标签: inversion

NumericalinversionofLaplacetransformsbyasimplecurvefittingtechnique

NUMERICALINVERSIONOFTHELAPLACETRANSFORM-ANEXPLICITCLOSED-FORMEXPRESSIONFORTHETIME-DOMAINSOLUTION

JustasanalogfiltersaredesignedusingtheLaplacetransform,recursivedigitalfiltersaredevelopedwithaparalleltechniquecalledthez-transform.Theoverallstrategyofthesetwotransformsisthesame:probetheimpulseresponsewithsinusoidsandexponentialstofindthesystem'spolesandzeros.TheLaplacetransformdealswithdifferentialequations,thes-domain,andthes-plane.Correspondingly,thez-transformdealswithdifferenceequations,thez-domain,andthez-plane.However,thetwotechniquesarenotamirrorimageofeachother;thes-planeisarrangedinarectangularcoordinatesystem,whilethez-planeusesapolarformat.Recursivedigitalfiltersareoftendesignedbystartingwithoneoftheclassicanalogfilters,suchastheButterworth,Chebyshev,orelliptic.Aseriesofmathematicalconversionsarethenusedtoobtainthedesireddigitalfilter.Thez-transformprovidestheframeworkforthismathematics.TheChebyshevfilterdesignprogrampresentedinChapter20usesthisapproach,andisdiscussedindetailinthischapter.CHAPTERThez-Transform33JustasanalogfiltersaredesignedusingtheLaplacetransform,recursivedigitalfiltersaredevelopedwithaparalleltechniquecalledthez-transform.Theoverallstrategyofthesetwotransformsisthesame:probetheimpulseresponsewithsinusoidsandexponentialstofindthesystem'spolesandzeros.TheLaplacetransformdealswithdifferentialequations,thes-domain,andthes-plane.Correspondingly,thez-transformdealswithdifferenceequations,thez-domain,andthez-plane.However,thetwotechniquesarenotamirrorimageofeachother;thes-planeisarrangedinarectangularcoordinatesystem,whilethez-planeusesapolarformat.Recursivedigitalfiltersareoftendesignedbystartingwithone……

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