The Z-Trasform Cotet ad Figures are from Discrete-Time Sigal Processig, e by Oppeheim, Shafer, ad Buck, 999- Pretice Hall Ic.
The -Trasform Couterpart of the Laplace trasform for discrete-time sigals Geeraliatio of the Fourier Trasform Fourier Trasform does ot exist for all sigals The -Trasform is ofte time more coveiet to use Defiitio: X Compare to DTFT defiitio: is a complex variable that ca be represeted as =r e j Substitutig =e j will reduce the -trasform to DTFT x j j x e X e Copyright (C) 5 Güer Arsla 5M Digital Sigal Processig
The -trasform ad the DTFT The -trasform is a fuctio of the complex variable Coveiet to describe o the complex -plae If we plot =e j for = to we get the uit circle Im X e j Uit Circle r= Re Copyright (C) 5 Güer Arsla 5M Digital Sigal Processig
Covergece of the -Trasform DTFT does ot always coverge j j x e X e Ifiite sum ot always fiite if x[] o absolute summable Example: x[] = a u[] for a > does ot have a DTFT Complex variable ca be writte as r e j so the -trasform j j x re x r X re DTFT of x[] multiplied with expoetial sequece r - For certai choices of r the sum maybe made fiite x r - e j Copyright (C) 5 Güer Arsla 5M Digital Sigal Processig 4
Regio of Covergece The set of values of for which the -trasform coverges Each value of r represets a circle of radius r The regio of covergece is made of circles Im Re Example: -trasform coverges for values of.5<r< ROC is show o the left I this example the ROC icludes the uit circle, so DTFT exists ot all sequece have a -trasform Example: x cos Does ot coverge for ay r o ROC, o -trasform But DTFT exists?! Sequece has fiite eergy DTFT coverges i the measquared sese o Copyright (C) 5 Güer Arsla 5M Digital Sigal Processig 5
Right-Sided Expoetial Sequece Example a u X a u a x For Covergece we require a Hece the ROC is defied as a a Im a o x Re Iside the ROC series coverges to X a a a Geometric series formula Regio outside the circle of radius a is the ROC a a a Right-sided sequece ROCs a exted outside a circle Copyright (C) 5 Güer Arsla 5M Digital Sigal Processig 6
Same Example Alterative Way a u X a u a x a a a a > For the term with ifiite expoetial to vaish we eed a a Determies the ROC (same as the previous approach) I the ROC the sum coverges to a X a Copyright (C) 5 Güer Arsla 5M Digital Sigal Processig 7
Copyright (C) 5 Güer Arsla 5M Digital Sigal Processig 8 Two-Sided Expoetial Sequece Example u - - - u x ROC : ROC : X Re Im oo x x
Copyright (C) 5 Güer Arsla 5M Digital Sigal Processig 9 Fiite Legth Sequece otherwise a x a a a a a a X
Properties of The ROC of Z-Trasform The ROC is a rig or disk cetered at the origi DTFT exists if ad oly if the ROC icludes the uit circle The ROC caot cotai ay poles The ROC for fiite-legth sequece is the etire -plae except possibly = ad = The ROC for a right-haded sequece exteds outward from the outermost pole possibly icludig = The ROC for a left-haded sequece exteds iward from the iermost pole possibly icludig = The ROC of a two-sided sequece is a rig bouded by poles The ROC must be a coected regio A -trasform does ot uiquely determie a sequece without specifyig the ROC Copyright (C) 5 Güer Arsla 5M Digital Sigal Processig
Stability, Causality, ad the ROC Cosider a system with impulse respose h[] The -trasform H() ad the pole-ero plot show below Without ay other iformatio h[] is ot uiquely determied > or <½ or ½< < If system stable ROC must iclude uit-circle: ½< < If system is causal must be right sided: > Copyright (C) 5 Güer Arsla 5M Digital Sigal Processig