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Periodic Functions and Fourier Series. A function. is periodic. if it is defined for all real . such that. Periodic Functions. and if there is some positive number,. . be a periodic function with period. Fourier Series. The function can be represented by a trigonometric series as:.

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Periodic functions l.jpg

A function

is periodic

if it is defined for all real

such that

Periodic Functions

and if there is some positive number,

.


Fourier series l.jpg

be a periodic function with period

Fourier Series

The function can be represented by a trigonometric series as:



Slide10 l.jpg

and about?

.

We want to determine the coefficients,

Let us first remember some useful integrations.



Slide14 l.jpg

Determine about?

to

Integrate both sides of (1) from



Slide17 l.jpg

to about?

instead.

You may integrate both sides of (1) from

It is alright as long as the integration is performed over one period.


Slide20 l.jpg

Determine about?

Multiply (1) by

to

and then Integrate both sides from


Slide21 l.jpg

First term, about?

Second term,

Let us do the integration on the right-hand-side one term at a time.


Slide22 l.jpg

Second term, about?

Third term,


Slide23 l.jpg

Therefore, about?


Slide24 l.jpg

Determine about?

Multiply (1) by

to

and then Integrate both sides from


Slide25 l.jpg

First term, about?

Second term,

Let us do the integration on the right-hand-side one term at a time.


Slide26 l.jpg

Second term, about?

Third term,


Slide27 l.jpg

Therefore, about?



Slide29 l.jpg

We can write about?n in place of m:




Slide37 l.jpg

Therefore, the corresponding Fourier series is function.

In writing the Fourier series we may not be able to consider infinite number of terms for practical reasons. The question therefore, is – how many terms to consider?


Slide38 l.jpg

1.5 function.

1

0.5

f

(

q

)

0

0.5

1

1.5

q

When we consider 4 terms as shown in the previous slide, the function looks like the following.


Slide39 l.jpg

1.5 function.

1

0.5

f

(

q

)

0

0.5

1

1.5

q

When we consider 6 terms, the function looks like the following.


Slide40 l.jpg

1.5 function.

1

0.5

f

(

q

)

0

0.5

1

1.5

q

When we consider 8 terms, the function looks like the following.


Slide41 l.jpg

1.5 function.

1

0.5

f

(

q

)

0

0.5

1

1.5

q

When we consider 12 terms, the function looks like the following.


Slide42 l.jpg

1.5 function.

1

0.5

0

0.5

1

1.5

q

The red curve was drawn with 12 terms and the blue curve was drawn with 4 terms.


Slide43 l.jpg

1.5 function.

1

0.5

0

0.5

1

1.5

0

2

4

6

8

10

q

The red curve was drawn with 12 terms and the blue curve was drawn with 4 terms.


Slide44 l.jpg

1.5 function.

1

0.5

0

0.5

1

1.5

0

2

4

6

8

10

q

The red curve was drawn with 20 terms and the blue curve was drawn with 4 terms.


Slide45 l.jpg

Even and Odd Functions function.

(We are not talking about even or odd numbers.)


Slide46 l.jpg

Mathematically speaking - function.

Even Functions

The value of the function would be the same when we walk equal distances along the X-axis in opposite directions.


Slide47 l.jpg

Mathematically speaking - function.

Odd Functions

The value of the function would change its sign but with the same magnitude when we walk equal distances along the X-axis in opposite directions.


Slide48 l.jpg

5 function.

0

5

10

0

10

q

Even functions can solely be represented by cosine waves because, cosine waves are even functions. A sum of even functions is another even function.


Slide49 l.jpg

5 function.

0

5

10

0

10

q

Odd functions can solely be represented by sine waves because, sine waves are odd functions. A sum of odd functions is another odd function.


Slide50 l.jpg

The Fourier series of an even function function.

is expressed in terms of a cosine series.

The Fourier series of an odd function

is expressed in terms of a sine series.




Slide55 l.jpg

This is an even function. note.

Therefore,

The corresponding Fourier series is


Slide56 l.jpg

Functions Having Arbitrary Period note.

w is the angular velocity in radians per second.


Slide57 l.jpg

f note. is the frequency of the periodic function,

where

Therefore,






Slide65 l.jpg

when function.n is even.



Slide67 l.jpg

The Complex Form of Fourier Series function.

Let us utilize the Euler formulae.


Slide68 l.jpg

th harmonic component of (1) can be function.

The

expressed as:


Slide69 l.jpg

, function.

and

Denoting


Slide71 l.jpg

The Fourier series for function.

can be expressed as:


Slide72 l.jpg

The coefficients function.can be evaluated in the following manner.


Slide74 l.jpg

Note that function.

is the complex conjugate of

.

Hence we may write that

.




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