Goodwin Lecture1

8/16/2019 Goodwin Lecture1

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1

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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. onclusions


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&ultidimensional SignalsDigital Photography

Digital Video

x1

x2

x1 x2

x3 (time)


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. 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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1!

One 'imensional

$%ample

Sampling Lattice

0.20 10 20

/ / //

D

{ }. :n n Z = D Î

.10

/


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1"

Reciprocal attice and

!nit )ell

Unit ell

1

2w

p0 110

210

310


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1-

&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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. 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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. /pplication0 ideo ompression at Source. onclusions


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2!









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2"

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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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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3!

0 10 20

/ //

0 - 1-

/ //

0

/ //

1 0 x =

20.9 x = D

.1

.1

/ /

1- 20

/

- 10


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3"

)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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3-

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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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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4!

(enerall4 Q ## $%

?eed0 $ $% & # %

s.t. $&% & ' Q

Compromise:

spatial resolution $ $

temporal resolution % & # %

%UT&&&


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4"

volume determined by

1 1Q R F

Actual )apacit/ ('ataReadout"


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4-

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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5

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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1

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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2

'emo

$ull resolution

se'uence.econstructed

se'uence

/emporal decimation

pacial decimation


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!. "ndersampled Signals

#. $ilter %an&s





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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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5

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'


67/67

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

Documents

Goodwin Lecture1