Deprtmet of Electricl Egieerig Uiversity of Arkss ELEG 573L Digitl Sigl Processig Ch. The Z-Trsform Dr. Jigxi Wu wuj@urk.edu
OUTLINE The Z-Trsform Properties Iverse Z-Trsform Z-Trsform of LTI system
Z-TRANSFORM 3 Bilterl Z-trsform Uilterl Z-trsform X x X x 0 Z-trsform: C simplify the lysis of discrete-time LTI systems Alye the system i -domi isted of time domi Does t hve y physicl meig the frequecy domi represettio of discrete-time sigl c be obtied through discrete-time Fourier trsform Couterprt for cotiuous-time systems: Lplce trsform.
Z-TRANSFORM 4 Exmple: fid Z-trsforms. x. x u
Z-TRANSFORM 5 Exmple x u 3. Regio of covergece ROC
Z-TRANSFORM: CONVERGENCE 6 ROC of cusl sigl x u ROC of ti-cusl sigl x u
Z-TRANSFORM 7 Exmple: fid the Z-trsforms for the followig sigls x u u 3
Z-TRANSFORM 8 Exmple: fid the Z-trsforms for the followig sigls 3, x, 3
Z-TRANSFORM: TRANSFORM TABLE 9
Z-TRANSFORM: TRANSFORM TABLE 0
OUTLINE The Z-Trsform Properties Iverse Z-Trsform Z-Trsform of LTI system
PROPERTIES Lierity If The Zx X Zx X Z x x X X
PROPERTIES Time Shiftig Let be cusl sequece with the Z-trsform The 3 x X 0 0 0 0 0 m m m x X x Z 0 0 0 0 m m m x X x Z
PROPERTIES 4 Exmple Solve the differece equtio with iitil coditio y y y 3
PROPERTIES 5 Exmple Solve the differece equtio with iitil coditio y y y u 9 y y 0
PROPERTIES 6 Frequecy sclig If The Z x X x X Z
PROPERTIES 7 Exmple Fid the Z-trsform of u x cos
PROPERTIES 8 Differetitio with respect to If Z x X The Z d d x X
PROPERTIES 9 Exmple Fid the Z-trsform of y u
PROPERTIES 0 Iitil vlue lim X x0 Fil vlue lim X x
PROPERTIES Exmple Fid the iitil vlue d fil vlue of the followig sigl. 3 X 0.5
PROPERTIES Covolutio If Z x X Z h H The Exmple Z x h X H Fid the covolutio of the followig two sequeces x,,0, y,3,
PROPERTIES 3
OUTLINE 4 The Z-Trsform Properties Iverse Z-Trsform Z-Trsform of LTI system
INVERSE Z-TRANSFORM Review Iverse Z-Trsform by prtil frctio expsio Expd X i the form of,, etc. 5 u u X 0 0 X A X A 0 A A A X X A 0 A A A X
INVERSE Z-TRANSFORM 6 Exmple Fid the iverse Z-Trsform of X 4
INVERSE Z-TRANSFORM 7 Solve the differece equtio 3 y y y 4 8
INVERSE Z-TRANSFORM 8 Exmple Fid the covolutio u b u
INVERSE Z-TRANSFORM 9 Exmple Fid the followig covolutios u u
OUTLINE 30 The Z-Trsform Properties Iverse Z-Trsform Z-Trsform of LTI system
Trsfer fuctio of discrete-time LTI system Z-trsform o both sides: LTI SYSTEM Trsfer fuctio of discrete-time LTI system d 3 h x y H X Y X Y H M k k N k k k x b k y 0 0 M k k k N k k k X b Y 0 0 N k k k M k k k b H 0 0
LTI SYSTEM 3 Exmple Let the step respose of LTI system be s follows. Fid the trsfer fuctio 6 y u u u 5 5 4
LTI SYSTEM 33 Zeros d poles H p M N p M N p Zeros: Poles:,,, p, p,, M p N Stbility A discrete-time LTI system is stble if ll the poles re iside the uit circle. A discrete-time LTI system is ustble if t lest oe pole is o or outside the uit circle. Review: cotiuous-time LTI system is stble if ll the poles re o the left hlf ple.
LTI SYSTEM 34 Exmple Cosider LTI system described by the differece equtio. Fid the trsfer fuctio d the eros d poles. Is the system stble? y y y x x
LTI SYSTEM 35 Exmple Fid the trsfer fuctio of the system show i the followig digrm. If k =, is the system stble? H 0.8 0.5 0.8
LTI SYSTEM 36 Mtlb Exmple H 3 % umertor coefficiets b = [, 0, ]; % deomitor coefficiets = [, 3, ]; [r, p, k] = residueb, r = [4.5, -3], p =[-, -], k = 0.5 H r p r 4.5 3 k p 0.5
LTI SYSTEM 37 Mtlb Exmple Cot d H 3 % umertor coeffciets b = [, 0, ]; % deomitor coefficiets = [, 3, ]; % prtil frctio expsio [r, p, k] = residueb, % fid the eros = rootsb; % plot the poles d eros pleb, ; % fid the output to the system with iput x x = [,, -, 3]; y = filterb,, x;
LTI SYSTEM 38 Mtlb Exmple multiple poles H H % umertor coeffciets b = [, 0, ]; % deomitor coefficiets = [, 4, 4]; % prtil frctio expsio [r, p, k] = residueb, r = [-0.5,.5], p =[-, -], k = 0 r r 4 4 0.5.5 p p [ ] u h 0.5 u.5 u