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![Page 1: 1 Lecture 1 Sampling of Signals by Graham C. Goodwin University of Newcastle Australia Lecture 1 Presented at the “Zaborszky Distinguished Lecture Series”](https://reader035.fdocuments.us/reader035/viewer/2022081519/56649da05503460f94a8c2ef/html5/thumbnails/1.jpg)
1
Lecture 1Sampling of Signals
by
Graham C. GoodwinUniversity of Newcastle
Australia
Lecture 1Presented at the “Zaborszky Distinguished Lecture Series”
December 3rd, 4th and 5th, 2007
![Page 2: 1 Lecture 1 Sampling of Signals by Graham C. Goodwin University of Newcastle Australia Lecture 1 Presented at the “Zaborszky Distinguished Lecture Series”](https://reader035.fdocuments.us/reader035/viewer/2022081519/56649da05503460f94a8c2ef/html5/thumbnails/2.jpg)
2
Recall Basic Idea of Samplingand Quantization
Quantization
Sampling
t1 t3t2 t4
t0
123456
![Page 3: 1 Lecture 1 Sampling of Signals by Graham C. Goodwin University of Newcastle Australia Lecture 1 Presented at the “Zaborszky Distinguished Lecture Series”](https://reader035.fdocuments.us/reader035/viewer/2022081519/56649da05503460f94a8c2ef/html5/thumbnails/3.jpg)
3
In this lecture we will ignore quantization issues and focus on the impact of different sampling patterns for scalar and multidimensional signals
![Page 4: 1 Lecture 1 Sampling of Signals by Graham C. Goodwin University of Newcastle Australia Lecture 1 Presented at the “Zaborszky Distinguished Lecture Series”](https://reader035.fdocuments.us/reader035/viewer/2022081519/56649da05503460f94a8c2ef/html5/thumbnails/4.jpg)
4
Outline1. One Dimensional Sampling2. Multidimensional Sampling3. Sampling and Reciprocal Lattices4. Undersampled Signals5. Filter Banks6. Generalized Sampling Expansion (GSE)7. Recurrent Sampling8. Application: Video Compression at Source9. Conclusions
![Page 5: 1 Lecture 1 Sampling of Signals by Graham C. Goodwin University of Newcastle Australia Lecture 1 Presented at the “Zaborszky Distinguished Lecture Series”](https://reader035.fdocuments.us/reader035/viewer/2022081519/56649da05503460f94a8c2ef/html5/thumbnails/5.jpg)
5
Sampling: Assume amplitude quantization sufficiently fine to be negligible.
Question: Say we are given
Under what conditions can we recover
from the samples?
( );f t t Î ¡
( ) ;if t i ZÎ
![Page 6: 1 Lecture 1 Sampling of Signals by Graham C. Goodwin University of Newcastle Australia Lecture 1 Presented at the “Zaborszky Distinguished Lecture Series”](https://reader035.fdocuments.us/reader035/viewer/2022081519/56649da05503460f94a8c2ef/html5/thumbnails/6.jpg)
6
A Well Known Result (Shannon’s Reconstruction Theorem for Uniform Sampling)
Consider a scalar signal f(t) consisting of
frequency components in the range . If
this signal is sampled at period , then the
signal can be perfectly reconstructed from the
samples using:
[ ]( )
( )
sin2
( )
2
s
sk
t k
y t y kt k
w
w
¥
=- ¥
é ùæ ö÷çê ú- D÷ç ÷çê úè øë û=æ ö÷ç - D÷ç ÷çè ø
å
,2 2s sw wæ ö- ÷ç ÷÷çè ø
2s
pwD <
![Page 7: 1 Lecture 1 Sampling of Signals by Graham C. Goodwin University of Newcastle Australia Lecture 1 Presented at the “Zaborszky Distinguished Lecture Series”](https://reader035.fdocuments.us/reader035/viewer/2022081519/56649da05503460f94a8c2ef/html5/thumbnails/7.jpg)
7
Low pass filter recovers original spectrum
Hence
or
( )sY w
2sw-
2sw
sw
( ) ( ) ( )
( ) 12 2
0 otherwise
s s
s ss
Y H Y
H
w w w
w ww w
=
æ ö- ÷ç= £ £ ÷ç ÷çè ø
=( ) ( ) ( )
( ) [ ] ( )
[ ] ( )
ss
sk
sk
y t h y t d
h y k t k d
y k h t k
s s s
s d s s
¥
- ¥
¥¥
- ¥=- ¥
¥
=- ¥
= -
= - - D
= - D
ò
åò
å
Proof: Sampling produces folding
![Page 8: 1 Lecture 1 Sampling of Signals by Graham C. Goodwin University of Newcastle Australia Lecture 1 Presented at the “Zaborszky Distinguished Lecture Series”](https://reader035.fdocuments.us/reader035/viewer/2022081519/56649da05503460f94a8c2ef/html5/thumbnails/8.jpg)
8
A Simple (but surprising) Extension
where
[Recurrent Sampling]
is a periodic sequence of integers; i.e.,
Let
Note that the average sampling period is
e.g.
average 5
k kMD = D
{ }kM k N kM M+ =
1
N
kk
M K=
=å
T K= DK
N
D=D
1
2
3
4
9
1
9
1
D =
D =
D =
D =
![Page 9: 1 Lecture 1 Sampling of Signals by Graham C. Goodwin University of Newcastle Australia Lecture 1 Presented at the “Zaborszky Distinguished Lecture Series”](https://reader035.fdocuments.us/reader035/viewer/2022081519/56649da05503460f94a8c2ef/html5/thumbnails/9.jpg)
9
Non-uniform
Uniform
0 9-1 10 19 20
x x x xxx
0 5 10 15 20
x x x xx
![Page 10: 1 Lecture 1 Sampling of Signals by Graham C. Goodwin University of Newcastle Australia Lecture 1 Presented at the “Zaborszky Distinguished Lecture Series”](https://reader035.fdocuments.us/reader035/viewer/2022081519/56649da05503460f94a8c2ef/html5/thumbnails/10.jpg)
10
Claim:
Provided the signal is bandlimited to
where , then the signal can be
perfectly reconstructed from the periodic
sampling pattern.
where = average sampling period
Proof:We will defer the proof to later when we will use it as an illustration of Generalized Sampling Expansion (GSE) Theorem.
,2 2s sw wæ ö- ÷ç ÷÷çè ø
2s
pw =D
D
![Page 11: 1 Lecture 1 Sampling of Signals by Graham C. Goodwin University of Newcastle Australia Lecture 1 Presented at the “Zaborszky Distinguished Lecture Series”](https://reader035.fdocuments.us/reader035/viewer/2022081519/56649da05503460f94a8c2ef/html5/thumbnails/11.jpg)
11
Outline1. One Dimensional Sampling2. Multidimensional Sampling3. Sampling and Reciprocal Lattices4. Undersampled Signals5. Filter Banks6. Generalized Sampling Expansion (GSE)7. Recurrent Sampling8. Application: Video Compression at Source9. Conclusions
![Page 12: 1 Lecture 1 Sampling of Signals by Graham C. Goodwin University of Newcastle Australia Lecture 1 Presented at the “Zaborszky Distinguished Lecture Series”](https://reader035.fdocuments.us/reader035/viewer/2022081519/56649da05503460f94a8c2ef/html5/thumbnails/12.jpg)
12
Multidimensional SignalsDigital Photography
Digital Video
x1
x2
x1
x2
x3 (time)
![Page 13: 1 Lecture 1 Sampling of Signals by Graham C. Goodwin University of Newcastle Australia Lecture 1 Presented at the “Zaborszky Distinguished Lecture Series”](https://reader035.fdocuments.us/reader035/viewer/2022081519/56649da05503460f94a8c2ef/html5/thumbnails/13.jpg)
13
Outline1. One Dimensional Sampling2. Multidimensional Sampling3. Sampling and Reciprocal Lattices4. Undersampled Signals5. Filter Banks6. Generalized Sampling Expansion (GSE)7. Recurrent Sampling8. Application: Video Compression at Source9. Conclusions
![Page 14: 1 Lecture 1 Sampling of Signals by Graham C. Goodwin University of Newcastle Australia Lecture 1 Presented at the “Zaborszky Distinguished Lecture Series”](https://reader035.fdocuments.us/reader035/viewer/2022081519/56649da05503460f94a8c2ef/html5/thumbnails/14.jpg)
14
How should we define sampling for multi-dimensional signals?
Utilize idea of Sampling Lattice
Sampling Lattice
nonsingular matrix D DT Î ´¡ ¡
( ) { }: DLat T Tn n ZL = = Î
![Page 15: 1 Lecture 1 Sampling of Signals by Graham C. Goodwin University of Newcastle Australia Lecture 1 Presented at the “Zaborszky Distinguished Lecture Series”](https://reader035.fdocuments.us/reader035/viewer/2022081519/56649da05503460f94a8c2ef/html5/thumbnails/15.jpg)
15
Also, need multivariable frequency domainconcepts.
These are captured by two ideas
i. Reciprocal Latticeii. Unit Cell
![Page 16: 1 Lecture 1 Sampling of Signals by Graham C. Goodwin University of Newcastle Australia Lecture 1 Presented at the “Zaborszky Distinguished Lecture Series”](https://reader035.fdocuments.us/reader035/viewer/2022081519/56649da05503460f94a8c2ef/html5/thumbnails/16.jpg)
16
Unit Cell (Non-unique)
i.
ii.
Reciprocal Lattice
( ){ } ( ){ }1 1* 2 2 :T T DLat T T n n Zp p- -
L = = Î
( ) ( ){ } ( ) ( ){ }1 1* *1 2
1 2 1 2
2 2
,
T T
D
UC T n UC T n
n n Z n n
p p- -
L + Ç L + =Æ
Î ¹
( ) ( ){ }12
D
T D
n Z
UC T n Rp-
Î
L + =U
( )*UC L
![Page 17: 1 Lecture 1 Sampling of Signals by Graham C. Goodwin University of Newcastle Australia Lecture 1 Presented at the “Zaborszky Distinguished Lecture Series”](https://reader035.fdocuments.us/reader035/viewer/2022081519/56649da05503460f94a8c2ef/html5/thumbnails/17.jpg)
17
One Dimensional Example
Sampling Lattice
0-20 10 20
x x xx
D
{ }. :n n ZL = D Î
-10
x
![Page 18: 1 Lecture 1 Sampling of Signals by Graham C. Goodwin University of Newcastle Australia Lecture 1 Presented at the “Zaborszky Distinguished Lecture Series”](https://reader035.fdocuments.us/reader035/viewer/2022081519/56649da05503460f94a8c2ef/html5/thumbnails/18.jpg)
18
Reciprocal Lattice and Unit Cell
Unit Cell
12
wp0 1
102
103
10
![Page 19: 1 Lecture 1 Sampling of Signals by Graham C. Goodwin University of Newcastle Australia Lecture 1 Presented at the “Zaborszky Distinguished Lecture Series”](https://reader035.fdocuments.us/reader035/viewer/2022081519/56649da05503460f94a8c2ef/html5/thumbnails/19.jpg)
19
Multidimensional Example
x1
x2
1 2 3 4 5-4 -3 -2 -1-1
-2
-3
-4
5
4
3
2
1
2 1
0 2T
é ùê ú=ê úë û
![Page 20: 1 Lecture 1 Sampling of Signals by Graham C. Goodwin University of Newcastle Australia Lecture 1 Presented at the “Zaborszky Distinguished Lecture Series”](https://reader035.fdocuments.us/reader035/viewer/2022081519/56649da05503460f94a8c2ef/html5/thumbnails/20.jpg)
20
Reciprocal Lattice and Unit Cell for Example
1/4 1/2 3/4 1-1/4
-1/2
-3/4
-1
1/2
1/4
( )1
10
21 1
4 2
TT-
é ùê úê ú= ê úê ú-ê úë û
( )( )12 TUC Tp
-
1
2
wp
2
2
wp
![Page 21: 1 Lecture 1 Sampling of Signals by Graham C. Goodwin University of Newcastle Australia Lecture 1 Presented at the “Zaborszky Distinguished Lecture Series”](https://reader035.fdocuments.us/reader035/viewer/2022081519/56649da05503460f94a8c2ef/html5/thumbnails/21.jpg)
21
Outline1. One Dimensional Sampling2. Multidimensional Sampling3. Sampling and Reciprocal Lattices4. Undersampled Signals5. Filter Banks6. Generalized Sampling Expansion (GSE)7. Recurrent Sampling8. Application: Video Compression at Source9. Conclusions
![Page 22: 1 Lecture 1 Sampling of Signals by Graham C. Goodwin University of Newcastle Australia Lecture 1 Presented at the “Zaborszky Distinguished Lecture Series”](https://reader035.fdocuments.us/reader035/viewer/2022081519/56649da05503460f94a8c2ef/html5/thumbnails/22.jpg)
22
We will be interested here in the situation where the Sampling Lattice is not a Nyquist Lattice for the signal (i.e., the signal cannot be perfectly reconstructed from the original pattern!)
Strategy: We will generate other samples by ‘filtering’or ‘shifting’ operations on the original pattern.
![Page 23: 1 Lecture 1 Sampling of Signals by Graham C. Goodwin University of Newcastle Australia Lecture 1 Presented at the “Zaborszky Distinguished Lecture Series”](https://reader035.fdocuments.us/reader035/viewer/2022081519/56649da05503460f94a8c2ef/html5/thumbnails/23.jpg)
23
Consider a bandlimited signal .Assume the D-dimension Fourier transform has finite support, S.
Then for given D-dimensional lattice T, there always exists a finite set , such that support
( ), Df x x Î ¡
{ } *
1
P
iw Î L
( )( ) ( )( )*
1
ˆ .P
ii
f S UCw w=
Í = L +U
Heuristically: The idea of “Tiling” the area of interest in the frequency domain
![Page 24: 1 Lecture 1 Sampling of Signals by Graham C. Goodwin University of Newcastle Australia Lecture 1 Presented at the “Zaborszky Distinguished Lecture Series”](https://reader035.fdocuments.us/reader035/viewer/2022081519/56649da05503460f94a8c2ef/html5/thumbnails/24.jpg)
24
One Dimensional ExampleOur one dimensional example continued.Sampling Lattice { };k k ZL = D Î
Unit Cell
12
wp0 1
102
103
10( )f w
Bandlimited spectrum
Use1
2
0
2
10
w
pw
=
æ ö÷ç=- ÷ç ÷çè ø
( )( ) ( ) ( )* *2f UC UCw wé ù é ù= L L +ê ú ê úë û ë ûUSupport
112
112
- 2w
p
![Page 25: 1 Lecture 1 Sampling of Signals by Graham C. Goodwin University of Newcastle Australia Lecture 1 Presented at the “Zaborszky Distinguished Lecture Series”](https://reader035.fdocuments.us/reader035/viewer/2022081519/56649da05503460f94a8c2ef/html5/thumbnails/25.jpg)
25
Outline1. One Dimensional Sampling2. Multidimensional Sampling3. Sampling and Reciprocal Lattices4. Undersampled Signals5. Filter Banks6. Generalized Sampling Expansion (GSE)7. Recurrent Sampling8. Application: Video Compression at Source9. Conclusions
![Page 26: 1 Lecture 1 Sampling of Signals by Graham C. Goodwin University of Newcastle Australia Lecture 1 Presented at the “Zaborszky Distinguished Lecture Series”](https://reader035.fdocuments.us/reader035/viewer/2022081519/56649da05503460f94a8c2ef/html5/thumbnails/26.jpg)
26
Generation of Extra SamplesSuppose now we generate a data set as shown in below( ){ }{ }1 D
Q
q q n Zg Tn
= Î
( )Q P³
Q – Channel Filter Bank
( )f x
( )1h w
( )ˆqh w
( )ˆQh w
L
L
L
( )1g x
( )qg x
( )Qg x
M
M
![Page 27: 1 Lecture 1 Sampling of Signals by Graham C. Goodwin University of Newcastle Australia Lecture 1 Presented at the “Zaborszky Distinguished Lecture Series”](https://reader035.fdocuments.us/reader035/viewer/2022081519/56649da05503460f94a8c2ef/html5/thumbnails/27.jpg)
27
Outline1. One Dimensional Sampling2. Multidimensional Sampling3. Sampling and Reciprocal Lattices4. Undersampled Signals5. Filter Banks6. Generalized Sampling Expansion (GSE)7. Recurrent Sampling8. Application: Video Compression at Source9. Conclusions
![Page 28: 1 Lecture 1 Sampling of Signals by Graham C. Goodwin University of Newcastle Australia Lecture 1 Presented at the “Zaborszky Distinguished Lecture Series”](https://reader035.fdocuments.us/reader035/viewer/2022081519/56649da05503460f94a8c2ef/html5/thumbnails/28.jpg)
28
Define
Let
be the solution (if it exists) of
for
( ) ( ) ( )
( )
( ) ( )
1 1 2 1 1
1 2
1
ˆ ˆ ˆ
ˆ( )
ˆ ˆ
Q
P Q P
h h h
hH
h h
w w w w w w
w ww
w w w w
é ù+ + +ê úê ú
+ê ú= ê úê úê úê ú+ +ë û
L
M
( )( )
( )
1 ,
,
,Q
x
x
x
w
w
w
é ùFê úê úF = ê úê úFê úë û
M
( )*UCwÎ L
( ) ( )
1
,
T
TP
j x
j x
e
H x
e
w
w
w w
é ùê úê úF = ê úê úê úë û
M
![Page 29: 1 Lecture 1 Sampling of Signals by Graham C. Goodwin University of Newcastle Australia Lecture 1 Presented at the “Zaborszky Distinguished Lecture Series”](https://reader035.fdocuments.us/reader035/viewer/2022081519/56649da05503460f94a8c2ef/html5/thumbnails/29.jpg)
29
Conditions for Perfect Reconstruction
can be reconstructed from
if and only if has full row rank for all in the Unit Cell
where
( )H w
( )f x
( ) ( ) ( )1 D
Q
q qq k Z
f x g Tk x Tkf= Î
= -å å
( ) ( ),Tj x
q q
UC
x x e dwf w w= Fò
GSE Theorem:
w
![Page 30: 1 Lecture 1 Sampling of Signals by Graham C. Goodwin University of Newcastle Australia Lecture 1 Presented at the “Zaborszky Distinguished Lecture Series”](https://reader035.fdocuments.us/reader035/viewer/2022081519/56649da05503460f94a8c2ef/html5/thumbnails/30.jpg)
30
Proof:
Multiply both sides by where (theReciprocal Lattice). Then sum over ‘q’
Note that “tiles” the entiresupport S
Thus,
( ) ( ) ( )*, ;T T
D
j x j Tkq q
k Z
x e x Tk e UCw ww f w-
Î
F = - Î Lå
( )ˆq ih w w+
( ) ( ) ( ) ( )
( )
1 1
ˆ ˆ ,T T
D
Ti
Q Qj Tk j x
q i q q i qq qk Z
j x
h x Tk e h x e
e
w w
w w
w w f w w w-
= =Î
+
+ - = + F
=
å å å
from the Matrix identity that defines
( ) ( ) ( ) ( )
( )
( ) ( ) ( )( )
*
*
1
1 1
ˆ ˆ
ˆ ˆ
TTi
T
D
Pj xj x
iis UC
QPj Tk
i q i qi q k ZUC
f x f e d f e d
f h x Tk e d
w ww
w
w w w w w
w w w w f w
- +
= L
-
= = ÎL
= = +
= + + -
åò ò
å å åò
*iw Î L
( ), xwF
{ }*; and 1, ,i i Pw w w+ Î L = K
![Page 31: 1 Lecture 1 Sampling of Signals by Graham C. Goodwin University of Newcastle Australia Lecture 1 Presented at the “Zaborszky Distinguished Lecture Series”](https://reader035.fdocuments.us/reader035/viewer/2022081519/56649da05503460f94a8c2ef/html5/thumbnails/31.jpg)
31
where we have used the fact that
Since is the output of f(x) passing through ,then
Hence, we finally have
( ) ( ) ( ) ( )
( )( )
*11
ˆ ˆ Ti
D
Pj Tk
i q ii UC
Q
qq k Z
f x x Tkf h e dw ww w w fw w- +
== Î L
é ùê ú
= -ê úê úê úë û
+ +å å òå
( )1
2 for .T Di T Zw p
-= Îl l
( )qg x
[ ] ( )qg Tk=
( ) ( ) ( )1 D
Q
q qq k Z
f x g Tk x Tkf= Î
= -å å
( )ˆqh w
ÑÑÑ
![Page 32: 1 Lecture 1 Sampling of Signals by Graham C. Goodwin University of Newcastle Australia Lecture 1 Presented at the “Zaborszky Distinguished Lecture Series”](https://reader035.fdocuments.us/reader035/viewer/2022081519/56649da05503460f94a8c2ef/html5/thumbnails/32.jpg)
32
Outline1. One Dimensional Sampling2. Multidimensional Sampling3. Sampling and Reciprocal Lattices4. Undersampled Signals5. Filter Banks6. Generalized Sampling Expansion (GSE)7. Recurrent Sampling8. Application: Video Compression at Source9. Conclusions
![Page 33: 1 Lecture 1 Sampling of Signals by Graham C. Goodwin University of Newcastle Australia Lecture 1 Presented at the “Zaborszky Distinguished Lecture Series”](https://reader035.fdocuments.us/reader035/viewer/2022081519/56649da05503460f94a8c2ef/html5/thumbnails/33.jpg)
33
Special Case: Recurrent Sampling
(where is implemented by a “spatial” shift )This amounts to the sampling pattern:
where w.l.o.g.
Now, given the samples , our goal is to
perfectly reconstruct
( ){ }1
Q
Lat T x=
Y = +I
{ } ( )qx UC TÎ
( ){ }x
f xÎ Y%
%
( ).f x
qh qx
![Page 34: 1 Lecture 1 Sampling of Signals by Graham C. Goodwin University of Newcastle Australia Lecture 1 Presented at the “Zaborszky Distinguished Lecture Series”](https://reader035.fdocuments.us/reader035/viewer/2022081519/56649da05503460f94a8c2ef/html5/thumbnails/34.jpg)
34
Here , and
Thus
To apply the GSE Theorem we require
( )ˆ Tqj x
qh e ww = ( ) ( )q qg x f x x= +
( ){ } ( ){ }, 1,Pq xn Z q Q
g Tn f xÎ YÎ =
=%K
%
Nonsingular
( )( ) ( )
( ) ( )
11 1
1
11 1 1
1
0
0
TTQ
TTP QP
TT TQ
TTT QP QP
j xj x
j xj x
j xj x j x
j xj xj x
e eH
e e
e e e
ee e
w ww w
w ww w
ww w
www
w++
++
é ùê ú= ê úê úë ûé ùé ùê úê ú= ê úê úê úê úë ûë û
L
L
LO
L
![Page 35: 1 Lecture 1 Sampling of Signals by Graham C. Goodwin University of Newcastle Australia Lecture 1 Presented at the “Zaborszky Distinguished Lecture Series”](https://reader035.fdocuments.us/reader035/viewer/2022081519/56649da05503460f94a8c2ef/html5/thumbnails/35.jpg)
35
Something to think aboutThe GSE result depends on inversion of a
particular matrix, H(w). Of course we have assumed here perfect representation of all coefficients. An interesting question is what happens when the representation is imperfect i.e. coefficients are represented with finite wordlength (i.e. they are quantized)
We will not address this here but it is something to keep in mind.
![Page 36: 1 Lecture 1 Sampling of Signals by Graham C. Goodwin University of Newcastle Australia Lecture 1 Presented at the “Zaborszky Distinguished Lecture Series”](https://reader035.fdocuments.us/reader035/viewer/2022081519/56649da05503460f94a8c2ef/html5/thumbnails/36.jpg)
36
Return to our one-dimensional exampleRecall that we had
so that
support
Say we use recurrent sampling with
1
2
0
2
10
w
pw
=
=-
1
2
0
0.9 ; 10
x
x
=
= D D =
( )( ) ( ) ( )* *2f UC UCw wé ù é ù= L È L +ê ú ê úë û ë û
![Page 37: 1 Lecture 1 Sampling of Signals by Graham C. Goodwin University of Newcastle Australia Lecture 1 Presented at the “Zaborszky Distinguished Lecture Series”](https://reader035.fdocuments.us/reader035/viewer/2022081519/56649da05503460f94a8c2ef/html5/thumbnails/37.jpg)
37
0 10 20
x xx
0 9 19
x xx
0
x xx
1 0x =
2 0.9x = D
-1
-1
x x
19 20
x
9 10
![Page 38: 1 Lecture 1 Sampling of Signals by Graham C. Goodwin University of Newcastle Australia Lecture 1 Presented at the “Zaborszky Distinguished Lecture Series”](https://reader035.fdocuments.us/reader035/viewer/2022081519/56649da05503460f94a8c2ef/html5/thumbnails/38.jpg)
38
Condition for Perfect Reconstruction is
nonsingular
( )( )
1 1 1 2
2 1 2 2
0.9 2
1 1
1
j x j x
j x j x
j
e e
e e
e
w w
w w
p-
é ùê úê úë ûé ùê ú= ê úë û
Hence, the original signal can be recovered from the sampling pattern given in the previous slide.
![Page 39: 1 Lecture 1 Sampling of Signals by Graham C. Goodwin University of Newcastle Australia Lecture 1 Presented at the “Zaborszky Distinguished Lecture Series”](https://reader035.fdocuments.us/reader035/viewer/2022081519/56649da05503460f94a8c2ef/html5/thumbnails/39.jpg)
39
Summary We have seen that the well known
Shannon reconstruction theorem can be extended in several directions; e.g. Multidimensional signals
Sampling on a lattice
Recurrent sampling
Given specific frequency domain distributions, these can be matched to appropriate sampling patterns.
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Outline1. One Dimensional Sampling2. Multidimensional Sampling3. Sampling and Reciprocal Lattices4. Undersampled Signals5. Filter Banks6. Generalized Sampling Expansion (GSE)7. Recurrent Sampling8. Application: Video Compression at Source9. Conclusions
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Application: Video Compression SourceIntroduction to video cameras Instead of tape, digital cameras use 2D sensor
array (CCD or CMOS)
ImageProcessor
ImageProcessor
Memory
ImageProcessor
ImageProcessor
ImageProcessor
ImageProcessing Display
( TV or LCD )Pipeline
DVCDcontroller
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Image Sensor
A 2D array of sensors replaces the traditional tape
Each sensor records a 'point' of the continuous image
The whole array records the continuous image at a particular time instant
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2D Colours Sensor Array
Data transfer from array is sequentialand has a maximal rate of Q.
* Based on http://www.dpreview.com/learn/
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Uniform 3D sampling
a sequence of identical frames equally spaced in time
Current Technology
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The volume of ‘box’ depends on the capacity: pixel rate = (frame rate) x (spatial resolution)
xx
Video Bandwidth
depends on spatial resolutionof the frames
depends on the frame rate
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1. Data recording on sensor:
• Sensor array density - for spatial resolution
pixels
frame R
• Sensor exposure time - for frame rate
frames
sec. F
2. Data reading from sensor:
• Data readout time - for pixel rate
pixels
sec. Q
Hard Constraints
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Generally Q << RFNeed: R1< RF1 < F
s.t. R1F1 = Q
Compromise: spatial resolution R1< R
temporal resolution F1 < F
BUT...
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x
yt
1R 1F
volume determined by 1 1Q R F
Actual Capacity (Data Readout)
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Observation
Most energy of typical video scene is concentrated around the plane and the axis.
,x y
t
t x
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t x
uniform sampling - compromisein frame rate
uniform sampling - compromisein spatial resolution
uniform sampling- no compromise
The Spectrum of this Video Clip
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y
x
2 1N y
21
Lx
frame type A frame type B
t
t 22 1M t
12 1M t
Recurrent Non-Uniform Sampling
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yt
x
What Does it Buy?
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Schematic Implementation
t
Filter bank t t
non-uniform data from the sensor
uniform high def. video
'compression at the source'
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Recurrent Non-Uniform Sampling
A special case of
Generalized Sampling Expansion Theorem
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Sampling Pattern
tmULAT
xlULATULAT
M
Mm
L
l
0)(
0)()(
1
2
2
12
2
1
sML
s
xULAT
)(1)(2
1
The resulting sampling pattern is given by:
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Frequency Domain
rTML
r
UUCS
)2(
1)(2
1
where:
tMxLUUC tx
t
xT
)12(,
)12(:)2(
1
is the unit cell of the reciprocal lattice
2
1
:
)12(0
0)12(
)2( Znn
tM
xLULAT T
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Reciprocal Lattice
x
x
tM
M
)12(
12
1
tM
M
)12(
12
1
x
t
tM )12( 1
xL )12(
Unit cell
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Apply the GSE Theorem
)()(
1)(2
2
1
xH
ML
where: is uniquely defined by H1…H2(…) is a set of 2(L+M)+1 constraints
)()(),( 1 xHx If exists, we can find the reconstruction function 1H
( )H w
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Reconstruction Scheme
H1 1
H2L+1 2L+1
H2(L+M)+1 2(L+M)+1
I(x,t) Î(x,t)
1M)2(L
frequencyNyquist
r
rTr t
xj
sr eH
.
The sub-sampled frequency of each filter H is:
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Reconstruction functions
t
t1)(2Msin
))1((
x)1)-(r(sin)12(),(
xrx
xx
txLtxr
))12((
t)1)-2L-(r(sin
x
x1)(2Lsin
)12(),(tLrt
tt
txMtxr
for r = 2,3,…,2L+1
for r = 2(L+1),…,2(L+M)+1
Multidimensional ‘sinc like’ functions
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Demo
Full resolutionsequence
Reconstructedsequence
Temporaldecimation
Spacialdecimation
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Outline1. One Dimensional Sampling2. Multidimensional Sampling3. Sampling and Reciprocal Lattices4. Undersampled Signals5. Filter Banks6. Generalized Sampling Expansion (GSE)7. Recurrent Sampling8. Application: Video Compression at Source9. Conclusions
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Conclusions Nonuniform sampling of scalar signals
Nonuniform sampling of multidimensional signals
Generalized sampling expansion
Application to video compression
A remaining problem is that of joint design of sampling schemes and quantization strategies to minimize error for a given bit rate
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References One Dimensional Sampling
A. Feuer and G.C. Goodwin, Sampling in Digital Signal Processing and Control. Birkhäuser, 1996. R.J. Marks II, Ed., Advanced Topics in Shannon Sampling and Interpolation Theory. New Your: Springer-Verlag, 1993.
Multidimensional Sampling W.K. Pratt, Digital Image Processing, 3rd ed: John Wiley & Sons, 2001. B.L. Evans, “Designing commutative cascades of multidimensional upsamplers and downsamplers,” IEEE Signal Process Letters, Vol4, No.11, pp.313-316, 1997.
Sampling and Reciprocal Lattices, Undersampled Signals A.Feuer, G.C. Goodwin, ‘Reconstruction of Multidimensional Bandlimited Signals for Uniform and Generalized Samples,’ IEEE Transactions on Signal Processing, Vol.53, No.11, 2005. A.K. Jain, Fundamentals of Digital Image Processing, Englewood Cliffs, NJ: Prentice-Hall, 1989.
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References Filter Banks
Y.C. Eldar and A.V. Oppenheim, ‘Filterbank reconstruction of bandlimited signals from nonuniform and generalized samples,’ IEEE Transactions on Signal Processing, Vol.48, No.10, pp.2864-2875, 2000. P.P. Vaidyanathan, Multirate Systems and Filter Banks. Englewood Cliffs, NJ: Prentice-Hall, 1993. H. Bölceskei, F. Hlawatsch and H.G. Feichtinger, ‘Frame-theoretic analysis of oversampled filter banks,’ IEEE Transactions on Signal Processing, Vol.46, No.12, pp.3256-3268, 1998. M. Vetterli and J. Kovaĉević, Wavelets and Subband Coding, Englewood Cliffs, NJ: Prentice Hall, 1995.
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References Generalized Sampling Expansions, Recurrent Sampling
A. Papoulis, ‘Generalized sampling expansion,’ IEEE Transaction on Circuits and Systems, Vol.CAS-24, No.11, pp.652-654, 1977. A. Feuer, ‘On the necessity of Papoulis result for multidimensional (GSE),’ IEEE Signal Processing Letters, Vol.11, No.4, pp.420-422, 2004. K.F.Cheung, ‘A multidimensional extension of Papoulis’ generalized sampling expansion with application in minimum density sampling,’ in Advanced Topics in Shannon Sampling and Interpolation Theory, R.J. Marks II. Ed., New York: Springer-Verlag, pp.86-119, 1993.
Video Compression at Source E. Shechtman, Y. Caspi and M. Irani, ‘Increasing space-time resolution in video’, European Conference on Computer Vision (ECCV), 2002. N. Maor, A. Feuer and G.C. Goodwin, ‘Compression at the source of digital video,’ To appear EURASIP Journal on Applied Signal Processing.
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Lecture 1Sampling of Signals
by
Graham C. GoodwinUniversity of Newcastle
Australia
Lecture 1Presented at the “Zaborszky Distinguished Lecture Series”
December 3rd, 4th and 5th, 2007