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Transforms
by
B Kanmani
B Kanmani, BMSCE 2
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
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