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1、806IEEETRANSACTIONSONCIRCUITSANDSYSTEMSÐI:FUNDAMENTALTHEORYANDAPPLICATIONS,VOL.46,NO.7,JULY1999StabilityandMarginofStabilityTestsforMultidimensionalFiltersEugeneCurtinandSalimSabaAbstractÐInthispaperwegivesimplifiedproofsofthestabilityII.THESTABILITYTESTandthemarginofsta
2、bilitycriteriaofZehebandWalachforn-Wewillusethefollowingnotationalconventions:dimensionalrecursivedigitalfilters.WebaseourproofsonthemethodofLagrangemultipliers.Boththestabilityandmarginofstabilityquestionsareconvertedintotheproblemoflocatingthezerosofasystemofpolynomial
3、equations.ThesesystemscanthenbesolvedusingBuchberger'salgorithmforcomputingtheGroebnerbasisofapolynomialideal.IndexTermsÐDigitalfilterwordlengtheffects,multivariablesystems,polynomials,recursivedigitalfilters,signalprocessing,stabilitycriteria.I.INTRODUCTIONFthetransferfu
4、nctionrepresentingan-dimensionalre-IcursivedigitalfilterisgivenbyWerefertoasthepolydiskofradius,asthetorus,andasthediagonal.Weusefortheunitpolydiskandandforthecorrespondingtorusanddiagonal.Weusethesymbolforthecomplexunitwhereandareassumedtobeandforthecomplexconjugateofac
5、omplexnumbermutuallyprimepolynomials,thenthefilterisstableifandTheorem1wasfirstprovenbyZehebandWalachin[10].onlyifisanalyticontheunitpolydisk.ZehebHowever,theirproofisfairlycomplicatedandweofferaandWalachin[10]introducedcriteriaforcheckingwhethersimpleproofbasedonthemetho
6、dofLagrangemultipliers.hasanyzerosintheunitpolydiskandinItshouldbenotedthatinthecaseTheorem1below[11]theygavesomedefinitionsforthemarginofstabilityisanimmediateconsequenceofRudin'stheorem,aswhenofan-dimensionalrecursivedigitalfilter.Algorithmsfortheconditionsin3)reducetoa
7、ndthemarginofstabilityoftwo-dimensional(2-D)filtersareforsomewhichisequivalenttodiscussedin[1],[5],[8],and[9].ThecriteriaofZehebandstatinghasazeroonWalachtransformtheproblemintolocatingthezerosofaTheorem1:Apolynomialhasasystemofpolynomialequations.Thisproblemcanbesolvedz
8、eroonifandonlyifoneofthefollowingconditionsusingBuchberger'salgorithm.Muchrecentresearchhasbeenhold:doneonboth
