Digital Sigal Processig Z-trasform dftwave
-Trasform Backgroud-Defiitio - Fourier trasform j ω j ω e x e extracts the essece of x but is limited i the sese that it ca hadle stable systems oly. jω e coverges if x < i.e., stable system Fourier Trasform coverges - So, we wat to exted it such that it ca be used as a tool to aalye digital systems i geeral.
Let j j r e r x e ω ω the it coverges if Th diti f i l d! < r x The coditio for covergece is relaxed! e.g. u x g e e j j j j ω ω ω ω r r e r e j j r < ω ω coverges if > r
jω -This implies that r e ca hadle some systems that jω e caot due to divergece. - Therefore we defie -trasform to be jω r e re jw x Represetig the coditio for covergece of i terms of regio of covergece RoC. r e jω e.g. i case x u r e jω exists for r >. So, RoC is re jω >.
I geeral, if x a u C R o is > a - I terms of, e jw is a special case Where, or r causal
e.g. u a x a u a a a a u a 0 a a a a a a a C R o < a a < :, or
e.g. Two - sided sequece u u x 3 u u x 0 3 3 + /3 3 > <, / /3
Some Commo -Trasforms δ all u > 3 u < 4 δ m m all except 0, if all except, if m > 0, m < 0 5 a u > a a 6 a u < a a 7 a a u > a a
8 9 0 3
Properties of R o C i geeral 0 r < R C < r R o L e jw absolutely coverges UC R o C 3 R C caot cotai a pole R o 4 FIR sequece etire plae, may be except for 0 or 5 Right-sided sequece outward of the outermost pole 6 Left-sided sequece iward from the iermost pole 7 Two-sided d sequece a rig i betwee two adjacet rigs 8 is a coected regio C R o
e.g. g If x is a sum of 3 sequeces whose poles are a, b, c respectively, There exist A possible R o Cs as show below a b c All right-sided All left-sided a b c two left-sided two right-sided id d
-Trasform Properties Liearity ax + bx a + b Time shiftig e.g. o x o 4 4 > u, 4 4 4 u ad delay -
-Trasform Properties..cot. Trasform Properties..cot. 3 Multiplicatio by a Expoetial Sequece x jw e o o x e.g. o o o w w j jw jw e e x e g e.g. e e x e cos u e e r u w r jw jw o o o + u re re jw jw o o + cos + w r re re o jw jw o > cos + r w r o > r
-Trasform Properties..cot. 4 Differetiatio i i of x d x RoC Rx d e.g. + a log > a d a d + a d a d + a + a a x x a a u a u
-Trasform Properties..cot. Trasform Properties..cot. 5 Cojugatio of Complex Sequece o R R C x * * * 6 Time-Reversal o * * R C x * x R R C x o 7 Covolutio-Itegratio x 7 Covolutio Itegratio * x x
4-Ways: Iverse -Trasform
Iversio by Cotour Itegratio Cauchy itegral defiitio of the iverse - Trasform Example: Iverse DTFT Implies.. Cotour C is chose as uit circle
Iversio Method - Iversio Method Cocept of Partial Fractio Expasio-
Iversio Method Cocept of Partial Fractio Expasio Iversio-. Fid partial fractio expasio method di third equivalet form. Ivert by expasio
Doig the Partial Fractio Expasio
Doig the Partial Fractio Expasio-
Writig Dow x
deped o kowig the ROC
Example- ROC If x is a sum of 3 sequeces whose poles are a, b, c respectively, There exist A possible R o Cs as show below a b c All right-sided a b c two left-sided
Example- Partial Fractio
Log Divisio
Fidig the coefficiets of Poles
Writig Dow x
Partial Fractio Expasio i MATLAB
Selected -Trasform Theorems
A IIR System
IIR Frequecy Respose
System Fuctio Of a Differece Equatio
H ad h
Frequecy Respose of a DE
LTI System Characteriatio
Stability, Causality- illustratio x u 0 RoC : > Outward UC RoC Causal. Stable
Stability, Causality- illustratio..cot x u Ati Causal Ustable ROC : < Iward UC RoC
Stability, Causality- illustratio..cot 3 x u Causal Ustable RoC : 0 > Outward UC RoC
Stability, Causality- illustratio..cot 4 x u Ati Causal Stable RoC : < Iward UC RoC What do you fid?