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CH#3

Fourier Series and Transform

2nd 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: discrete

Fourier 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 ( or complex exponentials) .

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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Function s(x) (in red) is a sum of six sine functions of different

amplitudes and harmonically related frequencies. Their

summation is called a Fourier series .

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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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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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Examples determine the Fourier transform of the

following periodic signal

Given that

Given that x(t) has the Fourier transform X(ω), Express the Fourier transform of the following signal in terms of X(ω)

𝛿(𝑡)↔ 1

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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

jj ee 2

1

2

1)cos( jj e

je

j

2

1

2

1)sin(

1

sincos

j

je j

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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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Example 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

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

The Fourier series coefficients The Fourier transform of is

tjtjtjtj eeeetf 6644

4

1

4

1

2

1

2

1)(

2)6cos(2

1)4cos()( 0 tttf

4

1

2

13322 cccc

)(tf

)6(2

)6(2

)4()4()( F

)(tf