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1、806IEEETRANSACTIONSONCIRCUITSANDSYSTEMSDI:FUNDAMENTALTHEORYANDAPPLICATIONS,VOL.46,NO.7,JULY1999StabilityandMarginofStabilityTestsforMultidimensionalFiltersEugeneCurtinandSalimSabaAbstractDInthispaperwegivesimpli?edproofsofthestabilityII.THESTABILITYTESTandthemarginofsta
2、bilitycriteriaofZehebandWalachforn-Wewillusethefollowingnotationalconventions:dimensionalrecursivedigital?lters.WebaseourproofsonthemethodofLagrangemultipliers.Boththestabilityandmarginofstabilityquestionsareconvertedintotheproblemoflocatingthezerosofasystemofpolynomial
3、equations.ThesesystemscanthenbesolvedusingBuchberger'salgorithmforcomputingtheGroebnerbasisofapolynomialideal.IndexTermsDDigital?lterwordlengtheffects,multivariablesystems,polynomials,recursivedigital?lters,signalprocessing,stabilitycriteria.I.INTRODUCTIONFthetransferfu
4、nctionrepresentingan-dimensionalre-Icursivedigital?lterisgivenbyWerefertoasthepolydiskofradius,asthetorus,andasthediagonal.Weusefortheunitpolydiskandandforthecorrespondingtorusanddiagonal.Weusethesymbolforthecomplexunitwhereandareassumedtobeandforthecomplexconjugateofac
5、omplexnumbermutuallyprimepolynomials,thenthe?lterisstableifandTheorem1was?rstprovenbyZehebandWalachin[10].onlyifisanalyticontheunitpolydisk.ZehebHowever,theirproofisfairlycomplicatedandweofferaandWalachin[10]introducedcriteriaforcheckingwhethersimpleproofbasedonthemetho
6、dofLagrangemultipliers.hasanyzerosintheunitpolydiskandinItshouldbenotedthatinthecaseTheorem1below[11]theygavesomede?nitionsforthemarginofstabilityisanimmediateconsequenceofRudin'stheorem,aswhenofan-dimensionalrecursivedigital?lter.Algorithmsfortheconditionsin3)reducetoa
7、ndthemarginofstabilityoftwo-dimensional(2-D)?ltersareforsomewhichisequivalenttodiscussedin[1],[5],[8],and[9].ThecriteriaofZehebandstatinghasazeroonWalachtransformtheproblemintolocatingthezerosofaTheorem1:Apolynomialhasasystemofpolynomialequations.Thisproblemcanbesolvedz
8、eroonifandonlyifoneofthefollowingconditionsusingBuchberger'salgorithm.Muchrecentresearchhasbeenhold:doneonboth
