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 3/27 1. Notationsanddefinitions Ø1Dand2Dfunctions ü1D: ),(x d  )(n d  ),(ng ),(xf ü2D: ),,( yx d  ),( nm d  ),,( nmg),,( yxf ØSeparableformsof2Dfunctions üDirac: üKronecker: ürect(x,y),sinc(x,y),comb(x,y) )()(),( yxyx d d d × = )()(),( nmnm d d d × = ØSpecialfunctions 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  a x ax )( )( d d = î í ì = ¹ = 01 00 )( n n x d  )()()( nfmnmf = - å ¥ ¥ - 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 x xsign å ¥ ¥ - - = )()( nxxcom b d  x x xc p p sin )(sin = ï î ï í ì > £ - = 10 11 )( x x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 HanoiUniversityofScienceandTechnology SchoolofElectronicsandTelecommunications 2Dsystemsandmathematicalpreliminaries **0 1/2012 Introduction 5/27 ỉOutputofalinearsystem: [ ] ỳ ỷ ự ờ ở ộ - - H = H = ồồ )','()','(),(),( ' ' nnmmnmxnmxnmy m n d [ ] ồồ ồồ = - - H = ị ' '' ' )',',()','()','()','(),( m nm n nmnmhnmxnnmmnmxnmy 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 2Dsystemsandmathematicalpreliminaries **0 1/2012 Introduction 6/27 3. Fouriertransform ỉFTofa1Dfunction:f(x) [ ] [ ] ũ ũ Ơ + Ơ - - - +Ơ Ơ - - = = = = x x x x px px deFFxf dxexfxfF xj xj 211 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 ),(),(),(: ), (),(),(: x x x x x x x x x x p x x p ddeFFyxf dxdyeyxfyxfF yxj yxj 4 HanoiUniversityofScienceandTechnology SchoolofElectronicsandTelecommunications 2Dsystemsandmathematicalpreliminaries **0 1/2012 Introduction 7/27 Performingachangeofvariables: ỉPropertiesofFT ỹSpatialfrequencies ỹUniqueness ỹSeparability ũ ũ ũ ũ +Ơ Ơ - - +Ơ Ơ - - +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 - + - - = '')','(),( )''(2 21 yddxeyyxxhyxg yxj x x p ',' yyyxxx - = - = )(2 21 :where yxj e x x p + = F )(2 21 21 ),(),( yxj eyxg x x p x x + H = ị HanoiUniversityofScienceandTechnology SchoolofElectronicsandTelecommunications 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 - +Ơ Ơ - +Ơ Ơ - = = 2121 * 21 * ),(H),(F),(),( x x x x x x dddxdyyxhyxfI Settingh=fốParsevalenergyconservationformula ũ ũ ũ ũ +Ơ Ơ - +Ơ Ơ - +Ơ Ơ - +Ơ Ơ - = 21 2 21 2 ),(F),( x x x x dddxdyyxf 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 ± ± ),( 21 lk m m x x d  ),( 00 yyxx ± ± d  )( 22 yx e + - p  )( 2 2 2 1 y e + - x p  ),(s 21 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 Rotation Propertyof2DFT FouriertransformFunction ),F( 21 )(2 2010 x x x x p  yxj e + ± )()( 21 yfxf ),( yxf ± ± ),(F),(H),(G 212121 x x x x x x × = ),(),( 2211 yxfayxfa + ),(F),(F 21222111 x x x x  aa + ),( * 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 + ± ),(F 2211 h x h x m m ),(),(),( yxfyxhyxg * = ),(),(),( yxfyxhyxg × = ),(F),(H),( G 212121 x x x x x x * = ),(),(),( yxfyxhyxc · = ),(F),(H),(C 212121 x x x x x x × - - = ò ò +¥ ¥ - +¥ ¥ - = dxdyyxhyxfI ),(),( * ò ò +¥ ¥ - +¥ ¥ - 2121 * 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 1 ,z 2 arecomplexvariables ồ ồ +Ơ -Ơ = +Ơ -Ơ = - - = m n nm zznmxzzX 2121 ),(),( ỹRegionofconverge(RoC):thisseriesconvergesuniformlyinthisregion ỹZtransformofaLSIsystemiscalledtransferfunction ),( ),( ),( ),(),(),( 21 21 21 212121 zzX zzY zzH zzXzzHzzY = ị = ỉInverseZtransform: 11where,),( )2( 1 ),( 21 2 1 1 2 1 121 2 = = = ũũ - - zzdzdzzzzzX j nmx nm p ),( 21 zzX 1 1 w j ez = 2 2 w j ez = ),( nmx HanoiUniversityofScienceandTechnology SchoolofElectronicsandTelecommunications 2Dsystemsandmathematicalpreliminaries **0 1/2012 Introduction 12/27 ỉPropertiesof2DZtransform Multiplication Convolution Modulation Shifting Separability Conjungation Linearity Rotation Property FouriertransformFunction ),( 2121 00 zzXzz nm )()( 21 nxmx ),( nmx - - ),(),( 2121 zzFzzH ì ),(),( 2211 nmxanmxa + ),(),( 21222111 zzX azzXa + ),( * nmX ),( * 2 * 1 * zzX )( F)(F 2211 x x ),( yxf ),( 00 nnmmx ),( nmxba nm ),(),( nmxnmh * ),(),( nmynmx ),( 1 2 1 1 - - zzX ),(F 21 x x ữ ứ ử ỗ ố ổ b z a z 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 ' 1 ' 2 2 ' 1 1 2 ),(, 2 1 C C z dz z dz zzY z z z z X j p 7 HanoiUniversityofScienceandTechnology SchoolofElectronicsandTelecommunications 2Dsystemsandmathematicalpreliminaries **0 1/2012 Introduction 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 ồồ Ơ < m n nmh ),( ),( 21 zzH HanoiUniversityofScienceandTechnology SchoolofElectronicsandTelecommunications 2Dsystemsandmathematicalpreliminaries **0 1/2012 Introduction 14/27 5. Matrixtheoryandresults ỉVectorsandmatrices ỹColumnvectorofsizeN: NnnuU á = = 1),( ỹRowvectorofsizeM: MmmuU á = = 1),( ỹMatrixAofsizeMxNcontainingMrows,Ncolumns ỳ ỳ ỳ ỳ ỷ ự ờ ờ ờ ờ 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 üRoworderedvector(rowstacking) [ ] T T NMxMxNxxNxxxx ),( ,),1,(),2( ,),1,2(),,1( ,),2,1(),1,1( L = [ ] T T NMxMxMxxMxxxx ),( ,),,1()2,( ,),2,1(),1,( ,),1,2(),1,1( L = üColumnorderedvector(columnstacking) ØMatrixtheorydefinitions { } ),( nmaA = üMatrix: üTranspose: { } ),( mnaA T = { } ),( ** nmaA = üComplexconjungate: { } )( nmI - = d üConjungatetranspose: üIdentitymatrix: üNullmatrix: { } 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 1/2012 Introduction 16/27 üVectorinnerproduct: å = = )()(, ** nynxYXYX T :Scalarquantity,ifequal0 èXandYareorthogonal üVectorouterproduct: { } )()( nymxXY T = :X:Mx1,Y:Nx1,XY T :MxN üSymmetric: T AA = ü Hermitian: T AA * = :RealsymmetricmatrixisHermitan.Eigenvaluesarereal üDeterminant: A 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 17/27 ØTransposeandconjungaterules [ ] [ ] [ ] [ ] [ ] ** * 1 1 * * .4 .3 .2 .1 BAAB AA ABAB AA T T TT T TT = = = = - - 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 A N N N L L L L ØCirculantmatrixC ú ú ú ú ú ú û ù ê ê ê ê ê ê ë é = - - - - 0121 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 18/27 whereandareeigenvaluesandeigenvectorsofR üOtherform,whichisthesetofeigenvalueequations ØOrthogonalandunitarymatrices üOrthogonalmatrix: üUnitarymatrix: IAAAAAA TTT = = = - ***1 or èExp.2.5a ØDiagonalforms üIfRisHermitianmatrix,thereexistsaunitarymatrixΦsuchthat whereΛisadiagonalmatrixcontainingeigenvaluesofR L = F RΦ *T L = ΦRΦ Nk kk ,,2,1,ΦRΦ k L = = l { } k l  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 ê ë é = À nmmm n n AAA AAA AAA ,2,1, ,22,21,2 ,12 ,11,1 , , , L L L L wherearematrices ji A , )(, jiji AA - = Ø isblockcirculantifiscirculant nmAA 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 2Dsystemsandmathematicalpreliminaries **0 1/2012 Introduction 20/27 ü isanNx1vector 7. Randomsignals ØDefinitions:givenasequenceofrandomvariablesu(n) üMean: üVariance: üCovariance: üCrosscovariance: üAutocorrelation: üCrosscorrelation: [ ] )()()( nuEnn u = = m m [ ] 2 2 2 )()()()( nnuEnn u m s s - = = [ ] [ ] [ ] { } )'()'()()()',()'(),( ** 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 uu m m - = = = [ ] )'()()',()()()',( * * nnnnrnvnuEnna vuuvuv m m - = = ØForvectorofsizeNx1:u [ ] { } )(nE m = = μu ü isanNxNmatrix [ ] [ ] { } )',())(( ** nnrECov T = = = - - = RRμuμuu u ü isanNxNmatrix [ ] [ ] { } )',())((, * * nnrECov uv T = = - - = uvvu Rμvμuvu