Goodwin Lecture1
Transcript of Goodwin Lecture1
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Lecture 1Sampling of Signals
by
Graham C. GoodwinUniversity of Newcastle
Australia
Lecture 1
Presented at the “Zaborszky Distinguished Lecture Series”
December 3rd, 4th and 5th, 2007
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Recall Basic Idea ofSamplingand Quantization
Quantization
Sampling
t 1 t 3t 2 t 4t
0
1l2l3l4l
5l6l
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In this lecture we will ignore quantizationissues and focus on the impact of differentsampling patterns for scalar and
multidimensional signals
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Outline1. One Dimensional Sampling
2. Multidimensional Sampling
3. Sampling and Reciprocal attices
!. "ndersampled Signals#. $ilter %an&s
'. (eneralized Sampling )*pansion +(S),
-. Recurrent Sampling
. /pplication0 ideo ompression at Source
. onclusions
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Sampling: /ssume amplitude quantizationsufficientl4 fine to 5e negligi5le.
Question: Sa4 we are gi6en
"nder what conditions can we reco6er
from the samples7
( ) ; f t t Î ¡
( ) ;i f t i Z Î
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A Well Known Result (Shannon’sReconstruction heorem for
!niform Sampling"onsider a scalar signal f (t ) consisting of
frequenc4 components in the range . If
this signal is sampled at period 8 then the
signal can 5e perfectl4 reconstructed from the
samples using0
[ ]( )
( )
sin2
( )
2
s
sk
t k
y t y k
t k
w
w
¥
=- ¥
éæ ö÷çê - D÷ç ÷çêè øë=æ ö÷ç - D÷ç ÷çè ø
å
,2 2
s sw wæ-ççè2
s
pwD <
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Low pass filter recoversoriginal spectrum Hence
or
( ) sY w
2
sw-
2
sw
sw
( ) ( ) ( )
( ) 12 2
0 otherwise
s s
s s s
Y H Y
H
w w ww w
w w
=æ-ç= £ £ççè
=( ) ( ) ( )
( ) [ ] ( )
[ ] ( )
s
s
s
k
s
k
y t h y t d
h y k t k d
y k h t k
s s s
s d s s
¥
- ¥¥¥
- ¥=- ¥
¥
=- ¥
= -
= - - D
= - D
òåò
å
Proof: Sampling produces folding
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A Simple (#ut surprising"$%tension
where
[Recurrent Sampling]
is a #eriodic se$uence o% integers& i'e'(
Let
)ote that the average sa*#+ing #eriod is
e.g.
a,erage 5
k k M D = D
{ }k M k N k M M + =
1
N
k
k
M K =
=å
T K = D K
N D =D
1
2
3
4
9
1
9
1
D =
D =
D =
D =
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Non-uniform
Uniform
0 -.1 10 1- 20
/ / / ///
0 5 10 15 20
/ / / //
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Claim:
9ro6ided the signal is 5andlimited towhere 8 then the signal can 5e
perfectl4 reconstructed from the periodic
sampling pattern.
where : a6erage sampling period
Proof:
;e will defer the proof to later when wewill use it as an illustration of (eneralized
Sampling )*pansion +(S),
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Outline1. One Dimensional Sampling
2. Multidimensional Sampling
3. Sampling and Reciprocal attices
!. "ndersampled Signals#. $ilter %an&s
'. (eneralized Sampling )*pansion +(S),
-. Recurrent Sampling
. /pplication0 ideo ompression at Source
. onclusions
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&ultidimensional SignalsDigital Photography
Digital Video
x1
x2
x1 x2
x3 (time)
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Outline1. One Dimensional Sampling
2. Multidimensional Sampling
3. Sampling and Reciprocal attices
!. "ndersampled Signals#. $ilter %an&s
'. (eneralized Sampling )*pansion +(S),
-. Recurrent Sampling
. /pplication0 ideo ompression at Source
. onclusions
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=ow should we define sampling for multi>dimensional signals7
"tilize idea of Sampling attice
Sampling Lattice
nonsingular matrix D DT Î ´¡ ¡
( ) { }: D Lat T Tn n Z = = Î
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/lso8 need multi6aria5le frequency domain
concepts.
These are captured y two ideasi. Reciprocal attice
ii. "nit ell
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!nit Cell +?on>unique,
i.
ii.
"eciprocal Lattice
( ){ } ( ){ }1 1
* 2 2 :T T D Lat T T n n Z p p- -
= = Î
( ) ( ){ } ( ) ( ){ }1 1
* *
1 2
1 2 1 2
2 2
,
T T
D
UC T n UC T n
n n Z n n
p p- -
+ ! + =
Î #
( ) ( ){ }1
2 D
T D
n Z
UC T n R p -
Î
+ =$
( )*UC
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One 'imensional
$%ample
Sampling Lattice
0.20 10 20
/ / //
D
{ }. :n n Z = D Î
.10
/
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Reciprocal attice and
!nit )ell
Unit ell
1
2w
p0 110
210
310
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&ultidimensional
$%ample
x1
x2
1 2 3 4 54 3 2 11
2
3
4
5
4
3
2
1
2 1
0 2T
éê=êë
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Reciprocal attice and !nit)ell for $%ample
1!4 1!2 3!4 11!4
1!2
3!4
1
1!2
1!4
( )1
10
21 1
4 2
T T -
éê
ê=êê-êë
( )( )1
2 T UC T p -
1
2
w
p
2
2w p
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Outline1. One Dimensional Sampling
2. Multidimensional Sampling
3. Sampling and Reciprocal attices
!. "ndersampled Signals#. $ilter %an&s
'. (eneralized Sampling )*pansion +(S),
-. Recurrent Sampling
. /pplication0 ideo ompression at Source
. onclusions
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;e will 5e interested here in the situation wherethe Sampling attice is not a ?4quist attice forthe signal +i.e.8 the signal cannot 5e perfectl4reconstructed from the original pattern@,
Strategy
e i++ generate other sa*#+es by %i+tering
or shi%ting o#erations on the origina+ #attern'
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onsider a 5andlimited signal ./ssume the D>dimension $ourier transform has finite
support8 S.
dimensional lattice T 8 there alwa4se*ists a finite set 8 such that support
( ) , D f x x Î ¡
{ } *1
P
iw Î
( )( ) ( )( )*1
" . P
i
i
f S UC w w=
% = +$
Heuristically: he idea o% “i+ing” the
area o% interest in the %re$uency do*ain
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One 'imensional
$%ampleOur one dimensional e*ample continued.Sampling attice { };k k Z = D ÎUnit ell
12w p0 1
102
103
10
( )" f w!andlimited spectrum
Use
1
2
0
2
10
w
pw
=
æç=- ççè
( )( ) ( ) ( )* * 2" f UC UC w wé ù é= +ê ú êë û ë$Support
112
112
- 2w
p
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Outline1. One Dimensional Sampling
2. Multidimensional Sampling
3. Sampling and Reciprocal attices
!. "ndersampled Signals#. $ilter %an&s
'. (eneralized Sampling )*pansion +(S),
-. Recurrent Sampling
. /pplication0 ideo ompression at Source. onclusions
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Outline1. One Dimensional Sampling
2. Multidimensional Sampling
3. Sampling and Reciprocal attices
!. "ndersampled Signals#. $ilter %an&s
'. (eneralized Sampling )*pansion +(S),
-. Recurrent Sampling
. /pplication0 ideo ompression at Source. onclusions
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Define
Let
5e the solution +if it exists, of
for
( ) ( ) ( )
( )
( ) ( )
1 1 2 1 1
1 2
1
" " "
"( )
" "
Q
P Q P
h h h
h H
h h
w w w w w w
w ww
w w w w
é + + +êê
+ê=êêêê + +ë
'
( )( )
( )
1 ,,
,Q
x x
x
ww
w
éêê = êêêë
'
( )*UC wÎ
( ) ( )
1
,
T
T P
j x
j x
e H x
e
w
w
w w
éêê = êêêë
'
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2-
)onditions for +erfect
Reconstruction can 5e reconstructed from
if and onl4 if has full row ran& for all inthe "nit ell
where
( ) H w
( ) f x
( ) ( ) ( )1 D
Q
q q
q k Z
f x g Tk x Tk f = Î
= -å å
( ) ( ),T j x
q q
UC
x x e d wf w= ò
GS# Theorem:
w
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Proof:
Multipl4 5oth sides 54 where +the
Reciprocal attice,.
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where we ha6e used the fact that
Since is the output of fx! passing through 8then
=ence8 we finall4 ha6e
( ) ( ) ( ) ( )
( )
( )*11
" " T
i
D
P j Tk
i q i
iUC
Q
q
q k Z
f x x Tk f h e d w w
w w w f w w- +
== Î
é ùê ú
= -ê úê úê úë û
+ +å å òå
( ) 1
2 $or .T Di T Z w p-
= Îl l
( )q g x
[ ] ( )q g Tk =
( ) ( ) ( )1 D
Q
q q
q k Z
f x g Tk x Tk f = Î
= -å å
( )"
qh w
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Outline1. One Dimensional Sampling
2. Multidimensional Sampling
3. Sampling and Reciprocal attices
!. "ndersampled Signals#. $ilter %an&s
'. (eneralized Sampling )*pansion +(S),
-. Recurrent Sampling
. /pplication0 ideo ompression at Source. onclusions
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Special )ase, RecurrentSampling
+where is implemented 54 a Cspatial shift ,
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%ere 8 and
Thus
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Something to thin-
a#out
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Return to our one.dimensional e%ampleRecall that we had
so that
support
Sa4 we use recurrent sampling with
1
2
0
2
10
w
pw
=
=-
1
2
0
0.9 ; 10
x
x
=
= D D =
( )( ) ( ) ( )* * 2" f UC UC w wé ù é= / +ê ú êë û ë
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0 10 20
/ //
0 - 1-
/ //
0
/ //
1 0 x =
20.9 x = D
.1
.1
/ /
1- 20
/
- 10
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)ondition for +erfectReconstruction 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-
éê
êëéê=êë
=ence8 the original signal can 5e reco6eredfrom the sampling pattern gi6en in thepre6ious slide.
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Summar/ ;e ha6e seen that the well &nown
Shannon reconstruction theorem can 5ee*tended in se6eral directionsE e.g.
Multidimensional signals
Sampling on a lattice
Recurrent sampling
(i6en specific frequenc4 domaindistri5utions8 these can 5e matched toappropriate sampling patterns.
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Outline1. One Dimensional Sampling
2. Multidimensional Sampling
3. Sampling and Reciprocal attices
!. "ndersampled Signals#. $ilter %an&s
'. (eneralized Sampling )*pansion +(S),
-. Recurrent Sampling
. /pplication0 ideo ompression at Source. onclusions
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Application, 0ideo)ompression Source&ntroduction to 'ideo cameras
Instead of tape8 digital cameras use 2D sensorarra4 +D or MOS,
Image
Processor
Image
Processor
Memory
Image
Processor
Image
ProcessorImage
Processor
Image
Processing Display
( TV or LCD )Pipeline
DVCDcontroller
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Image Sensor
/ 2D arra4 of sensors replaces the traditionaltape
)ach sensor records a FpointF of thecontinuous image
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1' )olours Sensor Arra/
Data transfer from arra& is se'uential
and has a ma(imal rate of ")
0 23456 78 9::;>>>?6;@5AB5>?C7l53@8
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!niform (D sampling
a sequence of identical frames equall4 spaced intime
)urrent echnolog/
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The volume of ‘box’ epens on the capacity:
pixel rate ! "frame rate# x "spatial resolution#
x
0ideo Bandwidth
depends onspatial
resolutionof the frames
depends onthe framerate
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1' $ata recoring on sensor:
6 Sensor array density
. %or s#atia+ reso+utionpixels
frame R
6 Sensor e/#osure ti*e
. %or %ra*e rate
frames
sec. F
E? $ata reaing from sensor:6 Data readout ti*e
. %or #i/e+ ratepixels
sec. Q
2ard )onstraints
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(enerall4 Q ## $%
?eed0 $ $% & # %
s.t. $&% & ' Q
Compromise:
spatial resolution $ $
temporal resolution % & # %
%UT&&&
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volume determined by
1 1Q R F
Actual )apacit/ ('ataReadout"
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O#ser3ation
'74: 585@FG 7H :G;BC3l AB657 4C585B4 C78C58:@3:56 3@7I86 :95;l385 386 :95 3JB4?
, x y
t
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uniform sampling
' compromise
in frame rate
uniform sampling
' compromise
in spatial resolution
uniform sampling
' no compromise
he Spectrum of this
0ideo )lip
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frame type ( frame type %
Recurrent 4on.!niformSampling
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What 'oes it Bu/5
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SchematicImplementationnon.uni%or* data %ro* the sensor
uni%or* high de%' ,ideo
*compression at the source*
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Recurrent 4on.!niform
Sampling/ special case of
(eneralized Sampling )*pansion
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Sampling +attern
∆
+
∆+=Ψ ∪∪
+== t #U L$T
x% U L$T U L$T
M
M #
L
%
0)(
0)()(
1
2
2
12
2
1
{ } s M L
s xU L$T +=Ψ ∪
++
=)(
1)(2
1
he resu+ting sa*#+ing #attern is gi,en by
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5!
6re7uenc/ 'omain
{ }& T
M L
&
U UC S ω +∆= −++
=∪ )2(
1)(2
1
)here:
∆+<
∆+<
===
t M x LU UC t xt
xT
)12(,)12(:)2( 1
π
ω
π
ω ω
ω
ω π
is the unit ce++ o% the reci#roca+ +attice
∈
∆+
∆+=− 2
1
:
)12(0
0)12()2( Z nn
t M
x LU L$T T
π
π
π
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5"
Reciprocal attice
x∆−
π
x∆
π
t M
M
∆⋅
+
+−
π
)12(
12
1
t M
M
∆⋅
+
+ π
)12(
12
1
ωJ
ω:
t M ∆+ )12( 1
π
x L ∆+ )12(π
7nit ce++
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5-
Appl/ the *S$ heorem
)()(
1)(2
2
1
x H
M L
γ ω =
Φ
ΦΦ
++
)here: is uni$ue+y de%ined by H 18 H 298:
γ is a set o% 29L;
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0
Reconstruction Scheme
KL ΦL
KE+L ΦE+L
KE(+')+L ΦE(+')+L
∑ ,(JM:) Î(JM:)
1%)2(&
$re'uen 'uist
++
= &
& T
&
t
x j
s& e H ω
.
he sub.sa*#+ed %re$uency o% each %i+ter > is
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Reconstruction functions
t
t1)(2%sin
))1((
x)1)(r (sin
)12(),(π
π
π
π
ϕ
∆+∆−−
∆−∆∆∆+=
x& x
x x
t x Lt x&
))12((
t)1)2&(r (sin
x
x1)(2&sin
)12(),(t L& t
t t
t x M t x& ∆−−−
∆−∆
∆+
∆∆+=π
π
π
π
ϕ
for r ? 2(3(8(2L;1
for r ? 29L;1:(8(29L;
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'emo
$ull resolution
se'uence.econstructed
se'uence
/emporal decimation
pacial decimation
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Outline1. One Dimensional Sampling
2. Multidimensional Sampling
3. Sampling and Reciprocal attices
!. "ndersampled Signals
#. $ilter %an&s
'. (eneralized Sampling )*pansion +(S),
-. Recurrent Sampling
. /pplication0 ideo ompression at Source. onclusions
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)onclusions ?onuniform sampling of scalar signals
?onuniform sampling of multidimensional
signals
(eneralized sampling e*pansion
/pplication to 6ideo compression
/ remaining pro5lem is that of Goint design of
sampling schemes and quantization strategiesto minimize error for a gi6en 5it rate
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References *ne $imensional Sampling
@' Aeuer and B'C' Boodin( ampling in Digital ignal 1rocessing and ontrol '
irkhEuser( 1--' F'G'
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References
.ilter %an/s I'C' H+dar and @'J' M##enhei*( Ai+terbank reconstruction o% band+i*ited signa+s
%ro* nonuni%or* and genera+ized sa*#+es( /ransactions on ignal 1rocessing (
Jo+'4"( )o'10( ##'2"4.2"!5( 2000' P'P' Jaidyanathan( +ultirate &stems and $ilter !an%s' Hng+eood C+i%%s( )G
Prentice.>a++( 1--3' >' N+ceskei( A' >+aatsch and >'B' Aeichtinger( Ara*e.theoretic ana+ysis o%
o,ersa*#+ed %i+ter banks( /ransactions on ignal 1rocessing ( Jo+'4( )o'12(
##'325.32"( 1--"' a++( 1--5'
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!
References 0enerali1e Sampling 2xpansions- Recurrent Sampling
@' Pa#ou+is( Benera+ized sa*#+ing e/#ansion( /ransaction on ircuits and
&stems( Jo+'[email protected]( )o'11( ##'52.54( 1-!!' @' Aeuer( Mn the necessity o% Pa#ou+is resu+t %or *u+tidi*ensiona+ 9BSH:(
ignal 1rocessing Letters( Jo+'11( )o'4( ##'420.422( 2004' K'A'Cheung( @ *u+tidi*ensiona+ e/tension o% Pa#ou+is genera+ized sa*#+ing
e/#ansion ith a##+ication in *ini*u* density sa*#+ing( in @d,anced o#ics inShannon Sa*#+ing and =nter#o+ation heory( F'G'
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Lecture 1Sampling of Signals
by
Graham C. GoodwinUniversity of Newcastle
Australia
Lecture 1
Presented at the “Zaborszky Distinguished Lecture Series”
December 3rd 4th and 5th 2007