Fourier series and transforms

PSUT

Engineering Mathematics II

Fourier Series and Transforms

Dr. Mohammad Sababheh

4/14/2009

2

2 11.1 Fourier Series

Fourier Series and Transforms

Contents 11.1 Fourier Series ........................................................................................................................................ 3

Periodic Functions ..................................................................................................................................... 3

Fundamental Period .................................................................................................................................. 4

Period of Multiple Functions..................................................................................................................... 5

Fourier Series ............................................................................................................................................ 6

11.2 Functions of Any Period p = 2L .......................................................................................................... 11

11.6a Parseval's Identity ............................................................................................................................. 13

Applications ......................................................................................................................................... 14

11.7 Dirichlet's Theorem ............................................................................................................................. 17

11.4 Complex Fourier Series ....................................................................................................................... 18

11.6b Parseval's Identity ............................................................................................................................. 22

11.9 Fourier Transform ............................................................................................................................... 24

Fourier Transform ................................................................................................................................... 24

Fourier Sine and Cosine Transforms ....................................................................................................... 27

Inverse Fourier Transform ...................................................................................................................... 29

Applications ......................................................................................................................................... 30

3


11.1 Fourier Series

Periodic Functions

A function is said to be periodic of period if for all x

Example 1

cos

2 cos 2

cos 2 sin sin 2

cos

Hence cos is periodic of period 2

Example 2

sin 4

2sin 4

2

sin 4 2

sin 4 cos 2 cos 4 sin 2

sin 4

=

Hence sin is periodic of period , observe that 2 is also a period of sin 4

4


Useful Identities

• sin sin cos cos sin • cos cos cos sin sin

Notes

• Any function can be considered periodic with period zero, this period is trivial and is not considered as a period.

• If is a period of , then is a period for any integer .

Proof:

want:

we know that

2

3 2

• If is a period then is not necessarily a period.

Fundamental Period

The most interesting period for a periodic function is the smallest positive period , this period is called the Fundamental Period.

The fundamental period of

• sin is 2

• sin 3 is

5


Period of Multiple Functions

If and are periodic of period then so is

Proof

Denote by

want

is periodic of period

If is periodic of period then the graph of repeats itself every units

2

Therefore if we know the curve of a periodic function on , , then we can draw the entire graph.

Exercise

If is periodic of period then

,

00,20,40,60,81

1,2

‐2 ‐1 0 1 2 3 4 5

6


Fourier Series

Our purpose is to approximate periodic functions by sine and cosine.

we define Fourier series of the periodic function f(x) by:

cos sin

Fourier coefficients , can be obtained by Euler formulas.

Derivation:

Suppose cos sin 5

*

cos sin 5 2

12

*

cos cos cos sin 5 cos

1 cos

*

....

0

7


In general

Example 3

Find the Fourier series:

1 01 0

Solution:

12

0

‐1,5

‐1

‐0,5

0

0,5

1

1,5

‐4 ‐3 ‐2 ‐1 0 1 2 3 4

• When the phrase "Fourier series" is mentioned then we implicitly understand that is periodic.

• If the period is not given , then we implicitly understand that its 2

12

1 cos

1 sin

Fourier coefficients of f(x), given by the Euler formulas

8


cos 1

1 cos 0

1 sin

1 sin sin

1 cos

cos

1 1 cos cos 1

1 2 2 cos

21 1

0 4

Now Fourier series

cos sin

4

2 1 sin 2 1

9


Example 4

Evaluate:

2 3 cos 4 sin

Solution:

Denote function by

* We need to find

* Remember that

12

2

* We need to find to be able to find

* However isn't in Fourier form because of " " , so we need to simplify using identity

121 cos

so 2 3 cos 2 2 cos 2 0

* And now substitute to find ...

10


Notes

By a trigonometric polynomial we mean a finite part of the Fourier series. For instance:

• 1 sin 3 cos 5 • 2 sin sin 2 sin 3 • 2 sin sin 2 (Trigonometric but not Fourier form)

Notes

• .

• sin sin 0

• sin cos 0

• cos cos 0

11

11 11.2 Functions of Any Period p = 2L

11.2 Functions of Any Period p = 2L

Example 1

Find the Fourier series of

, 1 1

Solution:

In this example, p = 2 (period = 2 )

In this case when p = 2 L

Thus in our example L = 1

12

13

cos sin

12

1 cos

1 sin

, 2

In general

12

12 11.2 Functions of Any Period p = 2L

11 sin 0

11 cos

11 cos

2 cos

2

1 1

4 1

13

4 1 cos

cos

2 sin

2 cos

Integration by parts

0

13

13 11.6a Parseval's Identity

11.6a Parseval's Identity

Consider Fourier series and expand it

cos sin cos sin cos 2 sin 2 …

Square it

cos sin 2 cos sin 2 cos sin

2 cos cos sin 2 cos sin

Integrate

cos sin

2 0

2| | | | | |1

2| | | | | |1

Parseval's Identity

Standard form

General form

14


Applications

Example 1

From Chapter 11.1 , Example 1

∑ sin 2 1 1 01 0

* L.H.S of Parseval's

2 0 04

2 1

16 12 1

*R.H.S of Parseval's

11 2

*Therefore

12 1

8

Example 2

From Chapter 11.2 Example 1

13

4 1 cos ,

Apply Parseval's

213

16

25

16

25

29

845

1

π90

15


Example 3

Find

1

Now series is given but not unlike Example 2

Solution:

We need such that

1

1

We attempt with since when integrating by parts , we get in the denominator

Taking

= 0

0

1 sin

1cos |

1 1

2 1

cos

1 cos


0

16


Now apply Parseval's

4 1

41 2

3

16

Exercise

Find

1

1

Example 4

Evaluate

2 sin 5 cos 3 cos 10

Solution:

Let 2 sin 3 cos 3 cos 10

Want

According to Parseval's

2

2 1 1 3

We can't find any sum using this

method , like ∑

17

17 11.7 Dirichlet's Theorem

11.7 Dirichlet's Theorem

If is a nice function , then

lim lim

2

Suppose that is periodic of period 2 and that is piecewise continuous , that and both exist.

Example 1

Suppose

2 1

sin ;

Plug 0 , 0 = 0

Plug

2

2 1

sin2

2 1sin

2 2 ,

2 1 12 1

sin2 1

2 2

2 12 1 2

12 1 4

Plug

0 , lim lim

2

20

18

18 11.4 Complex Fourier Series

11.4 Complex Fourier Series

cos sin

is called Real Fourier series

Note

cos sin

cos sin

2 sin

2 cos

cos2

sin2

12

The Complex Fourier Series of is defined to be

19


Remark

12

121

cos1

sin

12

, 0

12

, 0

, 0

Example 1

Write the complex Fourier transform of

2 sin cos 10

Solution:

22 2

1 1 12

12

1 ,

12 ,

1 ,

12

Example 2

Find the real Fourier series of

5

Solution:

5 sin cos sin cos 2 2

20


Example 3

Find the complex Fourier series of

,

Solution:

12

12

12

12

1 1 1 1

1 , 0

For 0

12

12

0

Therefore complex Fourier series is

0 1

,

21


Note

By a complex trigonometric polynomial , we mean a finite part of

For example

• Trig. 1 5 • Not Trig. 1

• Trig. 11 sin 11!

Note that a complex Fourier series of a complex trigonometric polynomial is the same function.

Exercise

Show that

0 ,2 ,

22

22 11.6b Parseval's Identity

11.6b Parseval's Identity

Note

|2 3 | 4 9 13

|3 | |2 3 | 0 3 9

| | 0 1 1

Example 1

1

lets apply Parseval's

| |

1

,

1

,

21

21 1

2 3

16

| |

12

| |

| |

Parseval's Identity for complex Fourier series

23

23 11.6b Parseval's Identity

Example 2

Evaluate

1 3 1 cos 4

Solution:

Let

1 3 1 cos 4

Want

| | 2 | |

1 , 1 , 312 ,

12 , 1

| | 2 1 1 914

14

2

24

24 11.9 Fourier Transform

11.9 Fourier Transform

Fourier Transform

Example 1

Find the Fourier transform of

1 2 20

Then apply Parseval's identity and see what it gives

Solution:

1

√2

1√2

1√2

0

0,5

1

1,5

‐4 ‐2 0 2 4

1√2

| |

Let be defined on ∞,∞

We define its Fourier transform by

Parseval's Identity

25


Note here if we are

asked about 0 , we take the limit

1√2

cos 2 sin 2 cos 2 sin 2

1√2

2 sin 2

Let's apply Parseval's

2 sin 24

2 sin 22

1) Let's play with

2 sin 22

2) Let 2

sin

sin2

3) Let's find

sin

sin2

26


sin2

sin cos2

sin 22

Let 2

sin2

4)

sin

Note

is continuous regardless of

lim ∞ 0

lim lim lim 0

sin 1t


2 sin cos

27


Fourier Sine and Cosine Transforms

Where did these equations come from?

Recall Fourier transform

1√2

If is even

1√2

cos sin

1√2

cos √2

sin

2√2

cos

2cos

2cos

2sin

If is defined on 0,∞ , we define its Fourier Cosine Transform by

And Fourier Sine Transform

28


Note

Practically

when is even.

when is odd.

Note that when is defined on 0,∞ , we can consider it even or odd.

Example 2

Find and for

, 0 10 ,

Solution:

2cos

2cos

2 sin cos

2 sin cos 1

Using limits

021

12

12

2

29


Inverse Fourier Transform

Useful Rules

2

0

1√2

2cos

2sin

Fourier Inverse Transform

Fourier Inverse Cosine Transform

Fourier Inverse Sine Transform

30


Applications

Example 3

Find ;

Solution:

Using the rules

2

0

2

2

2

...

31


Example 4

You are given that

21

2 2

1sin

2 sin1

1 sin

1

2

0 1 0 ? ? ?

The formula of the Fourier Inverse Sine Transform sin is true when is

continuous at . Moreover , recall that is computed for odd function .

If we extend to be odd , we get

Not continuous at 0 when taking , so we use Dirichlet's Theorem.

Education

Fourier series and transforms