Post on 27-Jun-2020
052600 VU Signal and Image Processing
Fourier Transform 4: z-Transform (part 2) & Introduction to
2D Fourier Analysis
Torsten Möller + Hrvoje Bogunović + Raphael Sahann
torsten.moeller@univie.ac.at hrvoje.bogunovic@meduniwien.ac.at
raphael.sahann@univie.ac.at
vda.cs.univie.ac.at/Teaching/SIP/17s/
1© Raphael Sahann
Overview
• Sampling & Impulse Train • Fourier Transform (1D)
– properties – convolution theorem – sampling in the Fourier space
• z-Transform – inverse z-Transform
• 2D Fourier Transform
2© Raphael Sahann
3
Check Yourself
Two signals and two regions of convergence.
≠4≠3≠2≠1 0 1 2 3 4n
x[n] =!7
8"n
u[n]
78
z-planeROC
z
z ≠ 78
≠4 ≠3 ≠2 ≠1 0 1 2n
y[n] = ≠ !78"n
u[≠1 ≠ n]
78
ROC
z-plane
z
z ≠ 78
33© Prof. Dennis Freeman, MIT, 2011
4
ÿ
Properties of Z Transforms
The use of Z Transforms to solve di↵erential equations depends on several important properties.
Property x[n] X(z) ROC
Linearity ax1[n] + bx2[n] aX1(z) + bX2(z) ∏ (R1 fl R2)
Delay x[n ≠ 1] z ≠1X(z) R dX(z)
Multiply by n nx[n] ≠z dz
R
Convolve in n x1[m]x2[n ≠ m] X1(z)X2(z) ∏ (R1 fl R2)
m=≠Œ
45
Œÿ
© Prof. Dennis Freeman, MIT, 2011
Relationship between the z-Transform and the Fourier Transform
5© Raphael Sahann
or
Relationship between the z-Transform and the Fourier Transform
6© Raphael Sahann
This z-Transform can be interpreted as the Fourier transform of the product of the original sequence x[n] and the
exponential sequence r-n
r = 1 reduces to the Fourier transform of x[n].
7
Relationship between the z-Transform and the Fourier Transform
© Raphael Sahann
image source: http://flylib.com/books/2/729/1/html/2/images/0131089897/graphics/06fig13.gif
• Interpreting Fourier transform as the z-transform on the unit circle in the z-plane corresponds to wrapping the frequency axis around the unit circle.
• Inherent periodicity in frequency is captured naturally, since a change of angle of 2π radians in the unit circle corresponds to traversing the unit circle once and returning to the same point.
8
Relationship between the z-Transform and the Fourier Transform
© Raphael Sahann
• If the Region of Convergence (ROC) includes the unit circle, the Fourier transform and all its derivatives with respect to ⍵ must be continuous functions of ⍵.
9
Relationship between the z-Transform and the Fourier Transform
© Raphael Sahann
10
Example 3.5
© Raphael Sahann
Consider the sequence
with a = -1/3 we obtain
and using a = 1/2 yields
11
Example 3.5
© Raphael Sahann
Consider the sequence
with a = -1/3 we obtain
and using a = 1/2 yields
12
Example 3.5
© Raphael Sahann
by the linearity of the z-transform
image source: http://d2r5da613aq50s.cloudfront.net/wp-content/uploads/405300.image2.jpg
Example 3.7
• Since there is no overlap between |z|>1/2 and |z|<1/3, x[n] has no z-transform (nor Fourier transform) representation.
13
x[n] =
✓1
2
◆n
u[n]�✓�1
3
◆n
u[�n� 1]
X(z) =1
1� 12z
�1
| {z }|z|> 1
2
+1
1 + 13z
�1
| {z }|z|< 1
3
© Raphael Sahann
Example 3.7
• Since there is no overlap between |z|>1/2 and |z|<1/3, x[n] has no z-transform (nor Fourier transform) representation.
14
x[n] =
✓1
2
◆n
u[n]�✓�1
3
◆n
u[�n� 1]
X(z) =1
1� 12z
�1
| {z }|z|> 1
2
+1
1 + 13z
�1
| {z }|z|< 1
3
© Raphael Sahann
Overview
• Sampling & Impulse Train • Fourier Transform (1D) • z-Transform
– inverse z-Transform • 2D Fourier Transform
15© Raphael Sahann
Inverse z-Transform
• The inverse z-transform is a complex contour integral, where C represents a closed contour within the ROC of the z-transform.
• Too complex for a typical task in this domain, so we use less formal procedures
16
x[n] =1
2⇡j
I
CX(z)zn�1
dz
© Raphael Sahann
Inspection Method
• We “inspect” a z-transform by looking up its transform in a table of common z-transforms.
• Find the inverse z-transform for:
17© Raphael Sahann
X(z) =1
1� 12z
�1, |z| > 1
2
Common z-transform pairs
18
image source: http://www.dip.ee.uct.ac.za/~nicolls/lectures/eee401f/03_ztrans.pdf© Raphael Sahann
Inspection Method• Find the inverse z-transform for:
19© Raphael Sahann
X(z) =1
1� 12z
�1, |z| > 1
2
right-sided
x[n] =
✓1
2
◆n
u[n]
Inverse z-transform
• Inspection Method
• Partial Fractions —> build partial fractions until they can be interpreted by the inspection method
• Power Series Expansion —> Taylor series expansion of z-transform, interpret result with inspection method
20
21
ÿ
Properties of Z Transforms
The use of Z Transforms to solve di↵erential equations depends on several important properties.
Property x[n] X(z) ROC
Linearity ax1[n] + bx2[n] aX1(z) + bX2(z) ∏ (R1 fl R2)
Delay x[n ≠ 1] z ≠1X(z) R dX(z)
Multiply by n nx[n] ≠z dz
R
Convolve in n x1[m]x2[n ≠ m] X1(z)X2(z) ∏ (R1 fl R2)
m=≠Œ
45
Œÿ
© Prof. Dennis Freeman, MIT, 2011
Overview
• Sampling & Impulse Train • Fourier Transform (1D) • z-Transform • 2D Fourier Transform
22© Raphael Sahann
© Laurent Condat / Torsten Möller
How to represent an image?
• An image is made of pixels (=picture elements)
• the coordinate values are discretized
23
f(t) ⇡1X
n=�1cne
j 2⇡nT t
cn =1
T
Z T/2
�T/2f(t)e�j 2⇡n
T tdt
© Torsten Möller
What is a Fourier Transform? (1D)
• Let’s go back to (spatial) representation of functions:
• Fourier series — into Frequency Domain:
24
f(t) ⇡1X
n=�1
✓Z 1
�1f(t) (t� n�T )dt
◆�(t� n�T )
f(t) ⇡1X
n=�1c[n]�(t� n�T )
• Let’s go back to (spatial) representation of functions:
• Fourier series — into Frequency Domain:
f(x) ⇡1X
n=�1cn
ej2⇡nT ·x
cn
=1
Tx
Ty
ZT/2
�T/2f(x)e�j
2⇡nT ·xdx
© Torsten Möller
What is a Fourier Transform? (2D)
25
f(x) ⇡1X
n=�1
✓Z 1
�1f(s) (s� n�X)ds
◆�(x� n�X)
f(x) ⇡1X
n=�1c[n]�(x� n�X) �X =
✓�x 00 �y
◆
© Torsten Möller
There are 4 Fourier Transforms! (1D)
• Recall Fourier series:
• f(t) is periodic with period T! • General Fourier Transform requires no
periodicity:
26
cn =1
T
Z T/2
�T/2f(t)e�j 2⇡n
T t
f(t) ⇡1X
n=�1cne
j 2⇡nT t
F (!) =
Z 1
�1f(t)e�j2⇡!tdt
f(t) =
Z 1
�1F (!)ej2⇡!tdt
© Torsten Möller
There are 4 Fourier Transforms! (2D)
• Recall Fourier series:
• f(x) is periodic with period T! • General Fourier Transform requires no
periodicity:
27
f(x) ⇡1X
n=�1cn
ej2⇡nT ·x
cn
=1
Tx
Ty
ZT/2
�T/2f(x)e�j
2⇡nT ·xdx
F (!) =
Z 1
�1f(x)e�j2⇡!·xdx
f(x) =
Z 1
�1F (!)ej2⇡!·xdx
© Torsten Möller
DFT — the most important one (1D)
• Discrete Fourier Transform (DFT) requires periodicity in both transform pairs
28
Fm =M�1X
n=0
fne�j2⇡mn/M
fn =1
M
M�1X
m=0
Fmej2⇡mn/M
© Torsten Möller
DFT — the most important one (2D)
• Discrete Fourier Transform (DFT) requires periodicity in both transform pairs
29
Fab =M�1X
m=0
N�1X
n=0
fmne�j2⇡(am/M+bn/N)
fmn =1
MN
M�1X
a=0
N�1X
b=0
Fabej2⇡(am/M+bn/N)
F (!) =
Z 1
�1f(t)e�j2⇡!tdtf(t) =
Z 1
�1F (!)ej2⇡!tdt
f(t) =1X
n=�1cne
j 2⇡nT t
Fm =M�1X
n=0
fne�j2⇡mn/Mfn =
1
M
M�1X
m=0
Fmej2⇡mn/M
fn =1
2⇡
Z ⇡
�⇡F (!)ej!nd! F (!) =
1X
n=�1fne
�j!nd!
cn =1
T
Z T/2
�T/2f(t)e�j 2⇡n
T tdt
© Torsten Möller
All Fourier Transforms (1D)
30
FT
FS —Fourier Series
DFT —Discrete FT
DTFT —Discrete Time FT
Spatial Domain Frequency Domain
continuous continuous
discretecontinuous + periodic
discrete + periodic discrete + periodic
continuous + periodicdiscrete
© Torsten Möller
All Fourier Transforms (nD)
31
FT
FS —Fourier Series
DFT —Discrete FT
DTFT —Discrete Time FT
Spatial Domain Frequency Domain
continuous continuous
discretecontinuous + periodic
discrete + periodic discrete + periodic
continuous + periodicdiscrete
© Torsten Möller
What happens to an impulse? (1D)
• it is basically a constant!
32
cn =1
T
Z T/2
�T/2�(t)e�j 2⇡n
T t
cn =1
Te0
cn =1
T
© Torsten Möller
What happens to an impulse? (2D)
• it is basically a constant!
33
cn
=1
Tx
Ty
ZT/2
�T/2�(x)e�j
2⇡nT ·xdx
cn
=1
Tx
Ty
e0
cn
=1
Tx
Ty
© Torsten Möller
What about a shifted impulse? (1D)
• the shifts remain as frequencies
34
cn =1
T
Z T/2
�T/2�(t� t0)e
�j 2⇡nT t
cn =1
Te�j 2⇡n
T t0
© Torsten Möller
What about a shifted impulse? (2D)
• the shifts remain as frequencies
35
cn
=1
Tx
Ty
ZT/2
�T/2�(x� x
0
)e�j
2⇡nT ·xdx
cn
=1
Tx
Ty
e�j
2⇡nT ·x0
© Torsten Möller
What happens to an impulse train?• Impulse train is periodic — apply Fourier
series, will not do the math here, see book:
• distance between impulses grows inversely
36
s�T (t) =1X
�1�(t� n�T )
S�T (!) =1
�T
1X
n=�1�(! � n
�T)
• Impulse train is periodic — apply Fourier series, will not do the math here, see book:
• distance between impulses grows inversely
© Torsten Möller
What happens to an impulse train?
37
S�X(!) =1
|�X|
1X
n=�1�(! ��X�1n)
s�X(x) =1X
n=�1�(x��Xn)
�X =
✓�x 00 �y
◆
© Torsten Möller
What happens to a box?
• it is the well-known sinc function
38
sinc(t) =sin⇡t
⇡t
© Torsten Möller
What happens to a box?
• it is the well-known sinc function
39
sinc(Tt, Zz) =sin(T⇡t)
⇡t
sin(Z⇡z)
⇡z
© Torsten Möller
What is the Fourier Transform of a convolution?
• convolution <==> multiplication:
• multiplication <==> convolution:
40
f ⇤ h(x) () F (!)H(!)
f(x)h(x) () F (!) ⇤H(!)
© Torsten Möller
What is sampling in 2D? (mathematically speaking)
41
�(x, y) =
⇢1 if x = y = 00 if t 6= 0
Z 1
�1
Z 1
�1�(x, y)dxdy = 1
• modeled through an ‘impulse’
• not really a function, but a distribution:
© Torsten Möller
The sifting property
• picking a value off from f:
• more general:
42
Z 1
�1f(x)�(x)dt = f(0)
Z 1
�1f(x)�(x� x0)dx = f(x0)
f(t)⇥ s�T (t) =
1X
�1f(n�T )�(t� n�T )
1X
�1f [n]�(t� n�T )
© Torsten Möller
What is sampling?
43
© Torsten Möller
The 2D impulse train
• pick up multiple values of f at once:
44
1
�Tx
> 2⌫max
1
�Ty> 2µ
max
© Torsten Möller
Sampling in the Fourier Domain (2D)
45
© Torsten Möller
Aliasing
46
© Torsten Möller
Aliasing
47
© Torsten Möller
Moire patterns
48
© Torsten Möller
Moire patterns
49
50© Raphael Sahann
Moire patterns
© Laurent Condat / Torsten Möller
Periodic patterns
51
© Torsten Möller 52
Fourier Spectrum
© Torsten Möller
Moire patterns
53