3. 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).
4. Jean Baptiste Joseph Fourier (1768-1830)
• 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
5. A Sum of Sinusoids
• Our building block:
• 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?
x
Asin(
6. Fourier Transform
• We want to understand the frequency of our signal. So, let’s
reparametrize the signal by instead of x:
x
Asin(
f(x) F()
Fourier
Transform
F() f(x)
Inverse Fourier
Transform
• For every from 0 to inf, F() holds the amplitude A and phase
of the corresponding sine
– How can F hold both? Complex number trick!
)
(
)
(
)
(
iI
R
F
2
2
)
(
)
(
I
R
A
)
(
)
(
tan 1
R
I
18. FT: Just a change of basis
.
.
.
* =
M * f(x) = F()
19. IFT: Just a change of basis
.
.
.
* =
M-1 * F() = f(x)
20. Fourier Transform – more formally
Arbitrary function Single Analytic Expression
Spatial Domain (x) Frequency Domain (u)
Represent the signal as an infinite weighted sum
of an infinite number of sinusoids
dx
e
x
f
u
F ux
i
2
(Frequency Spectrum F(u))
1
sin
cos
i
k
i
k
eik
Note:
Inverse Fourier Transform (IFT)
dx
e
u
F
x
f ux
i
2
21. • Also, defined as:
dx
e
x
f
u
F iux
1
sin
cos
i
k
i
k
eik
Note:
• Inverse Fourier Transform (IFT)
dx
e
u
F
x
f iux
2
1
Fourier Transform
23. angular frequency ( )
iux
e
Note that these are derived using
Fourier Transform Pairs (I)
24. Fourier Transform and Convolution
h
f
g
dx
e
x
g
u
G ux
i
2
dx
d
e
x
h
f ux
i
2
dx
e
x
h
d
e
f x
u
i
u
i
2
2
'
' '
2
2
dx
e
x
h
d
e
f ux
i
u
i
Let
Then
u
H
u
F
Convolution in spatial domain
Multiplication in frequency domain
25. Fourier Transform and Convolution
h
f
g
FH
G
fh
g H
F
G
Spatial Domain (x) Frequency Domain (u)
So, we can find g(x) by Fourier transform
g f h
G F H
FT FT
IFT
26. Properties of Fourier Transform
Spatial Domain (x) Frequency Domain (u)
Linearity
x
g
c
x
f
c 2
1
u
G
c
u
F
c 2
1
Scaling
ax
f
a
u
F
a
1
Shifting
0
x
x
f
u
F
e ux
i 0
2
Symmetry
x
F
u
f
Conjugation
x
f
u
F
Convolution
x
g
x
f
u
G
u
F
Differentiation
n
n
dx
x
f
d
u
F
u
i
n
2
frequency ( )
ux
i
e
2
Note that these are derived using
28. Example use: Smoothing/Blurring
• We want a smoothed function of f(x)
x
h
x
f
x
g
H(u) attenuates high frequencies in F(u) (Low-pass Filter)!
• Then
2
2
2
2
1
exp
u
u
H
u
H
u
F
u
G
2
1
u
u
H
2
2
2
1
exp
2
1
x
x
h
• Let us use a Gaussian kernel
x
h
x
29. Does not look anything like what we have seen
Magnitude of the FT
Image Processing in the Fourier Domain
30. Image Processing in the Fourier Domain
Does not look anything like what we have seen
Magnitude of the FT
32. Low-pass Filtering
Let the low frequencies pass and eliminating the high frequencies.
Generates image with overall
shading, but not much detail
33. High-pass Filtering
Lets through the high frequencies (the detail), but eliminates the low
frequencies (the overall shape). It acts like an edge enhancer.
