Dorf, R.C., Wan, Z. “The z-Transfrom” The Electrical Engineering Handbook Ed. Richard C. Dorf Boca Raton: CRC Press LLC, 2000 8 The z-Transform 8.1 Introduction 8.2 Properties of the z-Transform Linearity?Translation?Convolution?Multiplication by a n ?Time Reversal 8.3 Unilateral z-Transform Time Advance?Initial Signal Value?Final Value 8.4 z-Transform Inversion Method 1?Method 2?Inverse Transform Formula (Method 2) 8.5 Sampled Data 8.1 Introduction Discrete-time signals can be represented as sequences of numbers. Thus, if x is a discrete-time signal, its values can, in general, be indexed by n as follows: x = {…, x(–2), x(–1), x(0), x(1), x(2), …, x(n), …} In order to work within a transform domain for discrete-time signals, we define the z-transform as follows. The z-transform of the sequence x in the previous equation is in which the variable z can be interpreted as being either a time-position marker or a complex-valued variable, and the script Z is the z-transform operator. If the former interpretation is employed, the number multiplying the marker z –n is identified as being the nth element of the x sequence, i.e., x(n). It will be generally beneficial to take z to be a complex-valued variable. The z-transforms of some useful sequences are listed in Table 8.1. 8.2 Properties of the z-Transform Linearity Both the direct and inverse z-transform obey the property of linearity. Thus, if Z{f(n)} and Z{g(n)} are denoted by F(z) and G(z), respectively, then Z{af(n) + bg(n)} = aF(z) + bG(z) where a and b are constant multipliers. Z{()} () ()xn Xz xnz n n == - =-￥ ￥ ? Richard C. Dorf University of California, Davis Zhen Wan University of California, Davis ? 2000 by CRC Press LLC Translation An important property when transforming terms of a difference equation is the z-transform of a sequence shifted in time. For a constant shift, we have Z{f(n + k)} = z k F(z) Table 8.1 Partial-Fraction Equivalents Listing Causal and Anticausal z-Transform Pairs z-Domain: F(z) Sequence Domain: f(n) Source: J.A. Cadzow and H.F. Van Landingham, Signals, Systems and Transforms, Englewood Cliffs, N.J.: Prentice-Hall, 1985, p. 191. With permission. 1a. for , . . . 1b. for . . . , 2a. for 1 101 11 1 12 1 32 2 za za aun aa za za aun aa a za za n n - >-= {} - <--= --- ì í ? ? ? ü y ? t ? - > - - , ( ) , , , , () , , () ,( **** **** **** n aun aa za za naun aaa za za n n --= {} - <---= ì í ? ? ? ü y ? t ? - > - - 110123 1 1 321 1 22 2 2 432 3 ) ( ) , , , () , ( ) () , , () , , . . . 2b. for . . . , 3a. for **** ***** **** 1 2 12 100136 11 2 12 631 32 3 3 543 ( )( ) ( ) , , , , () , ( )( ) () , , nnaun aa za za nnaun aaa n n -- -= {} - < - -- -= --- ì í ? ? ? ü y ? t ? - - , . . . 3b. for . . . , () , ()! ()() () , ()! ()() ,, 4a. for 4b. for 5a. for 11 1 1 11 1 0 1 1 1 1 za za m nkaun za za m nkaun zzm m nm k m m nm k m m - > - -- - < - - -- 13 - = - - = - - ? ? **** **** 0 00100 000100 d d ( ) ,, ,, ,, (),,, nm zzmnm m -= {} <￥ 3 + = + . . . , . . . , . . . , , . . . 5b. for . . . , . . . , . . . , , . . ., , . . .** ? 2000 by CRC Press LLC for positive or negative integer k. The region of convergence of z k F(z) is the same as for F(z) for positive k; only the point z = 0 need be eliminated from the convergence region of F(z) for negative k. Convolution In the z-domain, the time-domain convolution operation becomes a simple product of the corresponding transforms, that is, Z{f(n) * g(n)} = F(z)G(z) Multiplication by a n This operation corresponds to a rescaling of the z-plane. For a > 0, where F(z) is defined for R 1 < ?z? < R 2 . Time Reversal where F(z) is defined for R 1 < ?z? < R 2 . 8.3 Unilateral z-Transform The unilateral z-transform is defined as where it is called single-sided since n 3 0, just as if the sequence x(n) was in fact single-sided. If there is no ambiguity in the sequel, the subscript plus is omitted and we use the expression z-transform to mean either the double- or the single-sided transform. It is usually clear from the context which is meant. By restricting signals to be single-sided, the following useful properties can be proved. Time Advance For a single-sided signal f(n), Z + {f(n + 1)} = zF(z) – zf(0) More generally, This result can be used to solve linear constant-coefficient difference equations. Occasionally, it is desirable to calculate the initial or final value of a single-sided sequence without a complete inversion. The following two properties present these results. Z a n { ()} fn F z a aR z aR= ? è ? ? ? ÷ <<for 12 ** Z ±{ ( ) fn Fz R zR( )} for =< -- -1 2 1 1 1 ** Z + - = ￥ == > ? {()} () ()xn Xz xnz z R n n0 for ** Z + - += - - ---{( )} () () () ( )fnk zFz zf zf zfk kkk 01 1 1 . . . ? 2000 by CRC Press LLC Initial Signal Value If f(n) = 0 for n < 0, where F(z) = Z{f(n)} for *z* > R. Final Value If f(n) = 0 for n < 0 and Z{f(n)} = F(z) is a rational function with all its denominator roots (poles) strictly inside the unit circle except possibly for a first-order pole at z = 1, 8.4z-Transform Inversion We operationally denote the inverse transform of F(z) in the form f(n) = Z –1 {F(z)} There are three useful methods for inverting a transformed signal. They are: 1.Expansion into a series of terms in the variables z and z –1 2.Complex integration by the method of residues 3.Partial-fraction expansion and table look-up We discuss two of these methods in turn. Method 1 For the expansion of F(z) into a series, the theory of functions of a complex variable provides a practical basis for developing our inverse transform techniques. As we have seen, the general region of convergence for a transform function F(z) is of the form a < *z* < b, i.e., an annulus centered at the origin of the z-plane. This first method is to obtain a series expression of the form which is valid in the annulus of convergence. When F(z) has been expanded as in the previous equation, that is, when the coefficients c n , n = 0, ±1, ±2, … have been found, the corresponding sequence is specified by f(n) = c n by uniqueness of the transform. Method 2 We evaluate the inverse transform of F(z) by the method of residues. The method involves the calculation of residues of a function both inside and outside of a simple closed path that lies inside the region of convergence. A number of key concepts are necessary in order to describe the required procedure. fFz() ()0= T￥ lim z ffn Fz() () ( )()￥= = T￥ T￥ lim lim1–z nz –1 Fz cz n n n ()= - =-￥ ￥ ? ? 2000 by CRC Press LLC A complex-valued function G(z) has a pole of order k at z = z 0 if it can be expressed as where G 1 (z 0 ) is finite. The residue of a complex function G(z) at a pole of order k at z = z 0 is defined by Inverse Transform Formula (Method 2) If F(z) is convergent in the annulus 0 < a < *z* < b as shown in Fig. 8.1 and C is the closed path shown (the path C must lie entirely within the annulus of convergence), then where m is the least power of z in the numerator of F(z)z n–1 , e.g., m might equal n – 1. Figure 8.1 illustrates the previous equation. 8.5 Sampled Data Data obtained for a signal only at discrete intervals (sampling period) is called sampled data. One advantage of working with sampled data is the ability to represent sequences as combinations of sampled time signals. Table 8.2 provides some key z-transform pairs. So that the table can serve a multiple purpose, there are three items per line: the first is an indicated sampled continuous-time signal, the second is the Laplace transform of the continuous-time signal, and the third is the z-transform of the uniformly sampled continous-time signal. To illustrate the interrelation of these entries, consider Fig. 8.2. For simplicity, only single-sided signals have been used in Table 8.2. Consequently, the convergence regions are understood in this context to be Re[s] < s 0 FIGURE 8.1 Typical convergence region for a transformed discrete-time signal (Source: J. A. Cadzow and H. F. Van Landingham, Signals, Systems and Transforms, Englewood Cliffs, N.J.: Prentice-Hall, 1985, p. 191. With permission.) Gz Gz zz k () () () = - 10 0 Res 0 0 [()] ()! [( ) ()]Gz k d dz zzGz zz k k k zz= - - = = - - 1 1 1 1 0 fn Fzz Fz C m Fzz Fz Cm n n () () () , () () ), sum of residues of at poles of inside –(sum of residues of at poles of outside - - 3 < ì í ? 1 1 0 0 ? 2000 by CRC Press LLC and *z* > r 0 for the Laplace and z-transforms, respectively. The parameters s 0 and r 0 depend on the actual transformed functions; in factor z, the inverse sequence would begin at n = 0. Thus, we use a modified partial- fraction expansion whose terms have this extra z-factor. FIGURE 8.2 Signal and transform relationships for Table 8.2. Table 8.2z-Transforms for Sampled Data f(t), t = nT, n = 0, 1, 2, . . . F(s), Re[s] > s 0 F(z), * z * > r 0 Source: J. A. Cadzow and H. F. Landingham, Signals, Systems and Transforms, Englewood Cliffs, N.J.: Prentice-Hall, 1985, p. 191. With permission. 1.1 (unit step) 2.(unit ramp) 3. 4. 5. 6.sin 7. 1 1 1 1 21 1 1 1 21 22 2 3 2 3 22 22 2 22 s z z t s Tz z t s Tzz z e sa z ze te sa Tze ze t s zT zzT t s s zz at aT at aT aT - - + - + - +- +-+ + - - - - - () () () () ( ) sin cos cos ( w w w w w w w - -+ ++ - + + ++ - -+ - - -- - - -- cos ) cos sin () sin cos cos () ( cos ) cos w w w w w w w w w w w T zzT et sa ze T zze Te et sa sa zze T zze Te at aT aT aT at aT aT aT 2 22 2 2 22 2 2 21 2 2 8. 9. ? 2000 by CRC Press LLC Defining Terms Sampled data:Data obtained for a variable only at discrete intervals. Data is obtained once every sampling period. Sampling period:The period for which the sampled variable is held constant. z-transform:A transform from the s-domain to the z-domain by z = e sT . Related Topics 17.2 Video Signal Processing?100.6 Digital Control Systems References J. A. Cadzow and H. F. Van Landingham, Signals, Systems and Transforms, Englewood Cliffs, N.J.: Prentice- Hall, 1985. R. C. Dorf, Modern Control Systems, 7th ed. Reading, Mass.: Addison-Wesley, 1995. R. E. Ziemer, Signals and Systems, 2nd ed., New York: MacMillan, 1989. Further Information IEEE Transactions on Education IEEE Transactions on Automatic Control IEEE Transactions on Signal Processing Contact IEEE, Piscataway, N.J. 08855-1313 ? 2000 by CRC Press LLC