Filters in Image Processing
Fourier Transform in one dimension
David Svoboda
email: svoboda@fi.muni.cz
Centre for Biomedical Image Analysis
Faculty of Informatics, Masaryk University, Brno, CZ
September 20, 2019
David Svoboda (CBIA@FI) Filters in Image Processing autumn 2019 1 / 58
Outline
1 Discrete Fourier Transform
2 Continuous Fourier Transform
3 Fourier Transform Properties
Visualization
Common Properties
Discrete specific
4 Fast Fourier Transform
David Svoboda (CBIA@FI) Filters in Image Processing autumn 2019 2 / 58
Fourier Transform
Jean Baptiste Joseph Fourier (1768–1830)
David Svoboda (CBIA@FI) Filters in Image Processing autumn 2019 4 / 58
Fourier Transform
Basis functions
Common request:
the basis should be orthonormal, i.e. ϕk · ϕl = δk,l
ϕk(m) =
1
√
N
e
2πimk
N =
1
√
N
cos
2πmk
N
+ i sin
2πmk
N
the basis functions for N = 16:
0 10
−1
0
1
freq=0
0 10
−1
0
1
freq=1
0 10
−1
0
1
freq=2
0 10
−1
0
1
freq=3
0 10
−1
0
1
freq=4
0 10
−1
0
1
freq=5
0 10
−1
0
1
freq=6
0 10
−1
0
1
freq=7
0 10
−1
0
1
freq=8
0 10
−1
0
1
freq=9
0 10
−1
0
1
freq=10
0 10
−1
0
1
freq=11
0 10
−1
0
1
freq=12
0 10
−1
0
1
freq=13
0 10
−1
0
1
freq=14
0 10
−1
0
1
freq=15
0 5 10 15
−1
0
1
freq=0
0 5 10 15
−1
0
1
freq=1
0 5 10 15
−1
0
1
freq=2
0 5 10 15
−1
0
1
freq=3
0 5 10 15
−1
0
1
freq=4
0 5 10 15
−1
0
1
freq=5
0 5 10 15
−1
0
1
freq=6
0 5 10 15
−1
0
1
freq=7
0 5 10 15
−1
0
1
freq=8
0 5 10 15
−1
0
1
freq=9
0 5 10 15
−1
0
1
freq=10
0 5 10 15
−1
0
1
freq=11
0 5 10 15
−1
0
1
freq=12
0 5 10 15
−1
0
1
freq=13
0 5 10 15
−1
0
1
freq=14
0 5 10 15
−1
0
1
freq=15
David Svoboda (CBIA@FI) Filters in Image Processing autumn 2019 5 / 58
Fourier Transform
Basis functions
Properties
periodical:
ϕk+N (m) =
1
√
N
e
2πim(k+N)
N =
1
√
N
e
2πimk
N = ϕk (m)
symmetrical:
ϕN−k (m) =
1
√
N
e
2πim(N−k)
N
=
1
√
N
e− 2πimk
N =
1
√
N
e
2πi(−m)k
N = ϕk (−m)
orthonormal:
ϕk (m) · ϕl (m) =
N−1
m=0
1
√
N
e
2πmki
N
1
√
N
e
2πmli
N =
1
N
N−1
m=0
e
2πm(k−l)i
N = δk,l
David Svoboda (CBIA@FI) Filters in Image Processing autumn 2019 6 / 58
Fourier Transform
Definition
Given 1D discrete function f of N samples and a basis
(ϕk(m), k = {0, . . . , N − 1}), let us define:
forward 1D discrete Fourier transform:
F(k) ≡ f · ϕk =
1
√
N
N−1
m=0
f (m)e−2πimk
N
inverse 1D discrete Fourier transform:
f (m) ≡ F · ϕm =
1
√
N
N−1
k=0
F(k)e
2πimk
N
f · ϕk =
1
√
N
N−1
m=0
f (m)e
2πimk
N =
1
√
N
N−1
m=0
f (m)e−2πimk
N
David Svoboda (CBIA@FI) Filters in Image Processing autumn 2019 7 / 58
Fourier Transform
Matrix notation
If
ϕk(m) =
1
√
N
e
−2πimk
N =
1
√
N
e−2πi
N
mk
=
1
√
N
ψmk
then
A =
1
√
N







1 1 1 . . . 1
1 ψ1 ψ2 . . . ψN−1
1 ψ2 ψ2·2 . . . ψ(N−1)2
1
...
...
...
...
1 ψN−1 ψ2(N−1) . . . ψ(N−1)(N−1)







and
F = Af ⇒ f = A−1
F = A
T
F
Notice: ψ = e−2πi
N is called the Nth root of unity.
David Svoboda (CBIA@FI) Filters in Image Processing autumn 2019 8 / 58
Fourier Transform
The obvious question
When we apply FT, we usually say ”let us decompose our signal into the
sine waves . . . ” Why do we use another (so complicated) basis?
Basis function is a ”sine wave”
we avoid complex numbers
more intuitive if basis function is a
simple ”sine wave”
sine waves without phase shift do not
generate the whole space
possible basis function: sin (km − α)
α hidden in the sine function spoils the
linearity; matrix multiplication cannot
be used
Basis function is ϕk (m)
we have to use complex
numbers
ϕk (m) functions generate the
whole space (form basis)
this basis is orthonormal
transform is linear, i.e. realized
via matrix multiplication
David Svoboda (CBIA@FI) Filters in Image Processing autumn 2019 9 / 58
Fourier Transform
The meaning of Fourier coefficients
If you perform inverse FT
f (m) = F · ϕm =
N−1
k=0
F(k)ϕk(m) =
1
√
N
N−1
k=0
F(k)ei 2πkm
N
you literally compose the original signal f by combining the individual
basis functions.
Each basis function ϕk(m) = ei 2πmk
N defines only the frequency.
Fourier coefficient F(k) = |zk| (cos αk + i sin αk) = |zk|eiαk modifies
the corresponding basis function ϕk(m) by scaling it and shifting it.
F(k)ϕm(k) = |zk|eiαk
ei 2πmk
N = |zk|eiαk +i 2πmk
N
= |zk| cos
2πk
N
m + αk + i sin
2πk
N
m + αk
David Svoboda (CBIA@FI) Filters in Image Processing autumn 2019 10 / 58
Fourier Transform
The meaning of Fourier coefficients
The number of samples is equal, i.e. |f | = |F| = |ϕk|
−4 −3 −2 −1 0 1 2 3 4
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
m
f(m)
Original function
−4 −3 −2 −1 0 1 2 3 4
−8
−6
−4
−2
0
2
4
6
8
k
F(k)
Real part
David Svoboda (CBIA@FI) Filters in Image Processing autumn 2019 11 / 58
Fourier Transform
The meaning of Fourier coefficients
F(0) matches the lowest frequency in the signal f and corresponds to
the “mean” of f :
F(0) ≡
1
√
N
N−1
m=0
f (m)e−2πim0
N =
1
√
N
N−1
m=0
f (m)
F(0) is usually called DC term
DC . . . “direct current” (current of zero frequency)
F(1) . . . F(N − 1) are called AC terms
AC . . . “alternating current”
F(N−1
2 ) matches the highest frequency in the signal f
Exercise: Why does the component F(N−1
2 ) correspond to the highest
frequency?
David Svoboda (CBIA@FI) Filters in Image Processing autumn 2019 12 / 58
Fourier Transform
Continuous version
We again deal with a projection
the basis functions are:
ϕω(x) = e2πixω
= cos 2πxω + i sin 2πxω
the projection of the function f onto the basis function ϕω(x) is the
inner product as well as in the discrete case:
f · ϕω =
b
a
f (x)ϕω(x)dx
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Fourier Transform
Definition
Given 1D integrable function f and a basis (ϕω, ω ∈ R), let us define:
forward 1D continuous Fourier transform
F(ω) ≡
∞
−∞
f (x)e−2πixω
dx
inverse 1D continuous Fourier transform
f (x) ≡
∞
−∞
F(ω)e2πixω
dω
Notice: The sign flip explanation is beyond the scope of this course.
David Svoboda (CBIA@FI) Filters in Image Processing autumn 2019 15 / 58
Fourier spectrum visualization
Centering
Analysis of zero frequency
Notice: Verify in FTutor1D (https://cbia.fi.muni.cz/software/ftutor1d.html).
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Fourier spectrum visualization
Real and Imaginary part
Analysis of time shift
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Fourier spectrum visualization
Magnitude and phase
Analysis of time shift
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FT – Properties & Theorems
Oddness and evenness
1 Each function f (x) is sum of its odd and even part:
E(x) =
1
2
[f (x) + f (−x)]
O(x) =
1
2
[f (x) − f (−x)]
f (x) = E(x) + O(x)
2 sin (x) is an odd function
cos (x) is an even function
3 Any FT basis function is composed of sine and cosine waves
Corollary:
FT of even function misses imaginary part (sine waves)
FT of odd function misses real part (cosine waves)
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FT – Properties & Theorems
Oddness
−4 −3 −2 −1 0 1 2 3 4
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
m
f(m)
Original function
−4 −3 −2 −1 0 1 2 3 4
−8
−6
−4
−2
0
2
4
6
8
k
F(k)
Imaginary part
original function (odd) its transform pair
David Svoboda (CBIA@FI) Filters in Image Processing autumn 2019 21 / 58
FT – Properties & Theorems
Evenness
−4 −3 −2 −1 0 1 2 3 4
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
m
f(m)
Original function
−4 −3 −2 −1 0 1 2 3 4
−2
0
2
4
6
8
10
12
k
F(k)
Real part
original function (even) its transform pair
David Svoboda (CBIA@FI) Filters in Image Processing autumn 2019 22 / 58
FT – Properties & Theorems
Transform pairs
original function transform pair
even even
odd odd
real and even real and even
real and odd imaginary and odd
real complex and hermitian
Hermitian . . . conjugated symmetry, i.e. F(−x) = F(x)
Exercise: How does change the FT if the original function is “real and
even”? Prove your assertion.
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FT – Properties & Theorems
Transform pairs
If F(ω) = FT [f (x)] (ω) we usually write:
f (x) ⊃ F(ω)
and call f (x) and F(ω) the Fourier transform pair.
Keep in mind that:
not all the continuous functions are integrable, i.e. suitable for FT.
For example sin x, cos x, f (x) = 1,
any discrete function is suitable for FT, since it is finite,
if f is even then |F(ω)| = |Re(F(ω))|, i.e. imaginary part is null,
it is common to say that f (x) belongs to time domain.
David Svoboda (CBIA@FI) Filters in Image Processing autumn 2019 24 / 58
FT – Properties & Theorems
Transform pairs
Statement:
Gaussian ⊃ Gaussian
Proof:
FT e−ax2
(ω) =
∞
−∞
e−ax2
e−2πixω
dx
=
∞
−∞
e−ax2
(cos 2πxω − i sin 2πxω) dx
...
=
π
a
e
− π2
a
ω2
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FT – Properties & Theorems
Transform pairs
Gaussian ⊃ Gaussian
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FT – Properties & Theorems
“sinc” function and its importance
sinc(x) =
sin x
x
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FT – Properties & Theorems
“Π” function and its importance
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Π(x) =



1 |x| < 1
2
(1
2 |x| = 1
2)
0 |x| > 1
2
David Svoboda (CBIA@FI) Filters in Image Processing autumn 2019 28 / 58
FT – Properties & Theorems
Transform pairs
Statement:
Rectangle Π function ⊃ Sinc
Proof:
FT [Π(ax)] (ω) =
∞
−∞
Π(ax)e−2πixω
dx
=
1
2a
− 1
2a
1 · (cos 2πxω − i sin 2πxω) dx = · · · =
1
a
sinc
π
a
ω
David Svoboda (CBIA@FI) Filters in Image Processing autumn 2019 29 / 58
FT – Properties & Theorems
Transform pairs
Rectangle Π function ⊃ Sinc
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FT theorems
Scaling
Statement:
f (ax) ⊃
1
|a|
F
ω
a
Proof:
FT [f (ax)] (ω) =
∞
−∞
f (ax)e−2πixω
dx
=
1
|a|
∞
−∞
f (ax)e−2πi(ax)(ω/a)
d(ax)
=
1
|a|
F
ω
a
Notice: Stretch in time domain corresponds to shrinkage in Fourier
domain and vice versa.
David Svoboda (CBIA@FI) Filters in Image Processing autumn 2019 31 / 58
FT theorems
Scaling
Obstacles in discrete domain
Continuous domain: scaling
Discrete domain: upsampling/downsampling (loss of information)
f (am) ≈ downsamplea−times(f (m))
Sampling may introduce new frequencies
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FT theorems
Scaling
Fine sampling
Rough sampling
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FT theorems
Scaling/Reciprocity
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FT theorems
Scaling/Reciprocity
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FT theorems
Scaling/Reciprocity
David Svoboda (CBIA@FI) Filters in Image Processing autumn 2019 36 / 58
FT theorems
Scaling/Reciprocity
David Svoboda (CBIA@FI) Filters in Image Processing autumn 2019 37 / 58
FT theorems
Shift
Statement:
f (x − a) ⊃ e−2πiaω
F(ω)
Proof:
FT [f (x − a)] (ω) =
∞
−∞
f (x − a)e−2πixω
dx
=
∞
−∞
f (x − a)e−2πi(x−a)ω
e−2πiaω
d(x − a)
= e−2πiaω
F(ω)
Notice: Shift affects only phase. The higher the frequency ω is the more
the corresponding cosine wave is affected.
David Svoboda (CBIA@FI) Filters in Image Processing autumn 2019 38 / 58
FT theorems
Shift
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FT theorems
Shift
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FT theorems
Linearity
Statement:
αf (x) + βg(x) ⊃ αF(ω) + βG(ω)
Proof:
FT [αf (x) + βg(x)] (ω) =
∞
−∞
[αf (x) + βg(x)] e−2πixω
dx
= α
∞
−∞
f (x)e−2πixω
dx + β
∞
−∞
g(x)e−2πixω
dx
= αF(ω) + βG(ω)
David Svoboda (CBIA@FI) Filters in Image Processing autumn 2019 41 / 58
FT theorems
Linearity
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FT theorems
Convolution theorem
Statement:
f (x) ∗ g(x) ⊃ F(ω)G(ω)
f (x)g(x) ⊃ F(ω) ∗ G(ω)
Proof:
FT [f (x) ∗ g(x)] (ω) =
∞
−∞


∞
−∞
f (x )g(x − x )dx

 e−2πixω
dx
=
∞
−∞
f (x )


∞
−∞
g(x − x )e−2πixω
dx

 dx
=
∞
−∞
f (x )e−2πix ω
G(ω)dx = F(ω)G(ω)
David Svoboda (CBIA@FI) Filters in Image Processing autumn 2019 43 / 58
FT theorems
Convolution theorem
The meaning:
The convolution in time domain corresponds to point-wise
multiplication in the Fourier domain, and vice versa.
time
domain
domain
Fourier
=
=
f
F G
f*g
FT FT IFT
g
FT(f)FT(g)
faster?
slower?
*
.
David Svoboda (CBIA@FI) Filters in Image Processing autumn 2019 44 / 58
FT theorems
Rayleigh’s energy theorem (Parseval’s theorem)
Statement:
∞
−∞
|f (x)|2
dx =
∞
−∞
|F(ω)|2
dω
Proof:
∞
−∞
|f (x)|2dx =
∞
−∞
f (x)f (x)dx
=
∞
−∞
f (x)f (x)e−2πixω dx ω = 0
= F(ω ) ∗ F(−ω ) ω = 0
=
∞
−∞
F(ω )F(ω − ω )dω ω = 0
=
∞
−∞
F(ω)F(ω)dω
David Svoboda (CBIA@FI) Filters in Image Processing autumn 2019 45 / 58
FT theorems
Rayleigh’s energy theorem (Parseval’s theorem)
Rayleigh’s energy theorem – the integral of the square of a function is
equal to the integral of the square of its transform.
Notice: |F(k)|2 is called a power spectrum.
David Svoboda (CBIA@FI) Filters in Image Processing autumn 2019 46 / 58
DFT theorems
Convolution theorem
Let f and g be 1D discrete periodic signals of length N, then:
f ∗ g = IDFT [DFT(f ) · DFT(g)]
√
N
Proof:
f (m) ∗ g(m) =
N−1
k=0
f (k)g(m − k)
=
N−1
k=0
1
√
N
N−1
n=0
F(n)e2πikn/N 1
√
N
N−1
l=0
G(l)e2πi(m−k)l/N
=
1
N
N−1
n=0
F(n)
N−1
l=0
G(l)
N−1
k=0
e2πikn/N
e2πi(m−k)l/N
=
1
N
N−1
n=0
F(n)
N−1
l=0
G(l)e2πiml/N
N−1
k=0
e2πik(n−l)/N
=
1
N
N−1
n=0
F(n)
N−1
l=0
G(l)e2πiml/N
δ(n − l)N =
N−1
n=0
[F(n) · G(n)] e2πimn/N
David Svoboda (CBIA@FI) Filters in Image Processing autumn 2019 47 / 58
DFT theorems
Stretch
If f (m) is a 1D function of length N, p ∈ N, and stretchp{f } = {g}, where
g(n) =
f (n/p) n = 0, p, 2p, . . . , (N − 1)p
0 otherwise
then
G(k) =
1
√
p



F(k) k = 0, . . . , N − 1
F(k − N) k = N, . . . , 2N − 1
...
...
F(k − (p − 1)N) k = (p − 1)N, . . . , pN − 1
Notice: Stretch by a factor p in the time domain results in p-fold
repetition of F(k) in the frequency domain.
David Svoboda (CBIA@FI) Filters in Image Processing autumn 2019 48 / 58
DFT theorems
Stretch
An example of stretch for p = 2
0 1 2 3 4 5 6 7
0
2
4
6
8
10
m
f(m)
time domain
1 2 3 4 5 6 7 8
0
2
4
6
8
10
u
|F(u)|
Fourier (frequency) domain
0 5 10 15
0
2
4
6
8
10
m
g(m)
time domain
0 5 10 15
0
2
4
6
8
10
u
|G(u)|
Fourier (frequency) domain
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DFT theorems
Periodicity of DFT
Statement:
F(k + N) = F(k)
Proof:
F(k + N) =
1
√
N
N−1
m=0
f (m)e−
2πim(k+N)
N
=
1
√
N
N−1
m=0
f (m)e−2πimk
N e−2πimN
N
=
1
√
N
N−1
m=0
f (m)e−2πimk
N
= F(k)
David Svoboda (CBIA@FI) Filters in Image Processing autumn 2019 50 / 58
DFT theorems
Symmetry of real DFT
Statement:
F(−k) = F(k)
Proof:
F(−k) =
1
√
N
N−1
m=0
f (m)e−
2πim(−k)
N
=
1
√
N
N−1
m=0
f (m)e
2πimk
N
= F(k) iff f ∈ RN
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Fast Fourier Transform
Idea: N-point signal (N = 2m, m ∈ N) is decomposed into two N/2-point
signals:
one with all odd samples
one with all even samples
Example:
1 input signal: {0 1 2 3 4 5 6 7 } ⊃ ?
2 2× simpler DFT: {0 2 4 6} ⊃ {A B C D}
2× simpler DFT: {1 3 5 7} ⊃ {P Q R S}
3 stretch: {0 0 2 0 4 0 6 0} ⊃ 1√
2
{A B C D A B C D}
4 stretch: {1 0 3 0 5 0 7 0 } ⊃ 1√
2
{P Q R S P Q R S}
shift: {0 1 0 3 0 5 0 7} ⊃ 1√
2
{P ψQ ψ2R ψ3S ψ4P ψ5Q ψ6R ψ7S}
5 linearity: {0 1 2 3 4 5 6 7} = {0 0 2 0 4 0 6 0} + {0 1 0 3 0 5 0 7}
⇓
{0 1 2 3 4 5 6 7} ⊃ 1√
2
{(A + P) (B + ψQ) (C + ψ2R) . . . }
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Fast Fourier Transform
Derivation:
F(k) =
1
√
N
N−1
m=0
f (m)e−2πimk
N
=
1
√
N
N/2−1
m=0
f (2m)e−
2πi(2m)k
N +
1
√
N
N/2−1
m=0
f (2m + 1)e−
2πi(2m+1)k
N
=
1
√
N
N/2−1
m=0
f (2m)e
−2πimk
N/2 + e−2πik
N
1
√
N
N/2−1
m=0
f (2m + 1)e
−2πimk
N/2
= Fe
(k) + ψk
Fo
(k)
Notice: ψ = e−2πi
N
David Svoboda (CBIA@FI) Filters in Image Processing autumn 2019 54 / 58
Fast Fourier Transform
Idea: While it is possible repeat the division.
F(k) → Fe
(k), Fo
(k)
→ Fee
(k), Feo
(k), Foe
(k), Foo
(k)
→ Feee
(k), Feeo
(k), Feoe
(k), Feoo
(k), Foee
(k), Foeo
(k),
Fooe
(k), Fooo
(k)
→ . . .
After log2(n) divisions we have Feeeoe...eoeooee(k) which is just one point
long signal in Fourier domain.
You should know that: {X} ⊃ {X}
Exercise: What is the complexity of FFT?
David Svoboda (CBIA@FI) Filters in Image Processing autumn 2019 55 / 58
Fast Fourier Transform
One more illustration
150 1 2 3 4 5 6 7 10 11 12 13 1498
0 2 4 14121086 1 3 5 7 9 11 13 15
0 4 8 12 2 6 10 14 1 5 9 13 3 7 1511
0 8 4 12 2 10 6 14 1 9 5 13 11 7 153
0 8 4 12 2 10 6 14 1 9 5 13 3 11 15716 signals (1 point each)
2 signals (8 points each)
8 signals (2 points each)
4 signals (4 points each)
1 signal (16 points)
David Svoboda (CBIA@FI) Filters in Image Processing autumn 2019 56 / 58
Bibliography
Bracewell, R. N., Fourier transform and its applications / 2nd ed.
New York: McGraw-Hill, 474 pages, ISBN 0070070156
Gonzalez, R. C., Woods, R. E., Digital image processing / 2nd ed.,
Upper Saddle River: Prentice Hall, 2002, pages 793, ISBN
0201180758
Veit, J., Integr´aln´ı transformace, Praha, 1983
Smith, Steven W. Digital signal processing: A practical guide for
engineers and scientists; Amsterdam: Newnes, 2003, 650 pages;
on-line version: http://www.dspguide.com/
David Svoboda (CBIA@FI) Filters in Image Processing autumn 2019 57 / 58
You should know the answers . . .
Express the discrete Fourier transform as a matrix multiplication.
Derive this matrix.
How many Fourier basis functions do we need if we transform the
signal of length N into the frequency domain?
Formulate the forward discrete Fourier transform. Explain all the
variables and constants.
What does DC and AC terms mean?
Explain the meaning of one particular Fourier coefficient in inverse
Fourier transform.
What is the product of the projection of an even function into a sin
wave?
Why are the wide functions in time domain transformed into their
narrow counterparts in frequency domain and vice versa?
Derive FFT for a signal of length 3m, m ∈ N.
David Svoboda (CBIA@FI) Filters in Image Processing autumn 2019 58 / 58