Web Appendix O - Derivations of the Properties of the z Transform

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1 M. J. Roberts - 2/18/07 Web Appedix O - Derivatios of the Properties of the z Trasform O.1 Liearity Let z = x + y where ad are costats. The ( z)= ( x + y )z = x z + y z ad the liearity property is O.2 Time Shiftig x + y X( z)+ Y z. = X( z)+ Y( z) There are two differet cases to cosider, egative ad positive shifts i discrete time (Figure O-1). x[] Origial Fuctio x[- ] 0 Positive Time Shift x[+ ] 0 Negative Time Shift Figure O-1 Shifts i discrete time Case 1.Positive shifts i discrete time The sigal is causal. Therefore positive shifts i discrete time simply shift i leadig zeros. O-1

2 M. J. Roberts - 2/18/07 g 0 g 0 z = g 0 z, 0 0 Let m = 0. The = 0 g 0 g m z ( m+ ) 0 = z 0 g m z m = z 0 G ( z ) g 0 z 0 G ( z ), 0 0 (O.1) This property applies oly to causal sigals. Otherwise a positive shift could shift i ew o-zero sigal values ad the relatioship betwee the trasforms of the origial ad shifted sigals would ot be uique. Case 2.Negative shifts i discrete time I this case, usig the uilateral z trasform, we are i geeral trucatig some of the o-zero part of the sigal by shiftig it to the left because the trasform summatio begis at = 0. Therefore, if the property is to apply geerally we must fid a way to restore the missig iformatio. Otherwise the trasform of the ushifted sigal ad the shifted sigal caot be uiquely related. Start with the defiitio of the z trasform g + 1 g + 1 z = z g + 1 z ( +1). We have a summatio which does ot iclude the effect of g 0 i the origial fuctio. Let m = + 1, the g + 1 z g m z m = z g m z m g 0 = z ( G ( z ) g 0 ) m=1 By this process we ow are icludig the effect of g 0 ad the trasform of the origial sigal ad the shifted sigal are uiquely related. We ca exted this method to greater shifts, for example, a egative shift of 2 i discrete time. g + 2 g + 2 z = z 2 +2 g + 2 z Let m = + 2, the g + 2 z 2 g m z m = z 2 g m z m g 0 z 1 g1 m=2 g + 2 z 2 G ( z) g 0 z 1 g1 O-2

3 M. J. Roberts - 2/18/07 The, by iductio, for greater shifts, 0 1 g + 0 z 0 G ( z) g m z m, 0 > 0 (O.2) O.3 Chage of Scale If we compress or expad the z-trasform of a sigal i the z domai, the equivalet effect i the DT domai is a multiplicatio by a complex expoetial. g g z = g z / g = G ( z / ) G ( z / ) (O.3) O.4 Iitial Value Theorem The iitial-value theorem is similar to its couterpart i the Laplace trasform. If we take the limit as z approaches ifiity of the z trasform G ( z) of ay fuctio g all the terms except the g 0 z0 limg ( z)= lim z z term approach zero leavig oly the first term. g z = lim g 0 z + g1 z g 0 = lim G z z + g 2 z 2 + = g 0 (O.4) O.5 z-domai Differetiatio Differetiatio i the z domai is related to a multiplicatio by i the DT domai. G ( z)= g z d dz G ( z )= d dz 0 g z = g +1 z = z 1 g z 0 ( g ) O-3

4 M. J. Roberts - 2/18/07 Therefore g z d dz G ( z ) (O.5) O.6 Covolutio i Discrete Time We have see i the Fourier ad Laplace trasforms that there is a importat relatioship betwee covolutio i oe domai ad multiplicatio i the other domai. A similar relatioship exists for the z trasform. Start with the defiitio of the covolutio sum g h = g m h m. m= Take the z trasform of both sides, ( g h )= g m h m g h Usig the fact that g m= z g m h m z = g m m= z m H( z) m= is causal, g h g h g m H z z m = H( z)g ( z) z m H( z) H( z)g ( z) (O.6) I words, covolutio of two DT fuctios i the DT domai correspods to multiplicatio of their z trasforms i the z domai, exactly as was true for the Fourier ad Laplace trasforms. Cosideratio of the z trasform of the product of two DTdomai fuctios is beyod the scope of this text. O.7 Differecig Differecig is the DT operatio that is aalogous to CT differetiatio. The first backward differece of g is g g 1. Usig the time-shiftig property (for causal fuctios), the z trasform of this differece forms the pair Therefore g g 1 G ( z) z 1 G ( z)= ( 1 z 1 )G ( z). O-4

5 M. J. Roberts - 2/18/07 g g 1 ( 1 z 1 )G z. (O.7) O.8 Accumulatio Accumulatio is the DT operatio which is aalogous to CT itegratio ad the proof of the property ca be doe i a aalogous maer. First realize that accumulatio is equivalet to covolutio with a uit sequece u g = u m g m = g m. m= The last summatio has a upper limit of because g is causal. Therefore g m = u g g m z z 1 G ( z )= G ( z)u( z)= z z 1 G ( z ) 1 1 z G ( z ) (O.8) 1 O.9 Fial Value Theorem Begi the derivatio of the fial-value theorem by cosiderig the z trasform of the differece betwee a DT fuctio ad a shifted versio of the same fuctio (a first forward differece), ( g + 1 g )= lim ( g m + 1 g m )z m Takig the z trasform ad usig the time-shiftig property, z( G ( z) g 0 ) G ( z)= lim ( g m + 1 g m )z m. Now take the limit as z 1 o both sides, Takig the z limit first, lim ( z 1)G ( z) z g 0 z1 { }= lim lim z1 ( g m + 1 g m )z m. O-5

6 M. J. Roberts - 2/18/07 lim{ ( z 1)G ( z) } g 0 = lim ( g m + 1 g m ). z1 lim{ ( z 1)G ( z) } g 0 = lim z1 lim{ ( z 1)G ( z) } g 0 = lim z1 lim g1 g 0 + g 2 g1 ++ g + 1 g g = lim ( z 1)G z z1 g + 1 g 0 O-6