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CH#3

Fourier Series and Transform

1st semester 1434-1435

King Saud University College of Applied studies and Community Service1301CTBy: Nour Alhariqi

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Outline Introduction Fourier Series Fourier Series Harmonics Fourier Series Coefficients Fourier Series for Some Periodic Signals Example Fourier Series of Even Functions Fourier Series of Odd Functions Fourier Series- complex form

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Introduction The Fourier analysis is the mathematical tool that shows

us how to deconstruct the waveform into its sinusoidal components.

This tool help us to changes a time-domain signal to a frequency-domain signal and vice versa.

•Time domain: periodic signal•Frequency domain: discreteFourier

Series•Time domain: nonperiodic

signal•Frequency domain:

continuous

Fourier Transform

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Fourier Series Fourier proved that a composite periodic signal with

period T (frequency f ) can be decomposed into the sum of sinusoidal functions.

A function is periodic, with fundamental period T, if the following is true for all t: f(t+T)=f(t)

0

T

f

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Fourier Series A periodic signal can be represented by a Fourier series

which is an infinite sum of sinusoidal functions (cosine and sine), each with a frequency that is an integer multiple of f = 1/T 

tnbtmaatfn

nm

m sincos)(11

0

nftbmftaatfn

nm

m 2sin2cos)(11

0

T

ntb

T

mtaatf

nn

mm

2sin

2cos)(

110

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Fourier Series Harmonics

tkttt

tkttt

sin,3sin,2sin,sin

andcos,3cos,2cos,cos

ftftft

ftftft

6sin,4sin,2sin

and6cos,4cos,2cos

Fourier Series = a sum of harmonically related sinusoids

fundamental frequency the kth harmonic frequencythe 2nd harmonic frequency

fundamental the kth harmonic

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Fourier Series Harmonics

ωω ω

ω ω ω

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Fourier Series Coefficients

Are called the Fourier series coefficients, it determine the relative weights for each of the sinusoids and they can be obtained from

tnbtmaatfn

nm

m sincos)(11

0

dttfT

a T 001

,2,1cos2

0 mdttmtfT

a Tm

,2,1sin2

0 ndttntfT

b Tn

DC component or average value

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Fourier Series for Some Periodic Signals

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Example

The Fourier series representation of the square wave

Single term representation of the periodic square wave

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Example

The two term representation of the Fourier series of the periodic square wave

The three term representation of the Fourier series of the periodic square wave

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Example

Fourier representation to contain up to the eleventh harmonic

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Example

Since the continuous time periodic signal is the weighted sum of sinusoidal signals, we can obtain the frequency spectrum of the periodic square-wave as shown below

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Example

From the above figure we see the effect of compression in time domain, results in expansion in frequency domain. The converse is true, i.e., expansion in time domain results in compression in frequency domain.

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f( The value of the function would be the same when we

walk equal distances along the X-axis in opposite

directions.

tftf

Mathematically speaking-

Even Functions

t

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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. tftf

Mathematically speaking-

f(

Odd Functions

t

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

10 0 105

0

5

q

Fourier Series of Even Functions

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

10 0 105

0

5

q

Fourier Series of Odd Functions

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The Fourier series of an even function tf

is expressed in terms of a cosine series .

10 cos

nn tnaatf

The Fourier series of an odd function tf

is expressed in terms of a sine series .

1sin

nn tnbtf

Fourier Series of Even/Odd Functions

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Fourier Series- complex form The Fourier series can be expressed using complex

exponential function

n

tjnnectf

T tjnn dtetf

Tc 0

1

The coefficient cn is given as

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Fourier Transform

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Outline Fourier transform Inverse Fourier transform Basic Fourier transform pairs Properties of the Fourier transform Fourier transform of periodic signal

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Fourier Transform Fourier Series showed us how to rewrite any periodic

function into a sum of sinusoids. The Fourier Transform is the extension of this idea for non-periodic functions.

the Fourier Transform of a function x(t) is defined by:

The result is a function of ω (frequency). 

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Inverse Fourier Transform We can obtain the original function x(t) from the function

X(ω ) via the inverse Fourier transform.

As a result, x(t) and X(ω ) form a Fourier Pair:

( ) ( )x t X

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Example Let The called the unit impulse signal :

The Fourier transform of the impulse signal can be calculated as follows

So ,

)()( ttx )(t

1)()( )0(

jtj edtetX

( ) 1t

w

X(w)

t

x(t)

01

00)(

t

tt

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Basic Fourier Transform pairs Often you have tables for common Fourier transforms

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Example Consider the non-periodic rectangular pulse at zero with

duration τ seconds

Its Fourier Transform is:

2

sin)( cP

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properties of the Fourier Transform

Linearity:

Left or Right Shift in Time:

Time Scaling:

( ) ( )x t X ( ) ( )y t Y

( ) ( ) ( ) ( )x t y t X Y

00( ) ( ) j tx t t X e

1( )x at X

a a

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properties of the Fourier Transform

Time Reversal:

Multiplication by a Complex Exponential ( Frequency Shifting) :

Multiplication by a Sinusoid (Modulation):

( ) ( )x t X

00( ) ( )j tx t e X

0 0 0( )sin( ) ( ) ( )2

jx t t X X

0 0 0

1( )cos( ) ( ) ( )

2x t t X X

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Example: Linearity

2( ) 4sinc 2sincX

The Fourier Transform of x(t) will be :

Let x(t) be : )(2

1)()( 24

tptptx

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Example: Time Shift

2( ) ( 1)x t p t

( ) 2sinc jX e

The Fourier Transform of x(t) will be :

Let x(t) be :

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Example: Time Scaling

2 ( )p t

2 (2 )p t

2sinc

sinc2

time compression frequency expansion

time expansion frequency compression

1a 0 1a

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Example: Multiplication by a Sinusoid

Let x(t) be : 0( ) ( )cos( )x t p t t

The Fourier Transform of x(t) will be :0 01 ( ) ( )

( ) sinc sinc2 2 2

X

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Fourier Transform for periodic signal We learned that the periodic signal can be represented by

the Fourier series as:

We can obtain a Fourier transform of a periodic signal directly from its Fourier series

n

tjnnectx 0

T tjnn dtetx

Tc 0

01 the coefficient cn is given as

nn ncX )(2)( 0

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Fourier Transform for periodic signal The resulting transform consists of a train of impulses in

the frequency domain occurring at the harmonically related frequencies, which the area of the impulse at the nth harmonic frequency nω0 is 2π times nth the Fourier series coefficient cn

So, the Fourier Transform is the general transform, it can handle periodic and non-periodic signals. For a periodic signal it can be thought of as a transformation of the Fourier Series

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Example Let The Fourier series representation of

The Fourier series coefficients The Fourier transform of

So,

tjtj eet 00

2

1

2

1)cos( 0

)cos()( 0ttf )(tf

2

1

2

111 cc

)(tf

)()()( 00 F

0 0 0cos( ) ( ) ( )t

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Examples

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Example 1 Let

which has the fundamental frequency

Rewrite x(t) as a complex form and find the Fourier series coefficients ?

Its known from Euler’s relation that:

42coscos2sin1)( 000

ttttx

0

tjtjtjtj eeeet 0000

2

1

2

1

2

1)cos( 0

nc

tjtjtjtj ej

ej

eej

t 0000

2

1

2

1

2

1)sin( 0

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Solution Rewriting x(t) as a complex form, x(t) will be :

42

42

000

00

0000

2

1

2

1

2

1

2

11)(

42coscos2sin1)(

tjtj

tjtjtjtj

ee

eeej

ej

tx

ttttx

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Solution

Thus, the Fourier series coefficients are:

tjjtjj

tjtj

eeee

ej

ej

tx

00

00

24242

1

2

1

2

11

2

111)(

42

42

11

0

2

1

2

1

2

11

2

11

1

jj

ecec

jc

jc

c

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Example 2 Consider a periodic signal x(t) with fundamental

frequency 2π, that has the following Fourier series coefficients

Rewrite x(t) as a trigonometric form?

From the given coefficients, the x(t) in complex form

,3

1

,2

1,

4

1,1

33

22110

cc

ccccc

0

3

3

2)(n

tjnnectx

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Solution rewriting x(t) and collecting each of the harmonic

components which have the same fundamental frequency, we obtain

Using Euler’s relation, x(t) can be written as:

tjtj

tjtjtjtj

ee

eeeetx

66

4422

3

12

1

4

11)(

ttttx 6cos3

24cos2cos

2

11)(

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Example 3 A periodic signal x(t) with a fundamental period T = 8 has

the following nonzero Fourier series coefficients

Express x(t) in the trigonometric form?

The fundamental frequency is

jcccc 4,2 3311

4

20

T

tjtj

tjtjtjtj

ee

ejeeetx

66

44

3

44

3

1

422)(

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Example 3 Let , find its Fourier transform ? The Fourier series representation of is

The Fourier series coefficients The Fourier transform of is

tjtj ej

ej

ttf 00

2

1

2

1)sin()( 0

)sin()( 0ttf )(tf

jc

jc

2

1

2

111

)(tf

)()()( 00 jj

F