EE2006 Engineering Mathematics I
Assoc. Prof. Wang Han
Room : S2 –B2b-49
Phone: 6790-4506
Email: hw@ ntu.edu.sg
Slides Adopted from Prof. N. Sundararajan
Topics :
Fourier Analysis – Fourier Series ; Fourier Transform
Textbook :
1. Kreyzig, E., Advanced Engineering Mathematics, 8th edition, John Wiley.
Reference:
2. Glyn James, Modern Engineering Mathematics, Fourth Edition, 2008, Prentice Hall.
3. Ravish R Singh, Mukul Bhatt, Engineering Mathematics, A Tutorial Approach, 2010, McGraw Hill.
1. Fourier Analysis
1.1 Periodic Functions
1.2 Fourier Series & Euler Formulae
1.3 Fourier Series for Functions with Period p = 2L
1.4 Even and Odd Functions
1.5 Fourier Series for 'Even' and 'Odd' Functions
1.6 Half-range Expansions
1.7 Exponential Form of Fourier Series
Fourier Transform :
1.8 Fourier Transform & Properties
1.9 Convergence of Fourier Transform
1.10 Fourier Transform of Elementary Functions
1.11 Some Observations about Fourier Transform
1.12 Properties of Fourier Transform
1.13 Discrete Fourier Transform
1.1 Periodic Functions
A function f(t) is said to be periodic if it is defined for all real t, and if there is some positive number T such that;
f(t + T) = f(t), for all t
T
f
(
t
)
t
Periodic function
T
'PERIOD' of
f
(
t
)
Periodic Functions
Example: sin t, cos t, f = c, const.
Non-periodic Functions
Ex: f(t) = t, t2, t3, ln t.
Note:
i.e. for any integer n,
T "fundamental period"
2T, 3T, 4T, .... nT are also periods of f(t)
Note:
If f(t) and g(t) have periods T, then h(t) = a f(t) + b g(t); (a, b, are constants) also has the period T.
Periodic Functions with Period 2 (T = 2)
Ex: 1, cos x, sin x, cos 2x, sin 2x ..... cos nx, sin nx, .....
Trigonometric Series
a0 + a1 cos x + b1 sin x + a2 cos 2x + b2 sin 2x + ....
where (a0, a1, a2, ...., b0, b1, b2 .... are real constants)
where an, bn are coefficients.
Examples
2
cos
x
0
1
1
2
cos 2
x
0
1
1
2
sin
x
0
1
1
2
sin 2
x
0
1
1
2
cos
x
0
1
1
1.2 Fourier Series & Euler Formulae
Aim: To represent a given period function f(x) with period 2 by the trigonometric series.
Problem : given f(x), find a0, an and bn for n = 1, 2, ... ?
Fourier Series f(x) – periodic function with period 2.
Fourier Coefficients a0, an, bn for n = 1, 2, ...
Euler Formulae in the interval [-,+]: (p=2)
Example 1:
Given a periodic function
f(x + 2) = f(x)
Find its Fourier series representation and Fourier coefficients.
2
k
k
0
x
f
(
x
)Solution
Note: a0 is the average value of f(x) in [, ]
an= 0
; b2 = 0 ; ; b4 = 0
Fourier series of f(x) is f(x) = b1sinx + b2 sin2x + b3 sin3x + …
2
Period
Frequency=1/period
(2
/5)
(2
/3)
Try this at home:
This property of functions is called “orthogonal”.
Example 2: Find the Fourier series of
in the interval of [-,].
Solution:
Hence, f(x) = a0 + a1cosx + a2cos2x + a3cos3x + …
+ b1sinx + b2 sin2x + b3 sin3x + …
1.3 Fourier Series for Functions with Period p = 2L
Periodic functions in applications have (rarely) the period 2, but some other period
(L - length of a vibrating string
- rod in heat conduction)
i.e. p = 2L 2
Fourier Series for such function
f(x), L < x < L
and f(x + 2L) = f(x)
is given by
where Fourier Coefficients a0, an, bn are:
Examples
1. Function with period p = 2L. Find the Fourier Series of the function
3
2
1
1
2
3
k
f
(
x
)
x
The period p = 2L = 4
L = 2
f(x
)
;
n = 1, ,
n = 2, a2 = 0
n = 3, ,
n = 4, an = 0
an = 0, for n is even
4/5
4/3
4
Period
Frequency
2. Find the Fourier series of
Solution: The Fourier series of f(x) with period p=2l=2 is given by,
Hence,
1.4 Even and Odd Functions
· A function f(t) is even if
- graph of such a function will be symmetric about y axis.
t
f
(
t
)
Examples: f(t) = cos t since cos(t) = cos t
· A function g(t) is odd if
- graph of such a function is symmetric about x axis.
t
g
(
t
)
Examples: g(t) = sin t as sin(t) = sin t
· If f(t) is even and has period T, then
· If g(t) is odd and has period T, then
·
The product of even function f(t) and an odd function g(t) is odd.
Let g(t) = f(t)g(t)
q(t) = f(t)g(t) = f(t) – g(t)
= f(t)g(t) = q(t)
q(t) is odd.
·
Note:
odd . odd = even
even . even = even
odd . even = odd
Examples:
(1) x2 is even in (, )
sinx is odd
x2 sinx is odd
(2) sinx . sinx = sin2x = even
· Note: Most functions are neither "even" or "odd".
Example: ex, x2 – x ...
1.5 Fourier Series for 'Even' and 'Odd' Functions
Case 1: If f(x) is an even function with period p=2L (no sine terms), then
The product of two even functions is even,
and
Case 2: If f(x) is an odd function with period p=2L, then
a0=0, an=0,
…(1)
Example 2: Find the Fourier series of f(x)=x|x| in the interval (-1,1).
Solution:
f(x) = x|x|
f(-x)= -x|x| = -f(x)
Hence f(x) is an odd function.
So, a0=0, an=0
Theorem:
* The Fourier coefficients of a sum f1 + f2 are the sums of the corresponding coefficients of f1 and f2.
* The Fourier coefficients of Cf(t) are C x Fourier coefficients of f(t).
Example:
Find the Fourier Series of the function
f(x) = x + , < x <
and f(x + 2) = f(x)
Solution:
We can write f = f1 + f2 where f1 = x and f2 = .
Now f2 = form
f1 = x odd
:
1.6 Half-range Expansions
Question: What is a half-range expansion (Fourier Series) of the given function f(x)?
Answer:
A function is given in the interval
f(x) , 0 x L
L
0
f
(
x
)
To find a Fourier Series, the function has to be 'periodic'.
Hence, the given function f(x), 0 x L has to be made periodic.
Before doing it, we have to define the 'period'. Normally, the period is defined as p = 2L, i.e. the function is defined in L x L.
But the function is given only in
f(x), 0 x L
Hence, in order to get the period of 2L, we have to extend the function in L x 0. This extension can be done in two ways
(1) Odd extension in L x 0
Or
(2) Even extension in L x 0.
What is 'odd' extension?
L
0
f
(
x
)
LDefine
f(x) = f(x) in L x 0
L
0
f
(
x
)
L
x'Even' Extension:
f(x) = f(x), L x 0
Once the function f(x) is defined in L x L either by odd or even extension, these can be extended with a period 2L as shown below.
Fig. (a) Function f(x) given on an interval 0 x L, (b) its even extension to the full "range" (interval) L x L (heavy curve) and the periodic extension of period 2L to the x-axis, (c) its odd extension to L x L (heavy curve) and the periodic extension of period 2L to the x-axis.
For these periodic functions, Fourier Series can be found in the normal way. They are referred as "Half-range" Expansions and are called "odd" or "even" depending on the extension in L x 0.
"Even" (cosine) Half-range Expansion
where
"Odd" (sine) Half-range Expansion
where
Example 1:
The function f(x) is given below in 0 x L. Find the two possible half-range Fourier expansions for f(x)
k
L
x
f
(
x
)
'Triangle'
Answer:
'Even' Extension
k
x
f
(
x
)
'Odd' Extension
L
k
x
f
(
x
)
Solution:
(a) Even periodic extension:
Now
i.e. , ,
an = 0 if n 2, 6, 10…
half-range expansion (even)
(b) Odd periodic extension:
Example 2: Find half-range cosine series of f(x)=x(-x) in the interval (0,) and hence deduce that
And
Solution: The half-range cosine series of f(x) with period 2 is given by
(L=
1.7 Complex Form of Fourier Series
Euler formulae:
ej = cos + jsin; e-j = cos - jsin
So
Given that
Replace sin, cos with its complex form, we have
where,
Thus the alternative complex form of Fourier series is
Case 1: when c=0,
Case 2: when c=-L,
Example 1: Find the complex form of Fourier series f(x) = 2x in the interval ) .
Solution: L= , hence,
Example 2: Find the complex form of Fourier series of
Solution: 2L=2, hence L=1
For n=0,
Hence,
f
(
t
)
k
t
-L
LExample 3:
f(t + 2L) = f(t), hence period p=2L ;
Solution:
When n=0,
c
0
c
1
c
2
c
3
c
4
c
5
nw
0
0.2
c
n
w
0
2
w
0
3
w
0
4
w
0
5
w
0
0
Suppose that a = 5 and k = 1
The Fourier series is
Summary
· Non sinusoidal, but periodic, waveforms are frequently encountered in the operation of electrical systems.
· Most periodic waveforms can be represented as Fourier Series.
· Any symmetry that exists in the waveform of a periodic function greatly simplifies the task of finding the Fourier coefficients.
· Since the Fourier series translates the problem from time domain to the frequency domain, the frequency response of a system can be readily used to predict the waveform of the output.
1-36
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