The Fourier Series

1
The Fourier Series

• BET2533 Eng. Math. III
• R. M. Taufika R. Ismail
• FKEE, UMP

2
Introduction
A Fourier series is an expansion of a periodic
function f (t) in terms of an infinite sum of
cosines and sines
3
In other words, any periodic function can be
resolved as a summation of constant value and
cosine and sine functions
4

• The computation and study of Fourier series is
  known as harmonic analysis and is extremely
  useful as a way to break up an arbitrary periodic
  function into a set of simple terms that can be
  plugged in, solved individually, and then
  recombined to obtain the solution to the original
  problem or an approximation to it to whatever
  accuracy is desired or practical.

5
f(t)
Periodic Function

t





6
where
we can also use the integrals limit .
7
Example 1
Determine the Fourier series representation of
the following waveform.
8
Solution
First, determine the period describe the one
period of the function
T 2
9
Then, obtain the coefficients a0, an and bn
Or, since y f(t) over the interval a,b, hence
is the total area below graph
10
Notice that n is integer which leads
, since
Therefore, .
11
Notice that
or
Therefore,
12
Finally,
?
13
Some helpful identities
For n integers,
14
Supplementary

• The sum of the Fourier series terms can evolve
  (progress) into the original waveform

• From Example 1, we obtain

• It can be demonstrated that the sum will lead to
  the square wave

15
(a)
(b)
(c)
(d)
16
(e)
(f)
17
(No Transcript)
18
Example 2
Given
Sketch the graph of f (t) such that
Then compute the Fourier series expansion of f
(t).
19
Solution
The function is described by the following graph
T 2
We find that
20
Then we compute the coefficients
21
since
22
(No Transcript)
23
Finally,
?
24
Example 3
Given
Sketch the graph of v (t) such that
Then compute the Fourier series expansion of v
(t).
25
Solution
The function is described by the following graph
v (t)
2
t
0
2
4
6
8
10
12
T 4
We find that
26
Then we compute the coefficients
27
(No Transcript)
28
since
29
Finally,
?
30
Symmetry Considerations

• Symmetry functions
• (i) even symmetry
• (ii) odd symmetry

31
Even symmetry

• Any function f (t) is even if its plot is
  symmetrical about the vertical axis, i.e.

32
Even symmetry (cont.)

• The examples of even functions are

33
Even symmetry (cont.)

• The integral of an even function from -A to A is
  twice the integral from 0 to A

-A
A
34
Odd symmetry

• Any function f (t) is odd if its plot is
  antisymmetrical about the vertical axis, i.e.

35
Odd symmetry (cont.)

• The examples of odd functions are

36
Odd symmetry (cont.)

• The integral of an odd function from -A to A is
  zero

-A
A
37
Even and odd functions

• The product properties of even and odd functions
  are

• (even) (even) (even)
• (odd) (odd) (even)
• (even) (odd) (odd)
• (odd) (even) (odd)

38
Symmetry consideration

• From the properties of even and odd functions,
  we can show that

• for even periodic function

• for odd periodic function

39
How?? Even function
(even) (odd)
(even) (even)
(odd)
(even)
40
How?? Odd function
(odd)
(odd) (odd)
(even)
(odd) (even)
(odd)
41
Example 4
Given
Sketch the graph of f (t) such that
Then compute the Fourier series expansion of f
(t).
42
Solution
The function is described by the following graph
f (t)
1
t
0
-4
-6
2
4
6
-2
-1
T 4
We find that
43
Then we compute the coefficients. Since f (t) is
an odd function, then
and
44
since
45
Finally,
?
46
Example 5
Compute the Fourier series expansion of f (t).
47
Solution
The function is described by
T 3
T 3
and
48
Then we compute the coefficients.
Or, since f (t) is an even function, then
Or, simply
49

50
and
since f (t) is an even function.
Finally,
?
51
Function defines over a finite interval

• Fourier series only support periodic functions
• In real application, many functions are
  non-periodic
• The non-periodic functions are often can be
  defined over finite intervals, e.g.

y 2
y 1
y 1
y t2
y t
52

• Therefore, any non-periodic function must be
  extended to a periodic function first, before
  computing its Fourier series representation
• Normally, we prefer symmetry (even or odd)
  periodic extension instead of normal periodic
  extension, since symmetry function will provide
  zero coefficient of either an or bn
• This can provide a simpler Fourier series
  expansion

53
Periodic extension
Non-periodic function
T
Even periodic extension
T
Odd periodic extension
T
54
Half-range Fourier series expansion

• The Fourier series of the even or odd periodic
  extension of a non-periodic function is called as
  the half-range Fourier series
• This is due to the non-periodic function is
  considered as the half-range before it is
  extended as an even or an odd function

55

• If the function is extended as an even function,
  then the coefficient bn 0, hence

• which only contains the cosine harmonics.
• Therefore, this approach is called as the
  half-range Fourier cosine series

56

• If the function is extended as an odd function,
  then the coefficient an 0, hence

• which only contains the sine harmonics.
• Therefore, this approach is called as the
  half-range Fourier sine series

57
Example 6
Compute the half-range Fourier sine series
expansion of f (t), where
58
Solution
Since we want to seek the half-range sine
series, the function to is extended to be an odd
function
f (t)
f (t)
1
1
t
t
0
p
-p
2p
-2p
0
p
-1
T 2p
59
Hence, the coefficients are
and
Therefore,
?
60
Example 7
Determine the half-range cosine series
expansion of the function
Sketch the graphs of both f (t) and the periodic
function represented by the series expansion for
-3 lt t lt 3.
61
Solution
Since we want to seek the half-range cosine
series, the function to is extended to be an even
function
f (t)
f (t)
t
t
T 2
62
Hence, the coefficients are
63
Therefore,
?
64
Parsevals Theorem

• Parservals theorem states that the average power
  in a periodic signal is equal to the sum of the
  average power in its DC component and the average
  powers in its harmonics

65
Pdc
Pavg
f(t)

t
Pa1
Pb1


Pa2
Pb2



66

• For sinusoidal (cosine or sine) signal,

• For simplicity, we often assume R 1?,
• which yields

67

• For sinusoidal (cosine or sine) signal,

68
Exponential Fourier series

• Recall that, from the Eulers identity,

yields
and
69
Then the Fourier series representation becomes
70
,
Here, let we name
Hence,
and .
c-n
c0
cn
71
Then, the coefficient cn can be derived from
72

• In fact, in many cases, the complex Fourier
  series is easier to obtain rather than the
  trigonometrical Fourier series
• In summary, the relationship between the complex
  and trigonometrical Fourier series are

or
73
Example 8
Obtain the complex Fourier series of the
following function
f (t) e t
74
Solution
Since , . Hence

75
since
76
Therefore, the complex Fourier series of f (t) is
?
Notes Even though c0 can be found by
substituting cn with n 0, sometimes it doesnt
works (as shown in the next example). Therefore,
it is always better to calculate c0 alone.
77
cn is a complex term, and it depends on
n?. Therefore, we may plot a graph of cn vs n?.
In other words, we have transformed the
function f (t) in the time domain (t), to the
function cn in the frequency domain (n?).
78
Example 9
Obtain the complex Fourier series of the function
in Example 1.
79
Solution
80
But
Thus,
Here notice that .
Therefore,
?
81
The plot of cn vs n? is shown below
0.5