Transforms

by

B Kanmani

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Sequence of presentation

1. Fourier Series: FS

2. Fourier Transform: FT

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1. Fourier Series

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The

Fourier representation

of

continuous time periodic signals:

Fourier Series

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0 0 0

1 1

0

0

( ) cos sin

cos

n n

n n

n n

n

x t a a n t b n t

c n t

The Fourier Series representation

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Example-I

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Continuous time periodic signal

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Time domain representation

without using Fourier series

( )

1 0 / 2

1 / 2

ONE

n

ONE

x t x t nT

where

t Tx

T t T

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Without Fourier Series

X-one: Equation for signal in one

period T

x(t): Sum of Time shifted X-one

Infinite Sum

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Without Fourier Series

SET of TWO equations

Difficult to perform operations like: multiplication, differentiation, addition

with another signal

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With Fourier Series

SINGLE EQUATION

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Same Example

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Time domain representation

using Fourier series

01,3,5,...

4( ) sin

n

x t n tn

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04

( ) sinx t t

ONE term

Fourier series

representation

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0

0

4( ) sin

4sin 3

3

x t t

t

TWO term

Fourier series

representation

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0

0

0

4( ) sin

4sin 3

3

4sin 5

5

x t t

t

t

THREE term

Fourier series

representation

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11

0

1,3,5,...

4( ) sin

n

x t n tn

Eleven term

Fourier series

representation

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01,3,5,...

4( ) sin

n

x t n tn

Infinite term

Fourier series

representation

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Some observations

Exact representation when infinite terms are considered

Signal and the cosine basis functions are

assumed to extend to infinity

Original signal is the weighted sum of

harmonics of sinusoidal signals

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Additional Information

Frequency Information

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Frequency Spectrum

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Time-domain

Frequency-domain

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Time-domain

T=1 sec

fo=1Hz

Frequency-domain

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Time-domain

T=0.5 sec

fo=2 Hz

Frequency-domain

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Observation

Compression

in time domain

leads to

expansion

in frequency domain

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Example-II

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Half-wave rectified wave

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ONE TERM

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TWO TERMS

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FIVE TERMS

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TWENTY TERMS

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Fourier Series

0

022,4,6,...

1( ) 0.5sin( )

1cos

( 1)n

x t t

n tn

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Time-domain

Frequency-domain

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Example-III

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THE PERIODIC SIGNAL

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ONE TERM

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TWO TERMS

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THREE TERMS

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FIVE TERMS

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TWENTY TERMS

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Fourier Series

0 0 0

1 1 1( ) sin( ) sin(2 ) sin(3 ) ...

2 3x t t t t

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Time-domain

Frequency-domain

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Fourier Series

Exact time-domain representation

Periodic continuous time signals

From FS, we can get its spectrum

The spectrum is always discrete

The spectrum contains harmonics of the fundamental

Reducing time period increases frequency

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Fourier Series

Continuous time

Periodic signals

Discrete spectrum

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Fourier Series: Application

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Fourier Series: Application

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Fourier Series: Application

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In all cases: output is sine-wave

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What should be the filter cut-off?

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In all cases: cut-off is about 10 fc

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In general

more than 98% of energy

is contained in the

first TEN harmonics

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Another Example

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Time-domain

T=1 sec + 1 sec

Frequency = 0.5Hz

Frequency-domain

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Time-domain

T=1 sec + 3 sec

Frequency = 0.25 Hz

Frequency-domain

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Time-domain

T=1 sec + 7 sec

Frequency = 0.125 Hz

Frequency-domain

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2. Fourier Transform

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Fourier Transform

Continuous time

Non-Periodic signals

Continuous spectrum

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( ) ( )

1( ) ( )

2

j t

j t

X x t e

x t X e d

The Fourier Transform

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Example - I

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Example - II

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Example - III

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Example - IV

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>> fm=500; % sine wave frequency

>> Fs=20*fm; %Actual sampling rate

>>Ts=1/Fs; %Sampling interval

>> time=0:Ts:4.0;

>> x_1=0.5*cos(2*pi*fm*time);

>> sound(x_1,Fs);

Matlab Command

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Example - V

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Example - VI

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Thank you