1
HanoiUniversityofScienceandTechnology SchoolofElectronicsandTelecommunications
2Dsystemsandmathematicalpreliminaries
**0
1/2012
Introduction
1/27
Twodimensionalsystems
&
Mathematicalpreliminaries
HanoiUniversityofScienceandTechnology SchoolofElectronicsandTelecommunications
2Dsystemsandmathematicalpreliminaries
**0
1/2012
Introduction
2/27
1. Notationsanddefinitions
2. Linearsystemsandshiftinvariance
3. FourierTransform
4. ZTransformorLaurentseries
5. Matrixtheoryandresults
6. BlockmatricesandKroneckerproducts
7. Randomsignals
8. Someresultsfromestimationtheory
9. Someresultsfrominformationtheory
2
HanoiUniversityofScienceandTechnology SchoolofElectronicsandTelecommunications
2Dsystemsandmathematicalpreliminaries
**0
1/2012
Introduction

üDiracdelta:
üShifting:
üScaling:
üKroneckerdelta:
üShifting:
üRectangle:
üSignum:
üSinc:
üComb:
üTriagle:
1)(lim;0,0)(
0
= ¹ =
ò
+
-
®

e
e
e
d d
 dxxxx
)(')'()'( xfdxxxxf = -
ò
+
-

e
e

d

ï
î
ï
í
ì
>
£
=
2/10
2/11
)(
x
x
xrect
ï
î
ï
í
ì
< -
=
>
=
01
00
01
)(
x

xtri
HanoiUniversityofScienceandTechnology SchoolofElectronicsandTelecommunications
2Dsystemsandmathematicalpreliminaries
**0
1/2012
Introduction
4/27
2. Linearsystemsandshiftinvariance
Ø2Dlinearsystems
[]
× H
[ ]
),(),( nmxnmy H =
),( nmx
üLinearsuperpositionproperty:
üImpulseresponse:
üImpulseresponseiscalledPSF:Inputandoutputarepositivequantities
üIngeneral:Impulseresponsecantakenegativeorcomplexvalues
üRegionofsupport(RoS)ofimpulseresponse
üFiniteimpulseresponse(FIR)andinfiniteimpulseresponse(IIR):When
RoSisfiniteorinfinite
[ ]
)','()', ';,( nnmmnmnmh - - H =
d

[ ] [ ] [ ]
),(),(),(),(),(),(
221122112211
nmyanmyanmxanmxanmxanmxa + = H + H = + H
3

d
ỉSpatiallyinvariantorshiftinvariantsystem:
[ ]
)0,0,(),( nmhnm = H
d

[ ]
)0,0','()','()',',( nnmmhnnmmnmnmh - - = - - H =
d

)','()',',( nnmmhnmnmh - - = ị
ỹTheshapeofimpulseresponsedoesnotchangeastheimpulse
responsemovesaboutthe(m,n)plan
ỹDiscreteconvolution:
),(*),()','()','(),(
' '
nmxnmhnmxnnmmhnmy
m n
= - - =
ồồ
ỹContinuousconvolution:
'')','()','(),(*),(),( dydxxxfyyxxhyxfyxhyxg
ũ ũ
Ơ
Ơ -
Ơ
Ơ -
- - = =
ốExp.2.1
HanoiUniversityofScienceandTechnology SchoolofElectronicsandTelecommunications

2
)()()(:
)()()(:
ỉFTofa2Dfunction:f(x,y)
[ ]
[ ]
ũ ũ
ũ ũ
Ơ +
Ơ -
Ơ +
Ơ -
+
- -
+Ơ
Ơ -
+Ơ
Ơ -
+ -
= =
= =
21
)(2
2121
11
)(2
21
21
21
),(),(),(:

+Ơ
Ơ -
+Ơ
Ơ -
+ -
ỳ
ỷ
ự
ờ
ở
ộ
= = dyedxeyxfdxdyeyxfF
yjxjyxj
2121
22)(2
21
),(),(),(

px px x x p
x x
ỹFrequencyresponseandeigenfunctionsofshiftinvariantsystems
),( yxh
FH
F
ũ ũ
+Ơ
Ơ -
+Ơ
Ơ -
+

2Dsystemsandmathematicalpreliminaries
**0
1/2012
Introduction
8/27
ỹConvolutiontheorem
ỹInnerproductpreservation
),(F),(),G(),(*),(),(
212121

x x x x x x
ì H = ị = yxfyxhyxg
ỹCorrelationbetween2realfunctions
ũ ũ
+Ơ
Ơ -
+Ơ
Ơ -
+ + = ã = '')','()','(),(),(),( dydxyyxxfyxhyxfyxhyxc
Performingachangeofvariables:
),F(),(),(C),(),(),(
212121

x x x x x x
ì - - H = ị ã - - = yxfyxhyxc
ũ ũ ũ ũ
+Ơ
Ơ -
+Ơ
Ơ -

5
HanoiUniversityofScienceandTechnology SchoolofElectronicsandTelecommunications
2Dsystemsandmathematicalpreliminaries
**0
1/2012
Introduction
9/27
üFouriertransformpairs
comb(x,y)
tri(x,y)
rect(x,y)
1
),( yxf ),(F
21

x x

),( yx
d

yljxkj
ee

p p
 22 ± ±
2010
22
x p x p
 yjxj
ee


x x
inc
),(s
21
2

x x
inc
),(
21

x x
comb
HanoiUniversityofScienceandTechnology SchoolofElectronicsandTelecommunications
2Dsystemsandmathematicalpreliminaries
**0
1/2012
Introduction
10/27
Innerproduct
Spatialcorrelation
Multiplication
Convolution
Modulation
Shifting
Scaling
Separability
Conjungation
Linearity

*
yxf ),(F
21
*

x x
- -
)( F)(F
2211

x x

),( yxf
),( byaxf
[ ]
abba /)/,/(F
21

x x

),(
00
yyxxf ± ±
),(
)(2
21
yxfe
yxj
h h p
+ ±

*
21
),(H),(F
x x x x x x
 dd
),(F
2
2
1
1

x x
m m
),(F
21

x x
6
HanoiUniversityofScienceandTechnology SchoolofElectronicsandTelecommunications
2Dsystemsandmathematicalpreliminaries
**0
1/2012
Introduction
11/27
Theevaluationofatand yieldsFTof
4. ZTransformorLaurentseries
ỉFouriertransformofsequences(Fourierseries):Selfreading
ỉGeneralizationofFTseries:Ztransform
ỹFor2Dsequencex(m,n):
wherez

= ị
=
ỉInverseZtransform:
11where,),(
)2(
1
),(
21
2
1
1
2
1
121
2
= = =
ũũ
- -
zzdzdzzzzzX
j
nmx
nm

p

),(
21
zzX
1
1

zzXzz
nm
)()(
21
nxmx
),( nmx - -
),(),(
2121
zzFzzH ì
),(),(
2211
nmxanmxa +
),(),(
21222111
zzX azzXa +
),(
*
nmX
),(
*
2
*
1
*
zzX
)( F)(F
2211

x x


X
21
,
ũ ũ
ữ
ữ
ứ
ử
ỗ
ỗ
ố
ổ
ữ
ữ
ứ
ử
ỗ
ỗ
ố
ổ
1 2
'
2
2
'
1
1
'
2
'

13/27
ỉCausality
ỹCausal:Impulseresponseforanditstransferfunction
musthaveaonesidedLaurentseries
0)( =nh 0 <n
ồ
Ơ
=
-
=
0
)()(
n
n
znhzH
ỹAnticausal:Impulseresponseforanditstransferfunction
musthaveaonesidedLaurentseries
0)( =nh
0 n
ỹNoncausal:Neithercausaloranticausal
ỉStability:Outputremainsuniformlyboundedforanyboundedinput
Ơ <
ồ
Ơ
=0
)(
n
nh
ỉCausalandstablesystem:polesofH(z)mustlieinsidetheunitcircle
ỉ2Dcase: RoCofmustincludetheunitcircles

ở
ộ
=
),()2,(),1,(
),2()2,2(),1,2(
),1()2,1(),1,1(
NMaMa Ma
Naaa
Naaa
A
L
L
L
L
ỹIndexnotation:
{ }
1,0),,( - Ê Ê =

NnmnmaA
NN
ỹAnimageisusuallyvisualizedasamatrix
ốExp.2.2
8
HanoiUniversityofScienceandTechnology SchoolofElectronicsandTelecommunications
2Dsystemsandmathematicalpreliminaries
**0
1/2012
Introduction
15/27
ØRowandcolumnordering

{ }
0 =O
{ }
),(
**
mnaA
T
=
üMatrixaddition:
{ }
),(),( nmbnmaBA + = +
:A,B:Samedimension
üScalarmultiplication:
{ }
),( nmaA
a a
=
üMatrixmultiplication:
å
=
=
K
k
nkbkmanmc
1
),(),(),(
A:MxK,B:KxN,C:MxN
HanoiUniversityofScienceandTechnology SchoolofElectronicsandTelecommunications
2Dsystemsandmathematicalpreliminaries
**0

üRankofA:Numberofindependentrowsorcolumns
üInversematrix:
IAAAA = =
- - 11
:Squarematrixonly
üSingular:A
1
doesnotexistand
0 =A
üEigenvalues:allrootsof
k

l

0 = - IA
k

l
üEigenvectors:allsolutionsof
k
F 0, ¹ F F = F
kkkk
A
l
9
HanoiUniversityofScienceandTechnology SchoolofElectronicsandTelecommunications
2Dsystemsandmathematicalpreliminaries
**0
1/2012
Introduction

-
ØToeplitzandcirculantmatrices
ú
ú
ú
ú
û
ù
ê
ê
ê
ê
ë
é
=
-
-
+ - -
+ - -
0121
12
2101
110
,,,
,,
,
tttt
tt
tttt
ttt

2
2101
1210
,,,
,,
,,
cccc 
c
cccc
cccc
C
N
NN
N
L
L
L
L
CisalsoToeplitzandc(m,n)=c((mn)moduloN)
èExp.2.3
èExp.2.4
t(i,j)=t
ij
:Constantelementsalongthe
maindiagonalandsubdiagonal
HanoiUniversityofScienceandTechnology SchoolofElectronicsandTelecommunications
2Dsystemsandmathematicalpreliminaries
**0
1/2012
Introduction

k
Φ
èExp.2.5b
IAAAAAA
TTT
= = =
-
or
1
10
HanoiUniversityofScienceandTechnology SchoolofElectronicsandTelecommunications
2Dsystemsandmathematicalpreliminaries
**0
1/2012
Introduction
19/27
Ø isblockToeplitzifisToeplitzor
6. BlockmatricesandKroneckerproducts
ØBlockmatricesofsize:eachelementisamatrixitself
ú
ú
ú
ú
ú
û
ù
ê
ê
ê
ê

njiji
= =
-
,
)ulomod)((,
èExp.2.6
èExp.2.7
ØKroneckerproducts:A:M
1
xM
2
,B:N
1
xN
2
:
ØSeparableoperations:selfreading
{ }
BnmaBA ),( = Ä
qp ´
À
ji
A
,
À
ji
A
,
nm´
HanoiUniversityofScienceandTechnology SchoolofElectronicsandTelecommunications

[ ] [ ]
[
]
{
}
)'()'()()()',()'(),(
**
nnunnuEnnrnunuCov
u

m m
- - = =
[ ] [ ]
[
]
{
}
)'()'()()()',()'(),(
*
*
nnvnnuEnnrnvnuCov
vuuv

m m
- - = =
[
]
)'()()',()()()',()',(
**
nnnnrnunuEnnanna

ü isanNxNmatrix
[ ]
[
]
{ }
)',())((,
*
*
nnrECov
uv
T
= = - - =
uvvu
Rμvμuvu
11
HanoiUniversityofScienceandTechnology SchoolofElectronicsandTelecommunications
2Dsystemsandmathematicalpreliminaries
**0
1/2012
Introduction
21/27
ỉGaussianrandomprocess:thejointprobabilitydensityofanyfinitesub
sequenceisaGaussiandistribution
ỉStationaryprocess
ỉGaussianornormaldistribution:forstandardnormaldistribution
ù
ỵ
ù
ý
ỹ

kllu Ê Ê1),(
{ }
klmlu Ê Ê + 1),(
ỹu(n)iswidesensestationary:
[ ]
constnuE = =
m
)(
[
]
)'()',()'()(
*
nnrnnrnunuE - = =
ỹSymmetry:
ỹNonnegativity:
',)',()',(
*
nnnnrnnr " =
nnxnxnnrnx
n n
" ạ
ồồ
,)(,0)'()',() (
'
*
HanoiUniversityofScienceandTechnology SchoolofElectronicsandTelecommunications
2Dsystemsandmathematicalpreliminaries
**0
1/2012
Introduction

-
1,,,
1,
,,1
1
12

r r
r
r
r r r
L
L
L
L
N
N
R
[ ] [ ]
npnununuprobnununuprob " - - = - - )(,),1(|)()2() ,1(|)( L L
12
HanoiUniversityofScienceandTechnology SchoolofElectronicsandTelecommunications
2Dsystemsandmathematicalpreliminaries
**0
1/2012
Introduction
23/27
ỉOrthogonalityandindependence
ỹxandyareindependent:
nypxpyxp

{ }
Nnnx Ê Ê1),( R
ỹ isanNxNunitarymatrix,whichreducestoitsdiagonalform
R
ỹKLTof:
x
xy
*T
=
ỹPropertyofKLT:
[ ] [ ]
{ }
Rxxyy = = =
TT ****
EE
[ ]
)()()(E
*
lklyky
k
- = ị
d l
HanoiUniversityofScienceandTechnology SchoolofElectronicsandTelecommunications
2Dsystemsandmathematicalpreliminaries
**0
1/2012
Introduction
24/27
8. Someresultsfromestimationtheory
ỉMeansquareestimates(MSE)

yx
)()(,),2(),1(||
|
L
ỹIfxandy(n)areindependent,thensimplyismeanofxbecause:

x
( )
[ ]
( )
xEyxEExE = =
ỳ
ỷ
ự
ờ
ở
ộ

|
ỹForzeromeanGaussianrandomvariables,becomeslineariny(n)

x
ồ
=

=
N
n
nynx
1


[ ] [ ]
NnnxyEnykyEk
N
k
,,1,)()()()(
1
L = =
å
=

a
Inmatrixform:
xyy
rRα
1 -
=
TheminimizedMSEisgiven:
xy
T
x
rα - =
22

s s
e
Ifxandy(n)arenonzeromeanrandomvariables:
[ ]
)()()(
1

pI
2
log - =
ØEntropy:Definedasaverageinfogeneratedbythesource(bit/message)
å
=
- =
L
k
kk
ppH
1
2
log
üForagivenL,maxentropyofasourceisdeterminedforuniform
distribution,i.e
L
LL
H
L
k
p
k
2
1
2
log
1
log
1

represent(orcode)itforafixeddistortionD
ỹx:Gaussianwithy:reproducedvalueMSE(xy):Distortionmeasure
2

s

[
]
2
)( yxED - =
ThenRDFofxisdefinedas:
( )
ỳ
ỷ
ự
ờ
ở
ộ
ữ
ữ
ứ
ử
ỗ
ỗ
ố
ổ
=
ù
ợ
ù

)1(,),1(),0( -Nxxx L
{ }
)1(,),1(),0( -Nyyy L
[ ]
ồ
-
=
- =
1
0
2
)()(
1
N
k
kykxE
N
D