3 4 5 6 7 8 9 10 CHAPTER 3 11 THE Z-TRANSFORM 31 INTRODUCTION The z-trasform ca be used to obtai compact trasform-domai represetatios of sigals ad systems It provides ituitio particularly i LTI system aalysis ad ca be used to calculate digital filter resposes efficietly I this chapter, we review the z-trasform ad its properties We defie the trasfer fuctio of a discretetime LTI system ad we show how to use the iverse z-trasform to calculate the trasiet ad steady-state resposes of a digital filter At the ed of the chapter, we provide several problems o the z-trasform alog with a series of ijdsp mobile simulatio exercises 32 DEFINITION Discrete-time sigals ca be represeted compactly with the z-trasform which is the couterpart of the Laplace trasform The z-trasform of the sigal x( is defied as X ( z x( z, ROC: R (111 x 70
where z is a complex variable ad R x deotes the regio of covergece (ROC For a causal (right-haded sigal, ie, for x(=0 for <0, the lower idex o the sum starts from zero The z-trasform describes the sigal i the z domai ad for fiite-legth causal sigals the trasform coverges everywhere i z except possibly for z=0 I order to defie a uique z-trasform pair, we eed to specify the rage of values of z for which the z-trasform exists, ie, the z- trasform regio of covergece For causal sigals, the ROC exteds outwards from the outermost pole of the z- domai fuctio Example 111: Determie the z-trasform of the uit impulse, ( ( z 1 hece ( 1 ROC : z Example 112: Determie the z-trasform of the fiite-legth sigal i Figure 31 This sigal is o-zero oly for ={-2,-1,0,1,2,3} Mathematically, this sigal ca be writte i terms of uit impulses as follows x( 05 ( 2 15 ( 1 2 ( ( 1 05 ( 2 25 ( 3 The z-trasform of this fiite-legth o-causal sigal is give by 2 1 2 3 X ( z 05 z 15z 2 z 05z 25 z The z-trasform coverges for all values of z except for z = 0 ad z =, ie, the ROC cosists of all values of z except: z = 0 ad z = Example 113: Fid the z-trasform of the decayig expoetial sigal give by 71
, 05 for 0 x ( 0, for 0 The z-trasform of this causal discrete-time sigal is give by, 05 X ( z x( z 05 z z 0 0 0 This is a geometric series sum that for z > 05 coverges to X( z z z 05 Hece, the ROC of X(z i this case is: z >05 Note that this simple ratioal fuctio becomes zero at z=0 ad ifiity at z=05 Therefore the fuctio has a zero at z=0 ad a pole at z=05 x( 15 2 05-2 1 3-1 0 2 4-05 -1-25 Figure 111 The fiite-legth sigal i Example 32 Example 114: Determie the z-trasform of the followig sigal 1 x(, 3 The sigal is show i Figure 32 72
x( 1 033 011-5 -4-3 -2-1 0 1 2 3 4 5 Figure 112 The two-sided expoetial i Example 34 The z-trasform of this sigal ca be writte as 1 z 1 X ( z z 3 1 3 0 3z This z-trasform coverges to z /3 1 X ( z, ROC:1 3 z 3 1 ( z/3 1 (1 3 z The ROC is show i Figure 33 z-plae 1/3 3 Figure 113 The ROC for Example 34 321 THE z-transform PROPERTIES The z-trasform is liear ad some of its basic properties are give i this sectio Startig from the trasform pairs, 73
we ca write the followig properties: The time shift property: x( X ( z, ROC: R y( Y( z, ROC: R, x y m x( m z X ( z ROC: same as R x except possible deletio of z = 0 or z = (112 The covolutio property: x( * y( X ( z Y( z ROC: Cotais the itersectio of R x ad R y (113 The scalig property: x( X ( z / ROC: α R x (11 4 Time reversal: * * * x ( X (1/ z ROC: 1/ R x (11 5 ijdsp Simulatio 31: I Figure 34, we show the ijdsp Simulatio 31 where a simple 7-sample (z -7 delay is applied o a siusoid of the form si(02π The 7-sample delay ca be implemeted i ijdsp usig the filter fuctio ad programmig a FIR filter with b 7 =1 ad with all the other coefficiets set to zero From the plot of the frequecy respose of this filter i Figure 34 (top ad bottom ceter, oe otices that a pure delay is associated with uity amplitude gai ad piecewise liear phase Note that the output siusoid, displayed i Figure 34 (top right, is delayed by 7 samples If the z-trasform is evaluated o the uit circle, ie, if j z e, 2 f T (36 74
where T is the samplig period, the we obtai the discrete-time Fourier trasform (DTFT, j j X( e x( e (37 Figure 114 ijdsp Simulatio 31: A 7-sample delay implemeted usig the ijdsp filter fuctio 33 THE TRANSFER FUNCTION The z-trasform of the impulse respose of a filter is called the trasfer fuctio ad is give by, H( z h( z (118 By trasformig the differece equatio, defied i Eq (22, i the z domai, ie, L M l m X ( z blz Y( z 1 amz l0 m1 (119 we ca defie the trasfer fuctio as the ratio of the output, Y(z, over the iput, X(z, 1 L b b z b z 0 1 L 1 M 1 M Y( z H( z X ( z 1 a z a z (1110 For a FIR filter, the trasfer fuctio is give by 75
l H( z b z L l0 l (1111 34 INVERSE z-transform Iverse z-trasforms for fiite-legth sigals ca be determied by ispectio For ratioal fuctios, a iverse z- trasform ca be determied usig several methods icludig factorig the polyomial usig partial fractios or itegratig the z-domai fuctio o a closed cotour usig the residue theorem Example 115: Determie the iverse z-trasform of 3 1 2 X ( z 3z z 5 z, 0 z The time-domai sequece ca be obtaied by ispectio, ie, x( 3 ( 3 ( 1 5 ( 2 For ratioal z-domai polyomials of the form H( z L L1 b0z b1z bl M M1 z a1 z am (1112 where L M, iverse z-trasforms ca be obtaied by writig H(z as a weighted sum of first-order ratioal fuctios This ca be doe usig partial fractio expasio Assumig distict poles, we ca write H(z as, z z H z c c c z p z p ( 0 1 M 1 M (1113 where p1, p2,, p M are the poles of H( z ad c0, c1,, c M are costats The time-domai sigal, h(, ca be defied uiquely if the ROC is specified If the ROC extets outwards from the outermost pole of H(z, the h( is causal ad we ca write it as h( c ( c p c p c p, 0 0 1 1 2 2 M M (1114 The coefficiets of the partial fractio ca be determied usig the followig expressios 76
( z pm H( z c0 H( z, c, 1,2,, z 0 m m M z z pm (1115 Example 116: Use the iverse z-trasform to determie the impulse respose of the causal digital filter i Figure 35 The trasfer fuctio ad its partial fractio expasio ca be writte as 2 z z z z H( z 9 8 z z z z 2 5 1 1 1 6 6 2 3 Sice the filter is causal the ROC exteds outwards from the outermost pole, oe ca the write the impulse respose as 1 1 h( 9 8, 0 2 3 Sice the above impulse respose is absolutely summable, this causal filter is BIBO stable x( + Σ + Σ y( - + z 1 + 5/6 z 1 1/6 Figure 115 Secod-order filter i Example 36 For ratioal fuctios with repeated poles, oe ca use the residue theorem to obtai the iverse z-trasform A causal sigal ca be represeted as a sum of residues, ie, M 1 ( Res[ ( ] z p, 0 m m1 h z H z (1116 For a distict pole, the residue is give by 77
1 1 Res[ z H( z] zp [( z p ( ] m m z H z zpm (1117 For r repeated poles (poles of multiplicity r the residue is give by r1 1 1 d r 1 Res[ z H( z] z p 1 ( z p ( m r i z H z ( r 1! dz z pm (1118 Example 117: Use residues to determie the trasiet ad steady state resposes of the causal filter i Figure 36 The excitatio is a uit-step sequece, ie, x(=u( y( x(=u( 3 + Σ z 1 + 07 Figure 116 First-order IIR filter i Example 37 The trasfer fuctio H(z ad the z-trasform of the iput are give respectively by 3z z H( z, X ( z z07 z 1 The output i the z domai is obtaied by multiplyig the trasfer fuctio with X(z 2 3z Y( z H( z X ( z ( z1( z07 Usig residues, we ca write the output i the time domai as y( Res[ z Y ( z] Res[ z Y ( z] 1 1 z1 z07 1 2 1 2 ( z 1 z z ( z 07 z z 3 3 ( z 07( z 1 ( z 1( z 07 10 7(07, 0 z1 z07 (319 78
Note that the trasiet compoet is due to the pole of the filter fuctio ad the steady-state compoet is due to the pole of the z-domai iput Example 118: Determie the impulse respose correspodig to the followig BIBO stable trasfer fuctio H( z z 2 2 z 16z064 Sice the filter is stable, the ROC of H(z icludes the uit circle Thus, ROC h : z >08 The filter has two poles, both at z=08 The impulse respose ca be determied by 1 2 1 d 2 z z h( Res[ z H( z] z08 ( z 08 ( 108 u( 2 dz ( z 08 z08 ijdsp Simulatio 32: I ijdsp Simulatio 32, we demostrate how ijdsp ca be used to obtai iverse z-trasforms This procedure ca be used oly for causal sigals that are absolutely summable, ie, fuctios whose poles are iside the uit circle To ivert a z-trasform usig ijdsp, the z-domai fuctio is realized as a digital filter excited by a uit impulse as show i Figure 37 I ijdsp, the coefficiets are etered usig the dialog of the Filter Coeff block The filter is excited by a uit impulse (Delta programmed i the dialog associated with the Sigal Geerator block, ad the output of the filter is the correspodig time-domai sigal I the ijdsp Simulatio 32 show i Figure 38, the iverse z-trasform of the secod-order fuctio below is calculated ad the resultig sigal is plotted 1 H( z 1 045z 055z 1 2 δ( H(z h( Figure 117 Obtaiig the impulse respose usig ijdsp 79
35 POLES AND ZEROS AND FREQUENCY RESPONSE The trasfer fuctio ca be writte i terms of its poles ad zeros as follows ( z l ( z ( z ( z l1 M ( z p1( z p2( z pm ( z p 1 2 L H ( z G G L m1 m (1120 where ζ l ad p m are the zeros ad poles of H(z, respectively ad G is a positive costat For a causal ad BIBO stable filter, the correspodig impulse respose is absolutely summable, ie, h ( (321 0 This will be the case if all the poles are iside the uit circle, ie, if p m <1 for m=1,2,,m The locatios of the poles ad zeros also affect the shape of the frequecy respose The magitude of the frequecy respose ca be writte as, j H ( e G L l1 M m1 e e j j l p m (1122 It is therefore evidet that whe a isolated zero is close to the uit circle, the the magitude frequecy respose will assume a small value at the frequecy of the zero; whe a isolated pole is close to the uit circle, it will give rise to a peak i the magitude frequecy respose at the frequecy of the pole The locatios of the poles ad zeros also affect the phase respose of the filter I fact, if ay of the poles or zeros of the z-domai fuctio is outside the uit circle, the the filter is called o-miimum phase If all the poles ad zeros of H(z are iside the uit circle, the the filter is called miimum phase Miimum phase causal systems are also BIBO stable Similar argumets ca be made about discrete-time sigals i geeral, that is, a miimum phase sigal is oe whose z-domai represetatio has all its poles ad zeros iside the uit circle Example 119: 80
Imagiary Part Magitude (db For the secod-order system whose trasfer fuctio is give below, provide a pole-zero plot ad relate the locatios of the poles ad zeros with its frequecy respose 1 2 113435z 09025z H( z 1 2 1045z 055z The poles ad zeros appear i complex cojugate pairs because all the coefficiets are real-valued H( z o o j45 j45 ( z 095 e ( z 095 e o o j7234 j7234 ( z 07416 e ( z 07416 e The pole zero diagram ad the frequecy respose are give i Figure 39 Poles ear the uit circle give rise to Figure 118 ijdsp Simulatio 32: Usig ijdsp to obtai iverse z-trasforms 1 z plae plot 10 Frequecy respose 05 0 0-10 -05-20 -1-1 -05 0 05 1 Real Part -30 0 05 15 2 81
Figure 119 z-plae represetatio ad frequecy respose spectral peaks ad zeros ear the uit circle create spectral valleys i the magitude of the frequecy respose The symmetry aroud π is due to the fact that roots appear i complex cojugate pairs Note that H(z is miimum phase because all of its poles ad zeros are iside the uit circle ijdsp Simulatio 33: I the ijdsp Simulatio 33 show i Figure 310, the poles ad zeros, the impulse respose, ad the frequecy respose of the filter fuctio below are calculated ad plotted 1 H( z 10 1 09z Note that i Figure 310 (top right, the poles are distributed evely iside the uit circle The impulse respose is also show i Figure 310 (bottom right This type of filter is commo is speech processig applicatios; specifically it is used for speech sythesis i digital cellular phoes It ca be used to geerate a pseudo-periodic sigal from radom excitatio I speech processig, this pseudo-periodic sigal ca be desiged to have properties that are similar to those of the glottal waveform 82
Figure 1110 ijdsp Simulatio 33: Poles ad zeros, impulse ad frequecy respose of H(z 36 CASCADE AND PARALLEL CONFIGURATIONS LTI systems i cascade or parallel cofiguratios ca be combied ito a sigle system For example, the system o the top i Figure 311 is equivalet to the system show o the bottom of the same figure Coversely, parallel structures for realizatio of digital filters ca be derived usig z-domai aalysis For example, partial fractio expasio of trasfer fuctios ca lead to parallel realizatios Parallel ad cascade realizatios are useful whe dealig with umerical problems, roud off errors, etc ijdsp Simulatio 34: Use ijdsp to simulate a LTI system similar to those show i Figure 311 Plot the impulse ad frequecy resposes The three trasfer fuctios are give by 1 H1( z 1 1 09z, 1 H ( z 2 1 09z, ad 1 H ( z 1 3 1 064z 2 83
The impulse respose of the combied fuctio is show i Figure 312 (bottom right ad the frequecy respose computed with a 256-poit FFT is give i Figure 312 (top right, respectively Note that sice the poles of the - composite filter are located at 0, /2, -/2, ad, we obtai peaks at those frequecies X(z H 1 (z H 2 (z + Σ Y(z H 3 (z + X(z H 1 (z H 2 (z + H 3 (z Y(z Figure 1111 Equivalet parallel ad cascade cofiguratios of a LTI system Figure 1112 ijdsp Simulatio 35: Parallel ad cascade cofiguratios simulated usig ijdsp 84
37 SUMMARY The followig are some of the key poits covered i this chapter: The z-trasform provides ituitio ad compact mathematical represetatios of discrete-time sigals ad systems Digital filter z-domai represetatios are aalogous to Laplace trasform s-domai aalog filter represetatios z-trasform pairs are defied uiquely whe the ROC is specified The ROC is cotiguous ad does ot cotai poles The ROC of a causal sigal exteds outwards from the outermost pole The DTFT is a special case of the z-trasform Poles close to the uit circle create arrowbad peaks i the frequecy domai Zeros close to the uit circle create sharp otches i the frequecy domai A causal filter with all of its poles iside the uit circle is BIBO stable A sigal with all of its z-domai roots iside the uit circle is called miimum phase The scalig property ca be used to stabilize a ustable filter Partial fractios ca be used to ivert ratioal z-domai fuctios The residue theorem ca be used to ivert z-trasforms Iverse z-trasforms ca be used to calculate the filter trasiet ad steady state resposes Causal FIR filters have all of their poles at zero IIR filters cotai o-zero poles i their trasfer fuctio Parallel structures for realizig digital filters ca be derived usig partial fractio expasio 371 FURTHER READING Stability tests usig the Jury s method are give i [Jury64] z-trasforms ad their applicatio i cotrols systems are give i [Bolt93] Trasfer fuctios of shelvig filters are give i [Lae01] Symbolic iversios of z-trasforms are give i Mathematica [Graf04] A detailed discussio o miimum ad maximum phase systems is give i [Oppe99] 85
Trasformatios of s-domai to z-domai fuctios are described later i this book ad i [Grov91] 372 IJDSP EXERCISES Exercise 1: Expoetial Sequeces Desig ad simulate with ijdsp a digital filter that has the followig impulse respose h( 09 u( a Give the simulatio diagram ad a trasfer fuctio, H(z, correspodig to h( b Use ijdsp to plot the impulse respose c Use ijdsp to plot the frequecy respose Is this a LPF or a HPF? Is the filter stable? Provide the poles ad zeros of the system What would be the closest aalog circuit approximatio to this first-order digital filter? d Repeat parts a through c for the filters with the two impulse resposes that follow h( ( 08 1 u( 1 h( ( 1(09 u( e Give the trasfer fuctio for a causal system, H 1 (z -1 2 3 1-3z 3z 2z Use the pole-zero plot capability of ijdsp ad check if this trasfer fuctio represets a stable system Exercise 2: Digital Oscillator Digital oscillators ca be desiged by selectig the coefficiets of the digital filter so that we obtai oscillatory behavior whe the excitatio is a uit impulse a Desig ad simulate a digital oscillator for a samplig frequecy of 8000 Hz ad a siusoidal frequecy of 687 Hz Note that its impulse respose must be of the form: h( cos( 0 u( b Use ijdsp to plot the frequecy respose ad the poles ad zeros of this digital oscillator 86
Exercise 3: Cacelig Siusoidal Compoets Filters ca be desiged to cacel siusoids Cosider a system with the followig impulse respose: h( ( 2cos( / 4 ( 1 ( 2 The iput sigal is give by x( si( u( with: i / 2, ad ii / 4 1 2 To geerate the iput sigal, use the followig parameters: Sigal Type : siusoid Amplitude : 1 Time Shift : 0 For this problem, address the followig: a Plot the frequecy respose ad observe the pole ad zero locatios b Simulate the covolutio y( = x(*h( by realizig h( as a FIR filter ad x( as a siusoid Cosider both frequecies 1 ad 2 Use ijdsp ad plot the output of the system for each case c Note the behavior of this system with the two siusoidal frequecies above Whe do we get siusoidal cacellatio? How does this form of siusoidal cacellatio relate to the locatio of poles ad zeros? Exercise 4: Symmetric Impulse Resposes ad Liear Phase Liear phase filters have costat group delay ad are importat i certai applicatios, eg, high-ed audio amplifiers Cosider the followig trasfer fuctio that is associated with a symmetric impulse respose: 1 2 3 4 5 H ( z 1 05z 025z 025z 05z z Simulate the filter with ijdsp ad plot the phase respose of H(e jω a Commet o the phase respose What is the slope? Calculate the group delay of the filter by measurig the egative of the derivative of the phase respose of the filter? 87
b Use pole-zero display ad check for symmetries i the locatios of zeros (Hit: The group delay is the egative of the derivative of the phase with respect to the frequecy The group delay is measured i terms of samples Exercise 5: Pole-Zero Plots Figure 313 shows the pole-zero cofiguratios of two LTI systems H 1 (z ad H 2 (z: For each system i Figure 313, a Determie the coefficiets of H 1 (z ad H 2 (z b Modify H 1 (z ad H 2 (z such that they are realizable causal filters ad obtai their impulse respose ad frequecy respose usig ijdsp c What are the effects of the poles ad zeros o the frequecy respose i part b? Im Im 05 05 025 025-025 025 04 Re 025 05 Re Exercise 6: Cascade ad Parallel System Cofiguratios Cosider the followig system (cascade cofiguratio: Figure 313 z-plae diagrams for ijdsp Exercise 5 k k y( c d k0 (Hit: Note that this is a covolutio of two expoetial sigals a Implemet the cascade system usig oe Sigal Geerator block, two Filter blocks, two Filter Coeff blocks, ad oe Plot block for c=05 ad d=025 Observe the output of the system 88
b Implemet the cascade system usig oe Sigal Geerator block, oe Filter block, oe Filter Coeff block, ad oe Plot block for c=05 ad d=025 Cosider also the followig system (parallel cofiguratio: h( [05 2( 09 ] u( c Implemet the parallel system usig two Sigal Geerator block, two Filter blocks, two Filter Coeff blocks, oe Adder block, ad oe Plot block d Implemet the parallel system usig oe Sigal Geerator block, oe Filter block, oe Filter Coeff block, ad oe Plot block The followig exercises focus o geeratig frequecy resposes ad pole-zero plots from a trasfer fuctio ijdsp cotais a PZ Placemet block that ca be used to place the poles ad zeros of a system The output of the PZ Placemet block ca be coected to a Freq Resp block to view the frequecy respose of the system whose poles ad zeros are at the locatios specified i the PZ Placemet block Exercise 7: Pole-Zero Plots Fid the poles ad zeros of the followig three trasfer fuctios, H 1 (z, H 2 (z, ad H 3 (z Use the ijdsp to plot the magitude ad phase of the frequecy respose for each oe Observe the structure of poles ad zeros i each system relative to the frequecy respose a Is the system H 1 (z stable? 1 112z H1( z 1 1 05z b Determie the locatios of the zeros of H 2 (z Commet o the umber of the filter coefficiets, the order of the filter, ad the structure of the frequecy respose H z z ( 1 3 2 c Note the pole locatios of the system H 3 (z Provide commets o the order of the filter ad the structure of the frequecy respose 89
1 H ( z 3 5 1 085z Exercise 8: Varyig the Magitudes of Poles ad Zeros Cosider a system with complex cojugate poles p 1,2 j /4 re ad a zero located at z1 07071r Cosider three cases for r, amely, i r = 096; ii r = 071; iii r = 014 a Derive aalytically the impulse respose of the system ad show its depedece o r b Plot the impulse respose for each case usig ijdsp c Plot the magitude ad phase respose for each case usig ijdsp d Use the PZ Placemet block of ijdsp to move the poles ad zero ad ote the differeces i the frequecy resposes relative to the positio of the poles Exercise 9: Lowpass/highpass Filter Desig by Pole-zero Placemet For this problem, plot the magitude i db a Use the Filter ad the PZ Placemet blocks of ijdsp to desig a lowpass filter Use three sets of zeros ad two sets of poles Desig the filter with a approximate cutoff frequecy of c /3 b Use the Filter ad the PZ Placemet blocks of ijdsp to desig a highpass filter Use two sets of zeros ad five sets of poles Desig the filter with a approximate cutoff frequecy of c / 2 Exercise 10: A Iterestig Frequecy Respose Cosider the followig allpass system: 3 2 1 z 18z 162z 0729 H( z 1 2 3 1 18z 162z 0729z a Use ijdsp to fid the poles ad zeros of the trasfer fuctio H(z 90
b Plot the magitude respose ad the phase respose of the system c Discuss the symmetry of the umerator of H(z relative to its deomiator Note that allpass filters are ofte used to desig the delay ad phase characteristics i a sigal without alterig its magitude spectrum 91
REFERENCES [Bolt93] W Bolto, Laplace ad z-trasforms, Pearso, 1993 [Graf04] UE Graf, Applied Laplace Trasforms ad z-trasforms for Scietists ad Egieers: A Computatioal Approach usig a Mathematica Package, Birkhäuser Verlag, 2004 [Grov91] AC Grove, A Itroductio to the Laplace Trasform ad the z-trasform, Pretice Hall, 1991 [Jury64] EI Jury, Theory ad Applicatio of the z-trasform Method, Wiley, New York, 1964 [Lae01] J Lae, J Data, B Karley, J Norwood, DSP Filters, Prompt Publicatios, 2001 [Oppe99] AV Oppeheim, RW Schafer, JR Buck, Discrete-Time Sigal Processig, Pretice Hall Secod Editio, 1999 92