Transcript Fourier (1) - Petra Christian University

Fourier (2)
Hany Ferdinando
Dept. of Electrical Eng.
Petra Christian University
Overview
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Fourier transform for continuous-time
signal
Fourier transform for discrete-time
signal
Properties
Convolution with Fourier
Introduction to DFT and FFT
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Fourier Analysis
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Fourier Series
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
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Exponential
Sinusoid
Discrete-time Fourier Transform
Fourier Transform
Discrete Fourier Transform (DFT)
Fast Fourier Transform (FFT)
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Discrete-time Fourier Transform (DTFT)
j
F (e ) 

f e
k  
1
fk 
2

 F (e
j
 jk
k
jk
)e d

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Exercise
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h(n) = 1 for 0 ≤ n ≤ N-1, find the DTFT
Proof that cos , has sequence
{½,0,½} in time domain
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Properties of DTFT
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Convergence
Linearity
Convolution
Time shifting
Parseval’s theorem
Frequency convolution
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Convolution with Fourier
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We have f = ½ for n=±1 and so is g,
what is the spectrum of y = fg?
We have two choices to solve it:
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Multiply f and g, then find the signal in
frequency domain, or
Transform both f and g to frequency
domain, then convolve them
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Convolution with Fourier
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F = cos  and G = cos  also
1
j
Y (e ) 
2
1
j
Y (e ) 
2

 cos cos(   )d


 cos (cos cos  sin  sin  )d

cos
Y ( e j ) 
2
Y ( e j ) 

2
cos
 d

cos
2
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Application of DTFT
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It is applied when one wants to design
a digital filter
The frequency response of the filter is
known then one has to calculate its
coefficients
For detail discussion, please refer to
Digital Signal Processing course for
filter design
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Continuous-time Fourier Transform

F ( j ) 
 f (t )e
 jt
dt

1
f (t ) 
2

F
(
j

)
e
d


jt

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Energy and Power Signal
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Energy signal is defined as


f 2 (t )dt  

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Power signal has infinite energy but
finite power

f

T /2
2
(t )dt  
lim
T 
Fourier (2) - Hany Ferdinando
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2
f
(t )dt  

T T / 2
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Exercise
1
0.8
f(t)
0.6
A
0.4
0.2
-T/2
T/2
t
0
-0.2
-0.4
-20
-15
-10
-5
Fourier (2) - Hany Ferdinando
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5
10
15
20
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Properties of Fourier Transform
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Symmetry
Linearity
Scaling
Time shifting (delay)
Frequency shifting (modulation)
Time convolution
Frequency convolution
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Properties of Fourier Transform
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Time differentiation
Time integration
Frequency differentiation
Frequency integration
Reversal
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The Energy Spectrum
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For periodic function, its power can
be associated with the power
contained in each harmonic
component of the signal
For non periodic function, the energy
over the interval (-∞,∞) is finite, while
the power is zero
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The Energy Spectrum
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Energy is defined as E   f 2 (t )dt

Proof that

1
E   f (t )dt 
2

2

 F  j 
2
d

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Numerical Fourier Transform
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How to calculate Fourier transform in
computer?
The integration is approximated with
sum and the technique is called DFT
(Discrete Fourier Transform)
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Numerical Fourier Transform
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Unfortunately, DFT offers problem in
computation speed
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For n values, we need n2 of multiplication and
n(n-1) of addition
To solve this problem, FFT (Fast Fourier
Transform) is used
For detail discussion about DFT and FFT,
please refer to Digital Signal Processing
course
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Next…
The Fourier transform is discussed and students have to
exercise how to use it.
For the next class, students have to read Z transform:
• Signals and Systems by A. V. Oppeneim ch 10, or
• Signals and Linear Systems by Robert A. Gabel ch 4, or
• Sinyal & Sistem (terj) ch 10
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