Chapter 7: The z-transform. Chih-Wei Liu

Chapter 7: The -Trasform Chih-Wei Liu

Outlie Itroductio The -Trasform Properties of the Regio of Covergece Properties of the -Trasform Iversio of the -Trasform The Trasfer Fuctio Causality ad Stability Determiig Frequecy Respose from Poles & Zeros Computatioal Structures for DT-LTI Systems The Uilateral -Trasform

Itroductio The -trasform provides a broader characteriatio of discrete-time LTI systems ad their iteractio with sigals tha is possible with DTFT Sigal that is ot absolutely summable -trasform Two varieties of -trasform: Uilateral or oe-sided Bilateral or two-sided DTFT The uilateral -trasform is for solvig differece equatios with iitial coditios. The bilateral -trasform offers isight ito the ature of system characteristics such as stability, causality, ad frequecy respose. 3

A Geeral Complex Expoetial Complex expoetial = re j with magitude r ad agle r cos( ) jr r: dampig factor : siusoidal frequecy si( ) Re{ }: expoetial damped cosie Im{ }: expoetial damped sie < 0 expoetially damped cosie expoetially damped sie is a eigefuctio of the LTI system 4

Eigefuctio Property of 5 Trasfer fuctio H() is the eigevalue of the eigefuctio Polar form of H(): H() = H()e j () LTI system, h[] x[] = y[] = x[] h[] ) ( ] [ ] [ ] [ ] [ ] [ ] [ ] [ H h h x h x h y H() amplitude of H(); () phase of H() The Let = re j The LTI system chages the amplitude of the iput by H(re j ) ad shifts the phase of the siusoidal compoets by (re j ). h H. j e H y si. cos j j j j re r re H j re r re H y

The -Trasform x Defiitio: The -trasform of x[]: Defiitio: The iverse -trasform of X(): x x, j d. 6 A represetatio of arbitrary sigals as a weighted superpositio of eigefuctios with = re j. We obtai H ( re Hece h j ) j h[ ] r e h[ ]( re j ) j j r H re e d h[ ] r = re j d = jre j d d = (/j) d DTFT H ( re j -trasform is the DTFT of h[]r h[ ] j H ( re H ( ) j ) )( re d j ) d

Covergece of Laplace Trasform -trasform is the DTFT of x[]r A ecessary coditio for covergece of the -trasform is the absolute summability of x[]r : The rage of r for which the -trasform coverges is termed the regio of covergece (ROC). Covergece example: x r.. DTFT of x[]=a u[], a>, does ot exist, sice x[] is ot absolutely summable.. But x[]r is absolutely summable, if r>a, i.e. ROC, so the -trasform of x[], which is the DTFT of x[]r, does exist. 7

The -Plae, Poles, ad Zeros 8 To represet = re j graphically i terms of complex plae Horiotal axis of -plae = real part of ; vertical axis of -plae = imagiary part of. Relatio betwee DTFT ad -trasform: -trasform X(): the DTFT is give by the -trasform evaluated o the uit circle pole; ero c = eros of X(); d = poles of X(). j e j e The frequecy i the DTFT correspods to the poit o the uit circle at a agle with respect to the positive real axis. 0 0 N N M M a a a b b b. ~ N M d c b 0 0 / gai factor b b a

Example 7. Right-Sided Sigal Determie the -trasform of the sigal x u. Depict the ROC ad the locatio of poles ad eros of X() i the -plae. <Sol.> By defiitio, we have u. 0 This is a geometric series of ifiite legth i the ratio α/; X() coverges if α/ <, or the ROC is > α. Ad,,,. There is a pole at = α ad a ero at = 0 Right-sided sigal the ROC is > α. 9

Example 7.3 Left-Sided Sigal Determie the -trasform of the sigal y u. Depict the ROC ad the locatio of poles ad eros of Y() i the -plae. <Sol.> By defiitio, we have 0 Y Y u,,, 0 Y() coverges if /α <, or the ROC is < α. Ad,. There is a pole at = α ad a ero at = 0 Left-sided sigal the ROC is < α. Examples 7. & 7.3 reveal that the same -trasform but differet ROC. This ambiguity occurs i geeral with sigals that are oe sided

Properties of the ROC. The ROC caot cotai ay poles If d is a pole, the X(d) =, ad the -trasform does ot coverge at the pole. The ROC for a fiite-duratio x[] icludes the etire -plae, except possibly =0 or = For fiite-duratio x[], we might suppose that x. X() will coverge, if each term of x[] is fiite. ) If a sigal has ay oero causal compoets, the the expressio for X() will have a term ivolvig for > 0, ad thus the ROC caot iclude = 0. ) If a sigal has ay oero ocausal compoets, the the expressio for X() will have a term ivolvig for < 0, ad thus the ROC caot iclude =. 3. x[]=c[] is the oly sigal whose ROC is the etire -plae Cosider x. If 0, the the ROC will iclude = 0. If a sigal has o oero ocausal compoets ( 0), the the ROC will iclude =.

Properties of the ROC 4. For the ifiite-duratio sigals, the x x x. The coditio for covergece is X() <. We may write That is, we split the ifiite sum ito egative- ad positive-term portios: x ad x. 0 X ( ) Note that If I () ad I + () are fiite, the X() is guarateed to be fiite, too. A sigal that satisfies these two bouds grows o faster tha (r + ) for positive ad (r ) for egative. That is, x[ ] A ( r ), 0 (7.9) x[ ] A ( r ), 0 (7.0)

If the boud give i Eq. (7.9) is satisfied, the r A r A A r I () coverges if ad oly if < r. If the boud give i Eq. (7.0) is satisfied, the r A r A 0 0 I + () coverges if ad oly if > r +. = Hece, if r + < < r, the both I + () ad I () coverge ad X() also coverges. Note that if r + > r, the the ROC = For sigals x[] satisfy the expoetial bouds of Eqs. (7.9) ad (7.0), we have (). The ROC of a right-sided sigal is of the form > r +. (). The ROC of a left-sided sigal is of the form < r. (3). The ROC of a two-sided sigal is of the form r + < < r. 3

A right-sided sigal has a ROC of the form > r +. A left-sided sigal has a ROC of the form < r. A two-sided sigal has a ROC of the form r + < < r. 4

Example 7.5 Idetify the ROC associated with -trasform for each of the followig sigal: / /4 ; / y u (/4) u x u u w / u (/4) u. <Sol.> 0.. 04 0 04 The first series coverge for </, while the secod coverge for >/4. So, the ROC is /4 < < /. Hece,. 5 Y, 4 4. 0 9 ; Poles at = / ad = /4 The first series coverge for >/, while the secod coverge for > /4. Hece, the ROC is > /, ad Y, Poles at = / ad = /4 4

0 0 4, 4 3. W 0 0 The first series coverge for < /, while the secod coverge for < ¼. So, the ROC is < /4, ad W 4, Poles at = / ad = /4 This example illustrates that the ROC of a two-side sigal (a) is a rig, i.e. i betwee the poles, the ROC of a right sided sigal (b) is the exterior of a circle, ad the ROC of a left-sided sigal (c) is the iterior of a circle. I each case the poles defie the boudaries of the ROC. 6

Properties of the -Trasform 8 Most properties of the -trasform are aalogous to those of the DTFT. Assume that Liearity: Time Reversal: Time Shift: Z, with ROC x Z, with ROC y x R y Y R Z, with ROC at least x y ax by a by R R The ROC ca be larger tha the itersectio if oe or more terms i x[] or y[] cacel each other i the sum. x [ ] X (/ ), with ROC / Time reversal, or reflectio, correspods to replacig by. Hece, if R x is of the form a < < b, the ROC of the reflected sigal is a < / < b, or /b < < /a. Z 0 x0, with ROC Rx, except possibly 0 or. Multiplicatio by o itroduces a pole of order o at = 0 if o > 0.. If o < 0, the multiplicatio by o itroduces o poles at. If they are ot caceled by eros at ifiity i X(), the the ROC of ox() caot iclude <. R x

Properties of the -Trasform Multiplicatio by a Expoetial Sequece: x, with ROC Rx Is a complex umber.if X() cotais a factor (d ) i the deomiator, so that d is pole, the X(/) has a factor (d ) i the deomiator ad thus has a pole at d.. If c is a ero of X(), the X(/) has a ero at c. The poles ad eros of X() have their radii chaged by i X(/), as well as their agles are chaged by arg{} i X(/) X() X(/) 9

Properties of the -Trasform Covolutio: Z, with ROC at least x y x y Y R R Covolutio of time-domai sigals correspods to multiplicatio of -trasforms. The ROC may be larger tha the itersectio of R x ad R y if a pole-ero cacellatio occurs i the product X()Y(). Differetiatio i the -Domai: d x R d Z, with ROC x Multiplicatio by i the time domai correspods to differetiatio with respect to ad multiplicatio of the result by i the -domai. This operatio does ot chage the ROC. 0

Example 7.6 Suppose 3 Z x u u, Z y u u Y 4 Evaluate the -trasform of ax[] + by[]. <Sol.> Z ax by a b 4 4 3 4 4 3,. ad I geeral, the ROC is the itersectio of idividual ROCs However, whe a = b: We see that the term (/) u[] has be caceled i ax[] + by[]. a ay a. 3 4 5 4 The pole at =/ is caceled!! the ROC elarges

Example 7.7 Fid the -trasform of the sigal x u u <Sol.> 4 u, with ROC First, we ow that Apply the -domai differetiatio property, we have d w u W, with ROC d Next, we ow that Apply the time-reversal property, we have y Z u [ ], with ROC 4 /4 4, with ROC 4 Z Y Last, we apply the covolutio property to obtai X(), i.e. Z, with ROC w y x w y W Y R R 4 4, X( ), with ROC 4,

Example 7.8 Fid the -trasform of x a cos 0u, where a is real ad positive. <Sol.> Let y[] = α u[]. The we have the -trasform Y, a Now we rewrite x[] as the sum with ROC a. j 0 j0 x e y e y The, apply the property of multiplicatio by a complex expoetial, we have j 0 j0 Y e Y e, with ROC a j0 j0 ae ae j0 j0 ae ae j 0 j 0 ae ae acos 0, acos a with ROC a. 3 0

Iversio of the -Trasform Direct evaluatio of the iversio itegral for iversio of the -trasform requires the complex variable theory We apply the method of partial-fractio expressio, based o -trasform pairs ad -trasform properties, to iverse trasform. The iverse trasform ca be obtaied by expressig X() as a sum of terms for which we already ow the time fuctio, which relies o the property of the ROC A right-/left- sided time sigal has a ROC that lies outside/iside the pole radius M B b0 b bm a ratioal fuctio of N A a a a M < N Suppose that 4 0 If M N, we may use log divisio to express X() as. Usig the time-shift property ad the pair We obtai M N 0 f [ ] M N 0 f N Z [ ] X ( ) ~ B( ) A( ) M N f 0

Iversio by Partial-Fractio Expasio. Factor the deomiator polyomial as a product of pole factors to obtai 5 b 0 b a N 0 b Case I, If all poles d are distict: N Case II: If a pole d i is repeated r times: Ai Ai A ir,,, d d d d M M, M < N. A d A A d u, with ROC d d A A d u, with ROC d d or r i i. If the ROC is of the form > d i, the the right-sided iverse -trasform is chose: i m m Z A A d u, with ROC d! m d i Appedix B!! i

Iversio by Partial-Fractio Expasio Case II: If a pole d i is repeated r times:. If the ROC is of the form < d i, the the left-sided iverse -trasform is chose: i m m A A d u, with ROC d! m d i i Example 7.9 Fid the iverse -trasform of, with ROC <Sol.> By partial fractio expasio, we obtai 6. right-sided left-sided x u u u.

Example 7.0 7 Fid the iverse -trasform of 3 0 4 4, with ROC 4 <Sol.> Sice 3>,, 4 4 0 3 First, covert X() ito a ratio of polyomials i. 5 3 3 5 3 4 4 0 3 3, with ROC W ) ( W left-sided. 3 3 u u w w[] Fially, apply the time-shift property to obtai w x. 3 3 u u x

Remars Causality, stability, or the existece of the DTFT is sufficiet to determie the iverse trasform Causality: If a sigal is ow to be causal, the the right-sided iverse trasforms are chose Stability: If a sigal is stable, the it is absolutely summable ad has a DTFT. Hece, stability ad the existece of the DTFT are equivalet coditios. I both cases, the ROC icludes the uit circle i the -plae, =. The iverse - trasform is determied by comparig the locatios of the poles with the uit circle. If a pole is iside the uit circle, the the right-sided iverse -trasform is chose; if a pole is outside the uit circle, the the left-sided iverse - trasform is chose 8

Iversio by Power Series Expasio 9 Oly oe-sided sigal is applicable! Express X() as a power series i or i. Ex 7. Fid the iverse -trasform of. If the ROC is < a, the we express X() as power series i, so that we obtai a right-sided sigal.. If the ROC is > a, the we express X() as power series i, so that we obtai a left-sided sigal., with ROC... 3 3 3... 3 X... 3 x This may ot lead to a closed-form expressio!!

Example 7. A advatage of the power series approach is the ability to fid iverse -trasforms for sigals that are ot a ratio of polyomials i. Fid the iverse -trasform of X( ) e, with ROC all except <Sol.> Usig the power series represetatio for e a : Thus ( ) ( ) X ( ) 0!!! [ ] [ x[ ] [ ]!! x[ ] ( 0, 0 or odd, )! otherwise e a a 0! 4] 30

The Trasfer Fuctio 3 Recall that the trasfer fuctio H() of a LTI system is If we tae the -trasform of both sides of y[], the From differece equatio: This defiitio applies oly at values of for which X[] 0 After Substitutig for x[] ad H() for y[], we obtai ratioal trasfer fuctio Kowledge of the poles d, eros c, ad factor completely determie the system LTI system, h[] x[] = y[] = x[] h[] = re j h H X H Y X Y H M K K x b y a 0 0 M K N K b H a 0 0 ) ( ) ( ) ( ~ ) ( d c b H N M N M a b H 0 0 ) ( 0 0 ~ a b b

Example 7.3 Fid the trasfer fuctio ad impulse respose of a causal LTI system if the iput to the system is x (/ 3) u ad the output is y 3( ) u (/ 3) u <Sol.> The -trasforms of the iput ad output are respectively give by 3 X ( ), ROC 3 ad Y ( ), ROC ( 3) ( 3) Hece, the trasfer fuctio is The impulse respose of the system is obtai by fidig the iverse -trasform of H(). Applyig a partial fractio expasio to H() yields H( ), with ROC (/3) 33 4( ( 3) ) H ( ), ( )( ( 3) ) h ROC ( ) u (/ 3) u

Examples 7.4 & 7.5 Determie the trasfer fuctio ad the impulse respose for the causal LTI system described by y ( / 4) y (3/ 8) y x x <Sol.> We first obtai the trasfer fuctio by taig the -trasform: H( ) H ( ) (/4) (3/8) (/ ) (3/ 4) The system is causal, so we choose the right-side iverse -trasform for each term to obtai the followig impulse respose: <Sol.> h ( / ) u (3/ 4) u Fid the differece-equatio descriptio of a LTI system with trasfer fuctio 5 H ( ) 3 We rewrite H() as a ratio of polyomials i : y 5 H ( ) ( 3 ) 3y[ ] y[ ] 5x x 34

Causality ad Stability The impulse respose h(t) is the iverse LT of the trasfer fuctio H() Causality right-sided iverse -trasform Expoetially decayig term Pole d iside the uit circle, i.e., d < Stability the ROC icludes uit circle i -plae Expoetially icreasig term Causal system Pole d iside the uit circle, i.e., d < 35 No-caucal system Pole d outside the uit circle, i.e., d >

Causal ad Stable LTI System To obtai a uique iverse trasform of H(), we must ow the ROC or have other owledge(s) of the impulse respose The relatioships betwee the poles, eros, ad system characteristics ca provide some additioal owledges Systems that are stable ad causal must have all their poles iside the uit circle of the -plae: 36

Example 7.6 A LTI system has the trasfer fuctio H ( ) j j 4 4 0.9e 0.9e Fid the impulse respose, assumig that the system is (a) stable, or (b) causal. (c) Ca this system both stable ad causal? <Sol.> a. If the system is stable, the the ROC icludes the uit circle. The two cojugate poles iside the uit circle cotribute the right-sided term to the impulse respose, while the pole outside the uit circle cotributes a left-sided term. j j 4 4 h( ) (0.9e ) u[ ] (0.9e ) u[ ] 3( ) u[ ] 4(0.9) cos( ) u[ ] 3( ) u[ ] 4 b. If the system is causal, the all poles cotribute right-sided terms to the impulse respose. j j 4 4 h( ) (0.9e ) u[ ] (0.9e ) u[ ] 3( ) 4(0.9) cos[ ] u[ ] 3( ) u[ ] c. the 37 LTI system caot 4 be both stable ad causal, sice there is a pole outside the uit circle. u[ ] 3

Example 7.7 The first-order recursive equatio y[ ] y[ ] x[ ] may be used to describe the value y[] of a ivestmet by settig ρ=+r/00, where r is the iterest rate per period, expressed i percet. Fid the trasfer fuctio of this system ad determie whether it ca be both stable ad causal. <Sol.> The trasfer fuctio is determied by usig -trasform: H ( ) This LTI system caot be both stable ad causal, because the pole at = ρ > 38

Iverse System Give a LTI system with impulse respose h[], the impulse respose of the iverse system, h iv [], satisfies the coditio h iv * h H iv ( ) H ( ) or H iv ( ) H ( ) the poles of the iverse system H iv () are the eros of H(), ad vice versa a stable ad causal iverse system exists oly if all of the eros of H() are iside the uit circle i the -plae. A (stable ad causal) H() has all of its poles ad eros iside the uit circle H() is miimum phase. 39 A omiimum-phase system caot have a stable ad causal iverse system.

Example 7.8 40 A LTI system is described by the differece equatio Fid the trasfer fuctio of the iverse system. Does a stable ad causal LTI iverse system exist? ] [ 8 ] [ 4 ] [ 4 ] [ x x x y y y <Sol.> The trasfer fuctio of the give system is 4 4 8 ) ( ) )( ( ) ( H Hece, the iverse system the has the trasfer fuctio Both of the poles of the iverse system, i.e. =/4 ad =/, are iside the uit circle. The iverse system ca be both stable ad causal. Note that this system is also miimum phase, sice all eros ad poles of the system are iside the uit circle. ) )( ( ) ( ) ( 4 H iv

Example 7.9 Recall a two-path commuicatio chael is described by y x[ ] ax[ ] Fid the trasfer fuctio ad differece-equatio descriptio of the iverse system. What must the parameter a satisfy for the iverse system to be stable ad causal? <Sol.> First fid the trasfer fuctio of the two-path multiple chael system: H ( ) a The, the trasfer fuctio of the iverse system is ( ) H ( ) a The correspodig differece-equatio represetatio is y H iv ay x The iverse system is both stable ad causal whe a <. 4