x
x
xf
What will we obtain as more terms are included in the series ?
2
y
4
x- 0
2
15
Graph of a Function
xxf cos8
2
4
O
x
16
Graph of a Function
xxxf 3cos
3
1cos
82
4
x O
17
Graph of a Function
xxxxf 5cos
5
13cos
3
1cos
82
4
x O
18
Graph of a Function
xxxxxf 7cos
7
15cos
5
13cos
3
1cos
82
4
x
O
The above figure show that the graph is merely to the shape …..
19
The above function is defined within the interval
As the number of terms increases, the graph of Fourier series gradually approaches the shape of the original square waveform.
. x
x-5 -3 - 0 3 5
20
Example 5.3:
Find the Fourier series for defined within the interval
Answer :
2xxf . x
12
2
cos1
43 n
n
nxn
xf
O
f(x)
x
21
Solution
The Fourier series is
where
1
0 sincos2 n
nn nxbnxaa
xf
.sin)(1
cos)(1
)(1
0
dxnxxfb
dxnxxfa
dxxfa
n
n
. Therefore, .2 and get We LLcc
22
.3
2
3
21
3
11
1
)(1
2
3
3
2
0
x
dxx
dxxfa
23
.4
)1(
.cos4
cos1
,cos)(1
2
2
2
na
nn
dxnxx
dxnxxfa
nn
n
.0
.0
sin1
sin)(1
2
n
n
b
dxnxx
dxnxxfb
24
.cos)1(
43
cos4
)1(3
2
2
1
)sincos(2
)(
is seriesFourier The
12
2
12
2
1
0
n
n
n
n
n
nn
nxn
nxn
nxbnxaa
xf
25
Find the Fourier series for defined within the interval
Answer :
xxf .11 x
1
1
sin12
n
n
xnn
xf
x
f (x)
5 3 1 1 3 5
O
Example 5.4:
26
Solution
The Fourier series is
.sincos2
sincos2
1
0
1
0
nnn
nnn
xnbxnaa
xL
nbx
L
na
axf
.1 Therefore, .12 and 1get We LLcc
27
.0
2
1
1
1
)(1
1
1
2
1
1
2
0
x
dxx
dxxfL
aLc
c
28
.0
.0
)cos(11
)cos(1
1
,cos)(1
1
1
1
1
2
n
Lc
c
n
a
xnnn
dxxnx
dxxL
nxf
La
.2
)1(
.cos2
)sin(1
1
sin)(1
1
1
1
2
nb
nn
dxxnx
dxxL
nxf
Lb
nn
Lc
c
n
29
.sin)1(2
sin2
)1(
)sincos(2
is seriesFourier theTherefore,
1
1
1
1
1
0
n
n
n
n
n
nn
xnn
xnn
xnbxnaa
xf
30
Exercise
Find the Fourier series for defined by
Answer :
xxf . x
1
1
sin1
2n
n
nxn
xf
31
5.5 : Odd and Even Functions
A function f(x) is said to be even if :
f(-x) = f(x)
i.e the function value for a particular negative value of x is the same as that for the corresponding positive value of x.
A function f(x) is said to be odd if :
f(-x) = - f(x)
i.e the function value for a particular negative value of x is numerically equal to that for the corresponding positive value of x but opposite in sign.
32
Example 5.5 :
Even function :
y = f(x) = x2 is an even function
because
f(-2) = 4 = f(2)
f(-6) = 36 = f(6)
33
Odd function :
y = f(x) = x3 is an odd function
because
f(-2) = -8 = -f(2)
f(-3) = -27 = -f(3)
34
Properties of An Even function
The graph of an even function is symmetrical about the y-axis.
Hence, the areas under curves is twice the area from 0 to a :
aa
a
dxxfdxxf0
2
y
x a 0 a
y = x2
.0
0
a
a
dxxfdxxf
35
Properties of An Odd function
The graph of an odd function is symmetrical about the origin.
Thus, the integral is 0 because the areas cancel.
0
a
a
dxxf
y
x 0 a
y = x3
a
.0
0
a
a
dxxfdxxf
36
Moreover :If n is positive integer, thus
x2n is an even function,and x2n+1 is an odd function.
Function is an even function.
Function is an odd function.
x
L
cos
x
L
sin
37
The product of two even functions is even.Example :
The product of two odd functions is even Example :
The product of an even function and an odd function is odd. Example :
.642 xxx
.43 xxx
.532 xxx
38
5.6 : Relationship between Even and Odd functions to Fourier series
We could simply obtain the Fourier series for a function, f(x) defined within the interval –a<x<a. In this case, c = -a and c+2L = a, thus L = a, if we could identify whether f(x) is an even or odd function and use the properties of these functions in order to find the coefficients of Fourier series, a0, an and bn
39
i. If f(x) is an even function,
and is an even function, thus
an even function.
and is an odd function, thus
an odd function.
xa
ncos
xa
nxf
cos)(
xa
nxf
sin)(
xa
nsin
40
If f(x) is an even function
aa
a
n dxxa
nxf
adxx
a
nxf
aa
0
cos2
cos1
aa
a
dxxfa
dxxfa
a0
0
21
.0sin1
a
a
n dxxa
nxf
ab
series. CosinesFourier called is
cos2
series,Fourier 1
0
nn x
a
na
axf
41
ii. If f(x) is an odd function, thus :
and an even function, thus
is an odd function.
and an odd function, thus
is an even function
xa
ncos
xa
nxf
cos)(
xa
nxf
sin)(
xa
nsin
42
If f(x) is an odd function
01
0
a
a
dxxfa
a
0cos1
a
a
n dxxa
nxf
aa
.sin2
sin1
0
aa
a
n dxxa
nxf
adxx
a
nxf
ab
series. SineFourier called is
sin series,Fourier 1
n
n xa
nbxf
