2.1.1 Definition The Z-transform of a sequence x [n] is simply defined as (2.1) X re x k re x k r

Transcription

1 Z-Trsforms. INTRODUCTION TO Z-TRANSFORM The Z-trsform is coveiet d vluble tool for represetig, lyig d desigig discrete-time sigls d systems. It plys similr role i discrete-time systems to tht which Lplce trsform plys i cotiuous time systems. I this uit the mi objective is to preset importt cocepts of the Z-trsform d their pplictio i fidig the stbility of the discrete time systems. Z-trsform is powerful tool for determiig the trsfer fuctio of system. This is useful to study bout stbility of system with respect to pole-ero ptter i -ple. The frequecy respose of system c be obtied by replcig by e jw withi the trsfer fuctio of the system... Defiitio The Z-trsform of sequece x [] is simply defied s ZÈÎx[ ] X -  x[ ] (. - d the iverse Z-trsform is defied s - - Z ÈÎX x[ ] X d pj C (. jw - jw  [ ]  [ ] < (.3 X e x e x - - jw  [ ] ( X x re - jw jw  [ ]  [ ] < (ROC X re x re x r - -. RELATION BETWEEN Z-TRANSFORM AND FOURIER TRANSFORM As we ow -  x[ ] - X

2 3 Digitl Sigl Processig - Â ( It is obvious tht the regio of covergece for the Z-trsform of d [] is the etire -ple. Tble. List of Z-Trsform pirs S. No. x[] for 0 X( ROC. [] Etire -ple. [ m] m All, except 0 (if m > 0 or (if m < 0 3. u [] 4. u [] 5. u [ ] - > > < 6. e T x [] X (e T ROC of X ( Â X ( H ( Itersectio of ROC of X ( d ROC of H ( 7. xmh [ ] [ - m] m 0 Tble. Properties of Z-trsform S. No. Properties Sequece Z-trsform ROC. Lierity x [] + bx [] X ( + bx ( Cotis R «R. Time-shiftig x [ 0 ] e m X ( R x expect for the possible dditio or deletio of the origi or ifiity 3. Multiplictio by expoetil sequece 4. Differetitio x [] x [] X ( Scled versio of R (i.e., R the set of poits { } for i R d - X d 5. Cojugte x * [] X * ( R 6. Time reversl x [ ] X ( R 7. Covolutio x [] ƒ x [] X ( X ( R «R 8. First differece x [] x [ ] ( X ( At lest the itersectio of R d > 0 R

3 Z-Trsforms 33.4 ROC AND ITS PROPERTIES The Z-trsform does ot coverge for ll vlues of. For y give sequece, the set of vlues of for which the Z-trsform coverges is clled regio of covergece (ROC, which is govered by coditio: Defiitio: X If r 0 i ROC, the:  [ ] - x r - Right-hded: x[] 0 " < N : If r > r 0, the: - < -  [ ]  [ ] X x x r < [ ]  x N  x [ ] N r 0 <  x [ ] r  x [ ] r0 N N the r lso i ROC, the ROC is exterior of circle Defiitio: - < < -  [ ]  [ ] X x x r < - Left-hded: x[] 0 " > N : N X  x[ ] - If r 0 i ROC, the: If r < r 0, the: - N  - [ ] x r < N N  x [ ] r  x [ ] r0 - - < < the r lso i ROC, the ROC is iterior of circle Properties of ROC for Z-trsform:. The ROC of X ( cosists of rig i the -ple cetered bout the origi. The ROC does ot coti y poles

4 34 Digitl Sigl Processig 3. If x [] is of fiite durtio, the ROC is the etire -ple, except possibly 0 d Fig.. Uit circle i -ple. Exmple. Obti the Z-trsform of x ( m Solutio: By defiitio of Z-trsform - ( - ( - ZÈÎx m  x m m let m substitutio of the bove qutity yields -m ZÈx m x ( - Î Â.5 TRANSFER FUNCTION 0 -m  x( 0 -m X (.7 Let the DTLTI system be chrcteried by the followig differece equtio Trsfer fuctio of the system p [ ]  ( - - ( - (.8 y x b y H 0 Y X + p 0 q I y trsfer fuctio of Discrete Time Lier Time Ivrit (DTLTI system, if is replced by e jw the required frequecy respose of the system c be obtied.   q b (.9

5 jw H e ROC: rig without poles iside. p  + 0 q  b - jw - jw Z-Trsforms 35 (.0 Exmple. Obti the trsfer fuctio of the followig differece equtio d obti its frequecy respose. y 0.5y- + x+ x- Solutio: Tig Z-trsform o both the sides - - Y 0.5 Y + X + X The trsfer fuctio H Y + X The frequecy respose c be obtied by replcig by e jwt jwt H e + e - 0.5e Amplitude respose c be represeted s - jwt - jwt + coswt - jsi wt - 0.5cos w T jsi w T ( cos si ( cos w T + ( 0.5 si wt jw T + w T + wt H e The phse respose c be represeted s si 0.5 si t T t T j w T - w - w H e - + coswt -0.5coswT where T is smplig itervl T fs fs is smplig frequecy Exmple.3 Obti the Z-trsform of the followig fuctio d fid its ROC.  0 Solutio: x [ ] u[ ] <  [ ]  Â( fi < > X x for (

6 36 Digitl Sigl Processig X fi oe ero t 0 d oe pole t ( - Fig.. Uit circle i -ple showig Regio of Covergece (ROC. Exmple.4 Obti the Z-trsform of the followig fuctio d fid its ROC Â 0 < Solutio: x[ ] - u[ --l] fi X Â x[ ] Â - Â -( - - -l- - l -fi -l l l ( - X - Â - Â - - for < - l X for < ( l - Fig..3 Uit circle i -ple showig Regio of Covergece (ROC. Exmple.5 Obti the Z-trsform of the followig fuctio [ ] 0.5 [ ] + (-0.3 [ ] x u u

7 Z-Trsforms 37 x[ ] + by[ ] X + by, ROC Rx «Ry [ ] u, ROC > ( - - X +, : ROC > - «> - + Solutio: ( ( ( ( X, ROC: > X, ROC: > ( ( ( - 0. ( - 0.5( X, ROC: > 0.5 Exmple.6 Obti the Z-trsform of the followig fuctio [ ] -0.5 [- - ] + (-0.3 [ ] x u u x[ ] + by[ ] X + by, ROC Rx «Ry [ ] u, ROC > [ ] ( - - -u--, ROC < ( - - X +, : ROC > - «< - + Solutio: ( ( ( ( X, ROC: 0.3< < X, ROC: 0.3 < ( ( ( - 0. ( - 0.5( X, ROC: 0.3< < 0.5 It hs poles d eros

8 38 Digitl Sigl Processig.6 INVERSE Z-TRANSFORM.6. Power Series.6. Iversio Itegrl - + ( +  (- X log +, ROC: > - - log - X  x[ ] - + [ ] x - u[ -] A powerful lyticl method determiig the iverse Z-trsform is the iversio itegrl method. The fuctio Y ( c be cosidered i the complex -ple. A give coefficiet i such series my be determied by itegrl reltioship. It c be show tht pplictio of this cocept to y ( yields for the iverse trsform. - y y d (. p j Ú c where c is cotour chose to iclude ll sigulrities of the itegrd. By Cuchy s residue theorem the itegrl c be reduced to y  Re sè Îy - (. p m m where p m represets pole of y ( d Res [] represets the residue t p m Exmple.7 Fid the iverse Z-trsform of y Solutio: y - - ( -( -0.5 This c be expressed s + È y  Re sí m Î( -( -5 pm For the poles t d 0.5, the residues re clculted s follows + + È È Re s Í Í Î( -( Î - + È + È Re s Í Í Í( -( -0.5 Î- 0.5 Î 0.5

9 y -( 0.5 Exmple.8 Determie the iverse trsform of Y Z-Trsforms 39 Solutio: Note tht the mximum egtive power of i the umertor is lrger th for the deomitor. Multiplictio of the umertor d the deomitor by 3 results i Y ( -( È y  Re sí m Î( -( -5 pm Accordig to eq. (., we my determie the iverse trsform from y  m ( + + È Re sí Í Î 3 - ( -( -0.5 We must exmie to see if there re y vlues of for which there is pole t the origi. Ideed, for 0 there is secod-order pole t 0, d for there is simple pole t 0. However, for, the oly poles re d 0.5. Let us first determie the iverse trsform pertiet to this ltter rge. We hve y Re s[ ] + Re s [ ] ( 0.5 for (.3 The vlues of y (0 d y ( c be determied from the expressios ( È y( 0  Res Í m Í ( -( -0.5 (.4 Î pm [ ] [ ] [ ] Re s + Re s + Re s ( + + È y(  Res Í m Í Î 3 - ( -( -0.5 [ ] [ ] [ ] Re s + Re s + Re s ( The reder is ivited to demostrte tht the sum of the lst two residues i ech of equtios (.4 & (.5 is the sme s would be obtied by tig eq. (.3 d evlutig it for 0 d respectively. Thus, isted of performig complete evlutio of ll the residues for 0 d, it is ecessry oly to determie the dditiol residues t 0 i ech cse. For eq. (.3, we hve

10 40 Digitl Sigl Processig For eq. (.5 we hve 3 ( + + È Re s Í 6 Í ( -( -0.5 Î 3 ( È Re s Í Í ( -( -0.5 Î This gives y( y( ( For, the expressio of eq. (.3 is pplicble. A lterte wy to write y ( for 0 i oe expressio is the equtio y 6 d + d A few vlues re tbulted i the followig Tble.3. Tble y ( Exmple.9 Fid the iverse Z-trsform of the followig X, > X, < Solutio: From the equtios u[ ], > -u [-- ], < Solutio usig the bove equtios by visulitio ʈ ʈ x Á u x u Á [ ] [ ] [ ] - [- -].6.3 Study of Some Exmples Usig Prtil Frctio Expsio X q q q -  0 ( p-q p p p - 0  0 ( b b -c -c -d -d 0 b0 0 (

11 Z-Trsforms 4 Prtil Frctio Expsio: q < p, Simple Roots X q - ( - c p A p  - - ( - d - d X, > x [ ] u[ ] - - X, < x - u [ ] [ ] - (( A -d X d Prtil Frctio Expsio: q < p, Simple Roots X b± b -4c - ± ± 5 i X X ( - ( + 3 A + A - - ( - ( ( - ( - ( ( - ( ( - ( - ( ( A ( X ( ( ( A ( X ( X /5 4/5 + - ( - - ( + 3 X x u - -, > [ ] [ ] 3 X, < x [ ] - u[ --] x u 3 u Ê ˆ > fi + - Á + -3 u [ ] [ ] [ ] [ ] 4 Ê 4 ˆ < fi x - u u -- - Á + -3 u [ ] [ ] [ ] [ ]

12 Z-Trsforms 43 3 A 4 - ( - ( ( -3 Ê3ˆ ʈ ʈ 3 4 Á 4 Á Á 4 4 Prtil Frctio Expsio: q < p, Simple Roots -/ / 3 / X X x u - -, > [ ] [ ], < [ ] - [- -] X x u ROC: < < 3 Ê ˆ Ê 7ˆ Ê3ˆ x Á u u u u 6 Á Á 3 [ ] - [ ] + ( 4 [ ] + - (-3 [- - ] + (-4 [- -] ˆ Ê3 ˆ Ê x Á u u 6 Á 3 [ ] - [ ] + - [- -] Prtil Frctio Expsio: q > p X , > X +, > A A X Ê - ˆ - ( - Á ± - ± ± 4 4 poles È A Í - Î - - / È - Í A Í 8 - Í / - Î

13 44 Digitl Sigl Processig.7 Z-DOMAIN STABILITY x Ê ˆ Á u u [ ] d[ ]- 9 [ ] + 8 [ ].7. Stbility A system is sid to be stble if it produces bouded output for bouded iput (BIBO A system is sid to be stble if its impulse respose vishes fter sufficietly log time. h [] Æ 0 s Æ Fig..4 Uit circle i -ple. System is relible system is stble Ÿ system is cusl  [ ] ( jv stble h < H e coverges is i ROC cusl h [ ] 0 for < 0 h [ ] right-hded ROC is exterior of circle: > relible ROC > d icludes uit circle ll poles iside uit circle.7. Stbility of DTLTI System As i the cse of cotiuous-time system discrete-time system is sid to be stble if every fiite iput produces fiite output. The stbility cocept my be redily expressed by coditios reltig to the impulse respose h (. These coditios re:

14 46 Digitl Sigl Processig Solutio: ( Tig the Z-trsforms of both sides of the give system differece equtio d solvig for H (, we obti - Y + H X The poles d eros re best obtied by mometrily rrgig umertor d deomitor polyomils i positive powers of. + ( + H The poles re locted t d 0.5, which re iside the uit circle. Thus, the system is stble. (b The impulse respose my be obtied by expdig H ( i prtil frctio expsio ccordig to the procedure of the precedig sectio. This yields H Iversio of the bove equtio yields h It c be redily see tht the impulse respose h ( vishes fter sufficietly log time s expected, sice this is stble trsfer fuctio. (c To obti the respose due to x (, we multiply X ( by H ( d obti ( + Y Prtil frctio expsio yields Y The iverse trsform is Y SOME TYPICAL EXAMPLES ON Z-TRANSFORM Exmple. Fid x [] for the followig system trsfer fuctio. X

15 Z-Trsforms 47 Solutio: X [ ] x Exmple.3 Fid Z-trsform of È Í Z Í Í- - Î - ʈ ʈ Á u + - Á [ ] u[ ] Ê ˆ Á È [ ] + [ - ] Îu u ʈ Á È u[ ] + u[ - ] - u[ - ] Î Ê ˆ Á È [ ] - [ - ] + [ - ] Îu u u ʈ x [ ] Á Èd [ ] + u[ - ] Î [ ] ( u[ -] x - Solutio: ZÈu[ ] Î - Z ÈÎu[ - ] Z È Î u[ - ] - - (

16 Z-Trsforms 49 u > - - Solutio: [ ] Usig the differetitio property of Z-trsform (refer Tble., Property 4 d u[ ] Ê ˆ - Á- d - - > - ( - Exmple.6 Fid x [] of the followig fuctio usig covolutio property of Z-trsform. X Ê - ˆ Ê - ˆ Á- + Á 4 Solutio: X X X X - ʈ x Á X - [ ] u[ ] x Ê ˆ Á u 4 [ ] - [ ] Usig covolutio property of Z-trsform (refer Tble., Property 7 x x * x [ ] [ ] [ ]  0 x x - ʈ Ê-ˆ  Á Á 4 0 ʈ ʈ Ê-ˆ Á  Á Á 4 0 ʈ Ê ˆ ʈ Á - Á Á

17 50 Digitl Sigl Processig Exmple.7 coditios. Solutio: For For For < 4 > 3 ʈ Ê-ˆ Á Á  0 + ʈ Ê Ê ˆ ˆ Á - - Á Á - Ê ˆ - -Á- Èʈ ʈ Ê ˆ Í Á - Á Á- ÍÎ Èʈ ʈ Ê ˆ Ê ˆ Í Á - Á Á- Á- ÍÎ3 3 Èʈ Ê ˆ Í Á + Á- u [ ] ÍÎ3 3 4 Fid iverse Z-trsform of the followig fuctio uder differet ROC < < 4 3 X + Ê -ˆ Ê -ˆ Á- - Á 4 3 ʈ ʈ x Á u u Á 4 3 [ ] [ ]- [- -] ʈ ʈ x Á u u Á 4 3 [ ] - [- -]- [- -] ʈ ʈ x Á u u Á 4 3 [ ] [ ] + [ ] Exmple.8 Fid the Z-trsform of x( cos (w u (. Solutio: Give tht x cos w u From defiitio ÈÎx x  x. - -

18 5 Digitl Sigl Processig We ow tht cos Êe w Á ( jw + e - jw ˆ fi cos jw - jw Êe + e ˆ w Á w - w e + e j j jw - jw ( e + ( e Form the defiitio of Z-trsform, - ÈÎx x  x 0 - \  ( w x.cos. 0 È jw - jw (. e + (. e  0 Í Í Î - j (. w - j + (. - w - e e  0 (. (. j j e w - + e - w -  0 È Í + jw - - jw - Î -e. -e. È j j Í + w - w -e -e Î - jw jw È( - e + ( -e Í jw - jw ÍÎ ( -e ( -e - jw jw È -. e + -. e Í - jw jw Î -. e -. e + jw - jw È Êe + e ˆ - ÍÁ Î È jw - jw È Ê e + e ˆ Í - ÍÁ + ÍÎ Î

19 54 Digitl Sigl Processig A Z - Z - + BZ Z - + C Z - Z - Z + DZ Z A Z - Z - 4Z B Z - 4Z + 4Z + C Z -Z Z - + D Z -Z A Z - 5Z + 8Z B Z - 4Z + 4Z + C Z - 3Z + Z + D Z -Z Comprig, Z 3 -terms fi A + B + C 0 Z -terms fi 5A 4B 3C + D 0 Z-terms fi 8A + 4B + C D 0 Costt fi 4A A - 4 By solvig the bove terms, we get B -3 C 4 D Substitutig A, B, C, D from the bove equtios, the Z Z Z - Z 3Z Z X Z Z - 4 Z - Z - ( -( - 4Z ( Z - 4( Z - ( - fi - Z 3Z Z X ( Z ( Z - 4( Z - ( Z - Applyig the iverse Z-trsform, we get, - 3 x d + u - ( u + ( u Exmple. Fid the iverse Z-trsform of x > Solutio: Give tht, x > (

20 4 7 ( ( Ê Á Z-Trsforms The result is ( \ From the defiitio of Z-trsform, - ÈÎx x  x. - \ ( 0 ( ( 3 x Z x x x x By comprig equtios ( d (, we get, x ( º º ( x (0 4 x ( 3 x ( 38

21 Z-Trsforms 57 Solutio: Ê ˆ X log Á, > - - (i Give tht, - \ X -log -, > The power series expsio for log( p is give s log ( - p - Â p p < The regio of covergece is >, i.e., < \ Power series expsio of X ( is give by, - X Â ( - Â ( \ From bove equtios, x ( c be defied s Ê ˆ Á, for x 0, for 0 u - or x X Z Ê ˆ log Á, < - (ii Give tht - - \ X Z -log -, < The regio of covergece is, < i.e., < \ Power series expsio of X ( is give by, - X Â ( Â - ( - \ From bove equtios, x ( c be defied s x ( 0, for 0 Ê ˆ - Á for (or - u ( - x

22 . Determie the iverse Z-trsform of the followig ( X Z-Trsforms Obti the trsfer fuctio for the system described by differece equtio y + y ( - x 4. Obti the trsfer fuctio of the system described by the differece equtio give by d hece, impulse respose y ( - -4y ( - + 3y ( - 3 x( - + x( - 5. Fid x (0 d x ( for the sequece whose Z-trsform is X - 3 Multiple-Choice Questios. The Z-trsform of the uit rmp is give by ( (b (c (d. The Z-trsform of x (0 is ( X ( (b X (0 (c X ( (d lim X Æ 3. For cusl sigls d systems, the Z-trsform is defied s - ( X Â x (b X Â x (c X Â x - (d X Â x - 4. Regio of covergece is defied s ( Set of -vlues for which the series coverges. (b Set of -vlues for which the series coverges. (c Set of -vlues for which the series diverges. (d Set of -vlues for which the series diverges. 5. If lower limit of the ROC is greter th the upper limit of ROC the series - X Â x -

23 Z-Trsforms 6 3. The Z-trsform of  x - is ( X ( (b X (0 (c X ( (d X ( 4. The system is sid to be stble if d oly if  h - ( Greter th ifiity (c Less th ifiity 5. The Z-trsform of  x l - is (b Equl to ifiity (d Equl to ero X ( X ( (b - - (c X ( (d X ( 6. The Z-trsform of x ( is Ê ˆ Ê ˆ ( X Á (b X Á (c X ( (d X ( 7. Uit smple respose of system is ( h ( [h(] (b h ( [H (] (c h ( [X(] (d h ( [Y (] 8. The system is sid to be cusl if d oly if ( h ( 0 for < 0 (b x ( 0 for < 0 (c y ( 0 for < 0 (d y ( 0 for > 0 9. The system is described by the followig differece equtio y ( y ( x ( If the excittio is the uit impulse, the system trsfer fuctio is ( (b - + (c 0. The iverse Z-trsform of X (d ( - - if X ( coverges bsolutely for some < ( ( + for 0 (b ( + for (c ( + for 0 (d ( + for