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Frequency domain analysis and
Fourier Transform
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How to Represent Signals?
• Option 1: Taylor series represents any function using
polynomials.
• Polynomials are not the best - unstable and not very
physically meaningful.
• Easier to talk about “signals” in terms of its “frequencies”
(how fast/often signals change, etc).
Jean Baptiste Joseph Fourier (1768-1830)
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• Had crazy idea (1807):
•
Any periodic function
can be rewritten as a
weighted sum of Sines and
Cosines of different
frequencies.
• Don’t believe it?
– Neither did Lagrange,
Laplace, Poisson and
other big wigs
– Not translated into
English until 1878!
•
But it’s true!
– called Fourier Series
– Possibly the greatest tool
used in Engineering
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A Sum of Sinusoids
• Our building block:
Asin(x   
• Add enough of them to
get any signal f(x) you
want!
• How many degrees of
freedom?
• What does each control?
• Which one encodes the
coarse vs. fine structure of
the signal?
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Fourier Transform
• We want to understand the frequency  of our signal. So, let’s
reparametrize the signal by  instead of x:
Fourier
Transform
f(x)
F()
• For every  from 0 to inf, F() holds the amplitude A and phase
 of the corresponding sine
Asin(x   
– How can F hold both? Complex number trick!
F ( )  R( )  iI ( )
A   R( )  I ( )
2
F()
2
Inverse Fourier
Transform
I ( )
  tan
R( )
1
f(x)
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Time and Frequency
• example : g(t) = sin(2pi f t) + (1/3)sin(2pi (3f) t)
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Time and Frequency
• example : g(t) = sin(2pi f t) + (1/3)sin(2pi (3f) t)
=
+
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Frequency Spectra
• example : g(t) = sin(2pi f t) + (1/3)sin(2pi (3f) t)
=
+
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Frequency Spectra
• Usually, frequency is more interesting than the phase
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Frequency Spectra
=
=
+
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Frequency Spectra
=
=
+
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Frequency Spectra
=
=
+
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Frequency Spectra
=
=
+
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Frequency Spectra
=
=
+
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Frequency Spectra

1
= A sin(2 kt )
k 1 k
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Frequency Spectra
FT: Just a change of basis
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M * f(x) = F()
*
.
.
.
=
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IFT: Just a change of basis
M-1 * F() = f(x)
*
.
.
.
=
Fourier Transform – more formally
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Represent the signal as an infinite weighted sum
of an infinite number of sinusoids
F u   


Note:
f  x e
i 2ux
dx
eik  cos k  i sin k
i  1
Arbitrary function
Single Analytic Expression
Spatial Domain (x)
Frequency Domain (u)
(Frequency Spectrum F(u))
Inverse Fourier Transform (IFT)

f  x    F u e

i 2ux
dx
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Fourier Transform
• Also, defined as:

F u    f x e

Note:
iux
dx
eik  cos k  i sin k
• Inverse Fourier Transform (IFT)
1
f x  
2



F u eiux dx
i  1
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Fourier Transform Pairs (I)
Note that these are derived using
angular frequency ( e  iux )
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Fourier Transform Pairs (I)
Note that these are derived using
angular frequency ( e  iux )
Fourier Transform and Convolution
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Let
g  f h

Then G u    g  x e i 2ux dx




 




f  hx   e i 2ux ddx
  f  e

i 2u
 

 f  e
i 2u
d


d h x   e i 2u  x  dx
 hx'e


i 2ux '

dx'
 F u H u 
Convolution in spatial domain
 Multiplication in frequency domain
Fourier Transform and Convolution
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Spatial Domain (x)
Frequency Domain (u)
g  f h
g  fh
G  FH
G  F H
So, we can find g(x) by Fourier transform
g

IFT
G
f

FT

F
h
FT

H
Properties of Fourier Transform
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Spatial Domain (x)
Frequency Domain (u)
c1 f x  c2 g x
c1F u   c2Gu 
Scaling
f ax
1 u
F 
a a
Shifting
f x  x0 
e i 2ux0 F u 
Symmetry
F x 
f  u 
Conjugation
f  x 
F   u 
Linearity
Convolution
f x   g x 
F u Gu 
Differentiation
d n f x 
dxn
i 2u n F u 
Note that these are derived using
frequency ( e  i 2ux )
Properties of Fourier Transform
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Example use: Smoothing/Blurring
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• We want a smoothed function of f(x)
g x  f x hx
• Let us use a Gaussian kernel
hx 
 1 x2 
1
h x  
exp
2
2

2 



x
• Then
 1

2
H u   exp  2u   2 
 2

Gu   F u H u 
H u 
1
2
u
H(u) attenuates high frequencies in F(u) (Low-pass Filter)!
Image Processing in the Fourier Domain
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Magnitude of the FT
Does not look anything like what we have seen
Image Processing in the Fourier Domain
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Magnitude of the FT
Does not look anything like what we have seen
Convolution is Multiplication in Fourier Domain
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|F(sx,sy)|
f(x,y)
*
h(x,y)
|H(sx,sy)|
g(x,y)
|G(sx,sy)|
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Low-pass Filtering
Let the low frequencies pass and eliminating the high frequencies.
Generates image with overall
shading, but not much detail
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High-pass Filtering
Lets through the high frequencies (the detail), but eliminates the low
frequencies (the overall shape). It acts like an edge enhancer.
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Boosting High Frequencies
Most information at low frequencies!
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Fun with Fourier Spectra