Chapter 7 -Trasform Itroductio Trasform Uilateral Trasform Properties Uilateral Trasform Iversio of Uilateral Trasform Determiig the Frequecy Respose from Poles ad Zeros
Itroductio Role i Discrete-Time Systems -Trasform is the discrete-time couterpart of the Laplace trasform. Respose of Discrete-Time Systems If the system y[] + 3y[-] + y[-] = u[] + u[-] - u[-] for = 0,, The respose of the system is excited by a iput u[] ad some iitial coditios. The differece equatios are basically algebraic equatios, their solutios ca be obtaied by direct substitutio. The solutio however is ot i closed form ad is difficult to develop geeral properties of the system. A umber of desig techiques have bee developed i the -Trasform domai.
-Trasform 3 Positive ad Negative Time Sequece A discrete-time sigal x[], where is a iteger ragig (- <<, is called a positive-time sequece if x[]=0 for < 0; it is called a egative-time sequece if x[] = 0 for > 0. We maily cosider the positive-time sequeces. -Trasform Pair The -trasform is defied as where is a complex variable, called the -trasform variable. Example X ( Z [ x [ ]] x [ ] x[] = {,, 5, 7, 0, }; x[] = (/ u{}
-Trasform (c. 4 Example f[] = b for all positive iteger ad b is a real or complex umber. If b, the the ifiity power series coverges ad F( b b The regio b is called the regio of covergece. Uit Step Sequece The uit sequece is defied as The -Trasform is Expoetial Sequece F( f [ ] b ( b f[] = e at 0 0 0 F for 0,,, q[ ] 0 for 0 Q( 0 at ( e at 0 e b
-Trasform (c. 5 Regio of Covergece For ay give sequece, the set of values of for which the -trasform coverges is called the regio of covergece. Viewpoits The represetatio of the complex variable re j Cosider the -trasform ROC icludes the uit circle ==> Fourier Trasform coverges j j X( re x[ ]( re Coverget Coditio x[ ] r Covergece of the -Trasform ==> The -trasform ad its derivatives must be cotiuous fuctio of.
Uilateral Trasform 6 7.. The -Plae. If x[] is absolutely summable, the the DTFT obtaied from the -trasform by settig r =, or substitutig = e j ito Eq. (7.4. That is, j j e. e (7.6 3. The equatio = e j describes a circle of uit radius cetered o the origi i the -plae. Fig. 7.4. Uit circle i the -plae. The frequecy i the DTFT correspods to the poit o the uit circle at a agle with respect to the positive real axis.
Uilateral Trasform 7 Determie the -trasform of the sigal x,,,, 0, 0 otherwise Use the -trasform to determie the DTFT of x[]. <Sol.>. We substitute the prescribed x[] ito Eq. (7.4 to obtai.. We obtai the DTFT from X( by substitutig = e j : j j j j e e e e.
Uilateral Trasform 7..3 Poles ad Zeros. The -trasform i terms of two polyomials i :. 0 0 N N M M a a a b b b. ~ N M d c b where 0 0 / gai factor b b a. The c are the roots of the umerator polyomial the eros of X(. The d are the roots of the deomiator polyomial the poles of X(. 3. Symbols i the -plae: poles; eros 8
Uilateral Trasform 9 Example 7. -TRANSFORM OF A CAUSAL EXPONENTIOAL SIGNAL Determie the -trasform of the sigal x u. Depict the ROC ad the locatio of poles ad eros of X( i the -plae. <Sol.>. Substitutig x[] = α u[] ito Eq. (7.4 yields u 0.. This is a geometric series of ifiite legth i the ratio α/; the sum coverges, provided that α/ <, or > α. Hece,,,. (7.7
Uilateral Trasform 0 3. There is thus a pole at = α ad a ero at = 0, as illustrated i Fig 7.5. the ROC is depicted as the shaded regio of the -plae. a
Uilateral Trasform Example 7.3 -TRANSFORM OF A ANTICAUSAL EXPONENTIOAL SIGNAL Determie the -trasform of the sigal y u. Depict the ROC ad the locatio of poles ad eros of X( i the -plae. <Sol.>. We substitutig y[] = -α u[ - -] ito Eq.(7.4 ad write Y u 0.. The sum coverges, provide that / <, or < α. Hece, Y,,, 3. The ROC ad the locatio of poles ad ero are depicted i Fig 7.6. (7.8
Uilateral Trasform,,. Y,,, a
Uilateral Trasform 3 Example 7.4 -TRANSFORM OF A TWO SIDED SIGNAL Determie the -trasform of. u u x Depict the ROC ad the locatio of poles ad eros of X( i the -plae. <Sol.>. Substitutig for x[] i Eq. (7.4, we obtai u u 0. 0 0 /,
Properties of ROC 4 Ratioal Fuctio Ex. X( P( Q( x[ ] a u[ ] x[ ] a u[ ]
Properties of ROC 5 Properties The ROC is a rig or dis i the -plae cetered at the origi, i.e., The Fourier trasform of x[] coverges absolutely if ad oly if the ROC of the -trasform of x[] icludes the uit circle. The ROC caot cotai ay poles. If x[] is a fiite-duratio sequece, i.e. a sequece that is ero except i a fiite iterval N N, the the ROC is the etire -plae except possibly =0 or =. If x[] is a right-sided sequece, i.e. a sequece that is ero for <N <, the ROC exteds outward from the outermost fiite pole i X( to =. If x[] is a left-sided sequece, i.e., a sequece that is ero for >N >-, the ROC exteds iward from the iermost (smallest magitude oero pole i X( to (ad possibly icludig =0. A two-sided sequece is a ifiite-duratio sequece that is either right-sided or left-sided. If x[] is a two-sided sequece, the ROC will cosist of a rig i the -plae, bouded o the iterior ad exterior by a pole, ad, cosistet with property 3, ot cotaiig ay poles. The ROC must be a coected regio. 0 r r R L
Properties 6 of ROC
7 Properties of ROC
Properties Uilateral Trasform 8. Assume that Z, with ROC x x R Z, with ROC y y Y R The ROC is chaged by certai operatios.. Liearity: The ROC ca be larger tha the itersectio if oe or more terms i x[] or y[] cacel each other i the sum. Z, with ROC at least x y ax by a by R R (7. Time Reversal x Z, with ROC R x (7. 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.
Properties Uilateral Trasform 9 Multiplicatio by a Expoetial Sequece. Let be a complex umber. The x,with ROC R. The otatio, R x implies that the ROC boudaries are multiplied by. 3. If R x is a < < b, the the ew ROC is a < < b. 4. 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. 5. If c is a ero of X(, the X(/ has a ero at c. 6. This idicates that the poles ad eros of X( have their radii chaged by, ad their agles are chaged by arg{}. x
The Uilateral -Trasform 0 Defiitio X ( Z [ x [ ]] x [ ] 0 Time Delay x [ ] X ( x [ ] X ( x [ ] i x [ ] X ( x [ ] 0
Figure 7. (p. 569 The effect of multiplicatio by o the poles ad eros of a trasfer fuctio. (a Locatios of poles ad eros of X(. (b Locatios of poles ad eros of X(/. Iversio of Uilateral Trasform Covolutio Z, with ROC at least x y x y Y R R (7.5. 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(.
The Iverse -Trasform Methods Direct Divisio Partial Fractio Expasio Direct Divisio F( = - +3 +3 - + 3-3 + 9-4 f[] = {-, 3, 0, 0, 3, 3, 9,...} Ex. 3/( - - F( 4 3 5 6 3 3 3 3 9 3 4 4 3 5 6 3 4 4 3 3 3 3 6 3 3 3 3 6 3
3 Iverse -trasform by Power Series Expasio
4 C m The Iverse -Trasform (c. Partial Fractio Expasio ad Table Looup X ( M N b0 X ( a ( c ( d If M<N ad the poles are all first order N b0 A X ( a 0 If M>= N ad the poles are all first order, the complete partial fractio expressio ca be M N X ( B r 0 ( c ( d If X( has multiple-order poles ad M>=N M N N s r A X ( Br ( s m!( d i s m s d dw m 0 M N ( d A ( d X ( r r N A ( d s [( d w X ( w ] s m i w d i r 0 d A ( d X ( d, i C ( d ( d m i m m
Trasfer Fuctio 5. Output of LTI system: h x Taig -trasform HX y *. Trasfer fuctio: Y H (7.0 X Y (7.9 This defiitio applies at all i the ROC of X( ad Y( for which X( is oero. x[] LTI system, h[] or H( y[] = x[] h[] Y H X
Trasfer Fuctio 6 Example 7.3 System Idetificatio The problem of fidig the system descriptio from owledge of iput ad output is ow as system idetificatio. 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 <Sol.> 3( u (/3 u. The -trasforms of the iput ad output are respectively give by X ( with ROC / 3 (/ 3 ad 3 Y( (/ 3 4 ( ( (/ 3, with ROC
Trasfer Fuctio 7. We apply Eq.(7.0 the obtai the trasfer fuctio: 4( (/ 3 ( ( (/ 3 H(, with ROC 3. 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 4. The impulse respose is thus give by ( u (/ 3 u h
8 Solvig Differece Equatios with Iitial Coditios Differetiatio i the -Domai d x R d Z, with ROC x (7.6. 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.
Trasfer Fuctio 9 7.6. Relatig the Trasfer Fuctio ad the Differece equatio. Nth-order differece equatio: K 0 a y M b x K 0. The trasfer fuctio H( is a eigevalue of the system associated with the eigefuctio. If x[] =, the the output of a LTI system is y[] = H(. Substitutig x[ ] = H( ito the differece equatio gives the relatioship N K 0 a H( 3. Trasfer fuctio: M K 0 b Y ( H ( X ( M 0 N 0 b a Ratioal trasfer fuctio (7. See page 58, textboo.
Trasfer Fuctio 30 Example 7.4 Fidig the Trasfer Fuctio ad Impulse Respose Determie the trasfer fuctio ad the impulse respose for the causal LTI system described by the differece equatio y ( / 4 y (3/8 y x x <Sol.>. We obtai the trasfer fuctio by applyig Eq.(7.: H( (/ 4 (3/ 8. The impulse respose is foud by idetifyig the iverse -trasform of H(. Applyig a partial-fractio expasio to H( give H ( (/ (3/ 4 3. The system is causal, so we choose the right-side iverse -trasform for each term to obtai the followig impulse respose: ( / u (3/ 4 u h
Trasfer Fuctio 3 Example 7.5 Fidig a Differece-Equatio Descriptio Fid the differece-equatio descriptio of a LTI system with trasfer fuctio 5 H( ( 3 <Sol.>. We rewrite H( as a ratio of polyomials i. Dividig both the umerator ad deomiator, we obtai 5 H( ( 3. Comparig trasfer fuctio with Eq.(7., we coclude that M =, N =, b 0 = 0, b = 5, b =, a 0 =, a = 3, ad a =. Hece, this system is described by the differece equatio y 3y[ ] y[ ] 5x x Trasfer fuctio i pole-ero form: where c eros; d poles; ad b b / a gai factor 0 0 ~ M b ( c H( N ( d (7.
Causality ad Stability 3. The impulse respose of a stable LTI system is absolutely summable ad the DTFT of the impulse respose exists. Pole d iside the uit circle, i.e., d < Left-sided iverse trasform The ROC must icludes the uit circle i the -plae. Expoetially decayig term Pole d outside the uit circle, i.e., d > Right-sided iverse trasform Expoetially icreasig term Fig. 7.5 Note that a stable impulse respose caot cotai ay icreasig expoetial or siusoidal terms, sice the the impulse respose is ot absolutely summable. LTI system are both stable ad causal must have all their poles iside the uit circle. Fig. 7.6
Causality ad Stability 33 Figure 7.5 (p. 583 The relatioship betwee the locatio of a pole ad the impulse respose characteristics for a stable system. (a A pole iside the uit circle cotributes a right-sided term to the impulse respose. (b A pole outside the uit circle cotributes a left-sided term to the impulse respose.
Causality ad Stability 34 Figure 7.6 (p. 584 A system that is both stable ad causal must have all its poles iside the uit circle i the -plae, as illustrated here.
Causality ad Stability 35 7.7. Iverse System. The impulse respose of a iverse system, h iv [], satisfies h iv * h where h[] is the impulse respose of the system to be iverted.. Taig -trasform: H iv ( H( H iv ( H( 3. If H( is writte i pole-ero form show i Eq. (7.3, the H iv ( ~ b p N l M p ( d ( c (7.4 Ay system described by a ratioal trasfer fuctio has a iverse system of this form.
Causality ad Stability 36 Figure 7.8 (p. 586 A system that has a causal ad stable iverse must have all its poles ad eros iside the uit circle, as illustrated here. 4. H iv ( is both stable ad causal if all of its poles are iside the uit circle. Sice the poles of H iv ( are the eros of H (, we coclude that a stable ad causal iverse of a LTI system H( exists if ad oly if all the eros of H( are iside the uit circle. 5. A system with all its poles ad eros iside the uit circle, as illustrated i Fig. 7.8, is termed a miimum-phase system. The phase respose of a miimum-phase system is uiquely determied by the magitude respose, ad vice versa.
Causality ad Stability 37 Example 7.8 Stable ad Causal Iverse System 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.>. We fid the trasfer fuctio of the give system by applyig Eq.(7. to obtai 4 4 8 ( ( ( ( H. The iverse system the has the trasfer fuctio ( ( ( ( 4 H iv
Determiig the Frequecy Respose from Poles ad Zeros Impulse Respose for Ratioal Fuctios M N H ( B r 0 r r N M N N A h[ ] Br [ r ] A d u[ ] d r 0 Ifiite Impulse Repose (IIR Systems The legth of the impulse respose is ifiite. Fiite Impulse Respose (FIR Systems The legth of the impulse repose is fiite. Examples M y [ ] a x [ ] 0 How to chec? M y [ ] ay [ ] x[ ] a x[ M ]
Determiig the Frequecy Respose from Poles ad Zeros A stable liear time-ivariat system Ratioal Fuctio Magitude Respose M j H ( e Gai (db b a 0 0 N c e d e j H ( e j j M j b e b 0 N j 0 a0 a e 0 j ( c e j ( d e H e j ( m is approximately 6m db, while H e j ( 0 m is approximately 0m db j H ( e b a 0 0 M N M N j j c e c * e j j d e d * e M N j b0 j 0 log 0 H ( e 0 log0 0 log0 c e 0 log0 d e a j 0 0 log Y ( e j 0 log H ( e j 0 log X ( e j 0 0 0
Determiig the Frequecy Respose from Poles ad Zeros Phase Respose (c. j H ( e M N b 0 j j [ c e [ d e a 0 The pricipal value of the phase is deoted as ARG[H(e jw ] j ARG [ H ( e ] j H ( e ARG [ H ( e ] r ( Pricipal Values = Sum of Idividual M PVs j 0 ARG H e ARG N j [ ( ] ARG b [ c e a 0 ARG [ d e r ( j j
Determiig the Frequecy Respose from Poles ad Zeros Phase Distortio ad Delay Observatio j h [ ] [ ] H ( e e id d The Ideal Lowpass Filter with liear phase H lp j ( e e 0, j d, c Delay Liear Phase Observatio -- A arrow bad sigal s[]cos( 0 The phase for the 0 ca be approximated as Group Delay-- A measure for the oliearity of the phase c id h lp [ ] y [ ] s[ ]cos( d 0 0 0 j d j ( grd [ H ( e ] H ( e d j d Ideal Filters with Causality? si c ( d, ( d j H ( e d 0 d
Determiig the Frequecy Respose from Poles ad Zeros Phase Respose Alterative relatio j j H I ( e ARG [ H ( e ] arcta j H R ( e Group Delay Derivative of the cotiuous phase fuctio That is j d j d j d j grd [ H ( e ] {arg[ H ( e ]} (arg[ d e ] (arg[ ce ] d d d grd [ H ( e ] N j Ca be obtaied from the priciple values except at discotiuities. N j M d Re{ d e } c Re{ ce } j j d Re{ d e } c Re{ c e } M j
Determiig the Frequecy Respose from Poles ad Zeros Sigle Pole or Zero The form p ( The magitude squared re e ( re e ( re e r r cos( The log magitude i db is The phase Group Delay j j j j j j j j 0 0 0log re e 0log [ r r cos( ] j j r si( ARG re e arcta r cos( j grd re e j r r cos( r r cos( r r cos( j j re e
Determiig the Frequecy Respose from Poles ad Zeros Sigle Pole or Zero Group Delay cos( cos( cos( ( si cos( cos( cos( cos( cos( cos( ( si cos( cos( cos( si( r r r r r r r r r r r r r r r r r r e re grd j j cos( si( arcta r r e re ARG j j
Determiig the Frequecy Respose from Poles ad Zeros Ex. ( p v v 3 3 3
Determiig the Frequecy Respose from Poles ad Zeros Frequecy Resposee for a Sigle Zero at
Determiig the Frequecy Respose from Poles ad Zeros Frequecy Respose for a Sigle Zero ear
Determiig the Frequecy Respose from Poles ad Zeros 48 Relatioship betwee the locatios of poles ad eros i the -plae ad the frequecy respose of the system:. Trasfer fuctio: Substitutig e j for i H(. Assume that the ROC icludes the uit circle. Substitutig = e j ito Eq. (7.3 gives ( ( ~ ( j l N jl j p M jp j e d e e c be e H ( ( ~ ( ( j l N j p M M N j j d e c e be e H (7.5 3. The magitude of H(e j at some fixed value of, say, 0, is defied by ( ( ~ ( 0 j l N j p M j d e c e b e H o e j 0 g product term, where g represets either a pole or a ero.
Determiig the Frequecy Respose 49 from Poles ad Zeros 4. Vector represetatio of e j g: Fig. 7. 9. e j 0 a vector from the origi to e j 0 ; g a vector from the origi to g. We assess the cotributio of each pole ad ero to the overall frequecy respose by examiig e j 0 g as 0 chages. Figure 7.9 (p. 589 Vector iterpretatio of e j 0 g i the -plae.
Determiig the Frequecy Respose 50 from Poles ad Zeros Figure 7.0 (p. 589 The quatity e j g is the legth of a vector from g to e j i the -plae. (a Vectors from g to e j at several frequecies. (b The fuctio e j g. 5. Fig. 7.0(a depicts the vector e j 0 g for several differet values of ; while Fig. 7.0(b depicts e j 0 g as a cotiuous fuctio of frequecy.
Determiig the Frequecy Respose 5 from Poles ad Zeros If = arg{g}, the e j g attais its miimum value of g whe g is iside the uit circle ad taes o the value g whe g is outside the uit circle. Hece, if g is close to the uit circle ( g, the e j g becomes very small whe = arg{g}. 6. If g represets a ero, the e j g cotributes to the umerator of H(e j. Thus, at frequecies ear arg{g}, H(e j teds to have a miimum. 7. If g represets a pole, the e j g cotributes to the deomiator of H(e j. Whe e j g decreases, H(e j icreases, with the sie of the icrease depedet o how far the pole is from the uit circle.
Determiig the Frequecy Respose 5 from Poles ad Zeros Example 7. Magitude Respose from Poles ad Zeros Setch the magitude respose for a LTI system havig the trasfer fuctio <Sol.> H( ( 0.9e j 4 ( 0.9e j 4 The system has a ero at = - ad poles at = 0.9 e jπ/4 ad = 0.9e jπ/4 as depicted i Fig.7.3(a. Hece, the magitude respose will be ero at = ad will be large at = ± /4, because the poles are close to the uit circle. Figures 7.3 (b-(d depict the compoet of the magitude respose associated with the ero ad each poles. Multiplicatio of these cotributios gives the overall magitude respose setched i Fig. 7.3(e.
Figure 7.3a (p. 59 53 Solutio for Example 7.. (a Locatios of poles ad eros i the -plae. (b The compoet of the magitude respose associated with a ero is give by the legth of a vector from the ero to e j. p ; v Sigals_ad_Systems_Simo Hayi & Barry Va Vee
p ; v 54 Figure 7.3a (p. 59 Solutio for Example 7.. (c The compoet of the magitude respose associated with the pole at = e j/4 is the iverse of the legth of a vector from the pole to e j.
55 Figure 7.3b (p. 593, cotiued (d The compoet of the magitude respose associated with the pole at is the iverse of the legth of a vector from the pole to e j. (e The system magitude respose is the product of the respose i parts (b (d. p ; v Sigals_ad_Systems_Simo Hayi & Barry Va Vee
Determiig the Frequecy Respose 56 from Poles ad Zeros The phase of H(e j may also be evaluated i terms of the phase associated with each pole ad ero.. Usig Eq.(7.5, we obtai M P N l j j j arg{ H( e } arg{ b ~ } ( N M arg{ e c } arg{ e d }. The phase of H(e j ivolves the sum of the phase agles due to each ero mius the phase agle due to each pole. Discussio: see pp. 59-594, textboo. Figure 7.5 (p. 593 The quatity arg{e j g} is the agle of the vector from g to e j with respect to a horiotal lie through g, as show here.
Remars Four System Descriptios Impulse Respose Differece Equatios Frequecy Respose Z-Trasform ( Trasfer Fuctio ad System Fuctio x[] LTI Systems y[] M y [ ] x[ ]* h[ ] h[ ] x[ ] N a y [ ] b x [ ] 0 0 Y(e j =H(e j X(e j Y(=H(X(
Remars 58 Itroductio Trasform Uilateral Trasform Properties Uilateral Trasform Iversio of Uilateral Trasform Determiig the Frequecy Respose from Poles ad Zeros