Fourier Series Problems and Solution
Fourier Series Problems and Solution
Fourier Series Problems and Solution
Example 1
Find the fundamental frequency of the
following Fourier series:
Solution to Example 1
(a ) : f (t ) 5 cos 40t cos 80t
SOLUTION :
1 2f 40
SOLUTION :
1 2f 20
2 f 40
f 20 Hz
2 f 20
f 10 Hz
Example 2
Find the amplitude and phase of the
fundamental component of the function:
f (t ) 0.5 sin 1t 1.5 cos 1t 3.5 sin 2 1t
................... 3 cos 3 1t
Recall:
b
tan
a
1
Solution to Example 2
Fundamental component:
Example 3
Sketch the graph of the periodic function
defined by
Solution to Example 3
f(t)
1
-1
Example 4
Write down a mathematical expression of
the function whose graph is:
f(t)
-2
-1
Solution to Example 4
t..............0 t 1..........T 2
f (t )
1.............1 t 2
Example 5:
Sketch the graph of the following periodic
functions:
(a) : f (t ) t ,1 t 1;T 2
2
0,0 t 2 ;T
(b) : f (t )
sin t , t
t ,2 t 0;T 3
(c) : f (t )
t ,0 t 1
Solution to Example 5
(a)
f(t)
-1
Solution to Example 5
(b)
f(t)
/2
3/2
Solution to Example 5
(c)
-2
-1
Example 6
Show that f(t) is even
a)
f(t)
Solution to Example 6
(a) f(t) = f(-t)
cos t = cos (t)
Example 6
Show that f(t) is even
f(t)
b)
Solution to Example 6
(b) f(t) = f(-t)
t2 = (-t)2
t2 = t2
Example 6
Show that f(t) is even
c)
f(t)
3
Solution to Example 6
(c.) f(t)
3
= f(-t)
= 3
Example 7
Show that f(t) is odd
f(t)
4
t
Solution to Example 7
f(-t) = -f(t)
sin(-/4) = -sin(/4)
sin (-t) = -sin(t)
Example 8
State the product of the following
functions:
(a) f(t) = t3 sin wt
(b) f(t) = t cos 2t
(c) f(t) = t + t2
Solution to Example 8
f(t) = t3 sin wt
= (odd)(odd) = even
f(t) = t cos 2t
= (odd)(even) = odd
f(t) = t + t2
= odd + even = neither
Example
Find the Fourier series of the function
f (t ) t on t
Answer:
a0 0
an 0
2
bn cos n
n
2
f (t ) cos n sin nt
n
n 1
Solution
1
a0
2
1
a0
2
f (t )dt
tdt
1 t
1 ( ) 2 ( ) 2 1
a0
(0)
2 2 2 2
2 2
a0 0
2
Solution
1
an
f (t ) cos ntdt
an
u t
t cos ntdt
dv cos ntdt
1
du dt v sin nt
n
Solution
1
1 t
an sin nt
sin ntdt
n
n
1 t
1
an sin nt 2 cos nt
n
n
1
( )
1
1
an sin n
sin n( ) 2 cos n 2 cos n( )
n
n
n
n
1
1
an 0
Solution
1
bn
f (t ) sin ntdt
bn
u t
t sin ntdt
dv sin ntdt
1
du dt v cos nt
n
Solution
1
t
1
bn cos nt cos ntdt
n
n
1
t
1
bn cos nt 2 sin nt
n
n
1
( )
1
1
bn cos n
cos n( ) 2 sin n 2 sin n( )
n
n
n
n
bn
1
1
cos
n
cos
n
sin
n
sin
n
n
n
n
n
1 2
2
bn
cos n cos n
n
n
Solution
2
f (t ) cos n sin nt
n
n 1
2 L
bn f (t ) sin ntdt
L 0
2
bn
(
t
)
sin
ntdt
0
u t dv sin ntdt
1
du dt v cos nt
n
1
2 t cos nt
bn
cos ntdt
n 0 0 n
2 t cos nt
1
bn
2 sin nt
n 0 n
0
2
(0)
1
1
bn cos n
cos n(0) 2 sin n 2 sin n(0)
n
n
n
n
bn cos n
n
2
bn cos n
n
2
f (t ) cos n sin nt
n
n 1
Example
Expand the given function into a Fourier
series on the indicated interval.
0t 5
4,
f (t )
4, 5 t 0
a
0
0
Answer:
an 0
8
bn
(1 cos n )
n
8
nt
f (t )
(1 cos n ) sin
5
n 1 n
Solution
a0
a0
a0
a0
a0
5
1 L
1 0
f
(
t
)
dt
4
)
dt
(
4
)
dt
0
2 L L
2(5) 5
1
0
5
4t 5 4t 0
10
1
4(0) 4(5) 4(5) 4(0)
10
1
20 20
10
0
Solution
1 L
nt
an f (t ) cos
dt
L
L
L
5
1 0
nt
nt
an (4) cos
dt (4) cos
dt
5
0
5
5
5
1
5
nt
5
nt
an (4)
sin
(4)
sin
5
n
5
n
5
0
5
1 20
nt
20
nt
an
sin
sin
5 n
5 5 n
5 0
1 20
n (0) 20
n (5) 20
n (5) 20
n (0)
an
sin
sin
sin
sin
5 n
5
n
5 n
5
n
5
1 20
20
an
sin n
sin n
5 n
n
an 0
Solution
1 L
nt
f
(
t
)
sin
dt
L L
L
5
1 0
nt
nt
bn (4) sin
dt (4) sin
dt
5
0
5
5
5
bn
1
5
nt
bn (4)
cos
5
n
5
1 20
nt
bn
cos
5 n
5
5
nt
( 4)
cos
n
5
20
nt
cos
n
5
bn
1 20
n (0) 20
n (5) 20
n (5) 20
n (0)
cos
cos
cos
cos
5 n
5
n
5 n
5
n
5
bn
1 20 20
20
20
cos
n
cos
n
5 n n
n
n
1 20
bn
(2)(1 cos n )
5 n
8
bn
(1 cos n )
n
Solution
8
nt
f (t )
(1 cos n ) sin
5
n 1 n
2 L
nt
f (t ) sin
dt
L 0
L
2 5
nt
(4) sin
dt
0
5
5
5
2
5
nt
cos
( 4)
5
n
5 0
5
2
20
nt
cos
5 n
5 0
cos
5 n
5
n
5
2 20
20
bn
cos n
5 n
n
2 20
bn
(1 cos n )
5 n
8
bn
(1 cos n )
n
8
nt
f (t )
(1 cos n ) sin
5
n 1 n
Example
Find the Fourier series of the function
f (t ) t
Answer:
a0
on
3
4
an 2 cos n
n
bn 0
4
f (t )
2 cos n cos nt
3 n 1 n
Solution
1
a0
2
1
a0
2
f (t )dt
1
a0
2
t
1 ( )3 ( )3 1 2 3
3 2 3 3 2 3
a0
2
3
t dt
Solution
1
an
f (t ) cos ntdt
an
u t2
t 2 cos ntdt
dv cos ntdt
1
du 2tdt v sin nt
n
2
2
2t
1 t
1 t
2
an sin nt
sin ntdt sin nt t sin ntdt
n
n
n
n
Solution
ut
dv sin ntdt
1
du dt v cos nt
n
1
1 t
2 t
an sin nt cos nt
cos ntdt
n
n
n n
1 t
2 t
1
an sin nt cos nt 2 sin nt
n
n n
n
1 t
2t
2
an sin nt 2 cos nt 3 sin nt
n
n
n
Solution
2
1 2
( ) 2
2( )
2
2
an sin n
sin n( ) 2 cos n
cos
n
(
sin
n
sin
n
(
3
n
n
n2
n3
n
n
2
1 2
2
2
2
2
an sin n
sin n 2 cos n 2 cos n 3 sin n 3 sin n
n
n
n
n
n
n
1 4
cos
n
n
4
an 2 cos n
n
an
Solution
1
bn
f (t ) sin ntdt
bn
u t2
sin ntdt
dv sin ntdt
1
du 2tdt v cos nt
n
1 t2
1 t2
2t
2
bn cos nt cos ntdt cos nt t cos ntdt
n
n
n
n
Solution
ut
dv cos ntdt
1
du dt v sin nt
n
1 t2
bn cos nt
n
1 t2
bn cos nt
n
1
2 t
sin nt
sin ntdt
nn
n
2 t
1
sin nt 2 cos nt
n n
n
1 t
2t
2
bn cos nt 2 sin nt 3 cos nt
n
n
n
Solution
2( )
1 2
( ) 2
2( )
2
2
cos n
cos n( ) 2 sin n
sin
n
(
cos
n
cos
n
(
)
3
2
3
n
n
n
n
n
n
2
1 2
2
2
2
2
cos n
cos n 2 sin n 2 sin n 3 cos n 3 cos n
n
n
n
n
n
n
0
4
f (t )
2 cos n cos nt
3 n 1 n
a0
f (t )dt
t dt
1 t
1 ( )3 (0)3 1 3
a0
3 0 3
3 3
3
a0
2
3
an
u t2
f (t ) cos ntdt
t 2 cos ntdt
dv cos ntdt
1
du 2tdt v sin nt
n
2
2
2t
2 t
2 t
2
an sin nt
sin ntdt sin nt t sin ntdt
0
n
n
n
n 0
0
0
dv sin ntdt
1
du dt v cos nt
n
1
2 t
2 t
an sin nt cos nt cos ntdt
0 n
n
n n
0
0
2 t
2 t
1
an sin nt cos nt 2 sin nt
n
n n
n
0
0
0
2 t
2t
2
an sin nt 2 cos nt 3 sin nt
n
n
n
0
0
0
an sin n
sin n(0) 2 cos n 2 cos n(0) 3 sin n 3 sin n(0)
n
n
n
n
n
n
2 2
cos
n
n 2
4
an 2 cos n
n
an
4
f (t )
2 cos n cos nt
3 n 1 n
Example
Write the sine series of f(t) = 1 on [0,5]
Answer:
a0 0
an 0
2
bn
(1 cos n )
n
2
nt
f (t )
(1 cos n ) sin
5
n 1 n
Solution
Solution
Half range sine series
a0 = 0
an = 0
Solution
bn: b 2 L f (t ) sin nt dt
n
0
L
L
2 5
nt
bn (1) sin
dt
5 0
5
2
5
nt
bn (1)
cos
5
n
5 0
Solution
bn: b 2 5 cos n (5) 5 cos n (0)
5 n
5
n
5
2 5
5
bn
cos n
5 n
n
2 5
bn
(1 cos n )
5 n
2
bn
(1 cos n )
n
Solution
2
nt
f (t )
(1 cos n ) sin
5
n 1 n
Example
Find the convergence of f(x) on [-2,2]
ex
f (t ) 2 x 2
9
,2 t 1
,1 t 2
,t 2
Solution
Solution
f (2) f (2)
x 2 : f (2)
2
2 x 2 e x
f (2)
2
2
( 2 )
2(2) e
8 7.4
f (2)
0.31
2
2
Solution
x 1:
f (1) f (1)
f (1)
2
e x 2 x 2
f (1)
2
e (1) 2(1) 2 0.37 2
f (1)
0.82
2
2
Solution
x 2:
f (2) f (2)
99
f (2)
9
2
2
Example
Express the function in terms of H(t) and find its
Fourier transform
0, t 0
f (t ) at
e , t 0
Solution
F ( )
F ( )
F ( )
F ( )
F ( )
f (t )e it dt
1
2
1
2
1
(0)e dt (e at )e it dt
i t
(e
at
i t
)e dt
( a i ) t
e
2 (a i )
1
1
2 a i
1
2
e ( a i ) t dt
1 e ( a i ) e ( a i )( 0)
Seatwork
1. Find the Fourier series representation of the
function with period T= 1/50 given by:
1.......0 t 0.01
f (t )
0......0.01 t 0.02
Seatwork
2. Find the Fourier series representation of
the function with period 2 defined by
f (t ) t ,0 t 2
2
Seatwork
3. Find the half range sine series of
f ( x)
(x )
0 x
Seatwork
4. Find the half range cosine series of
f ( x)
(x )
0 x