Dorf.R C. Wan.Z"The z-Transfrom The electrical Engineering Handbook Ed. Richard C. Dorf Boca raton crc Press llc. 2000
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 an· Time Reversal Richard C. dorf 8.3 Unilateral z-Transform Time Advance Initial Signal value Final val University of California, Davis 8.4 z-Transform Inver Zhe en wan 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 an, in general, be indexed by n as follows: x={…,x(-2),x(-1),x(O),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 {x(n)}=X(z) x(n 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 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 ff(n)) and zig(n)) are denoted by Fz) and G(z), respectively, then Zaf(n)+ bg(n))=aF()+bG(z) where a and b are constant multipliers. c 2000 by CRC Press LLC
© 2000 by CRC Press LLC 8 The z-Transform 8.1 Introduction 8.2 Properties of the z-Transform Linearity • Translation • Convolution • Multiplication by an • 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{x n( )} X(z) x(n)z n n = = - =-• • Â Richard C. Dorf University of California, Davis Zhen Wan University of California, Davis
Table 8.1 Partial-Fraction Equivalents Listing Causal and Anticausal z-Transform Pairs Domain: Rz) Sequence Domain: (n) for z >al for z <al for z>a (n-1)a"-u(n-1) (n-1)(n-2)a"-u(n-1)= (n-1)(n-2)an-n(-n) forz>a I(n-k)a-mu(n-1) z|< ∏I(n-k)d"=m-n forz≠0,m≥0 5zm,f|<,m2080+m)={…00-…,…0…,0 Source: J.A. Cadzow and H.E. Van Landingham, Signals, Systems and Transforms, Englewood Cliffs N J. Prentice-Hall, 1985, P. 191. With permission. Translation An important property when transforming terms of a difference equation is the z-transform of a sequence hifted in time. For a constant shift we have Zlf(n+k)=zF(z) e 2000 by CRC Press LLC
© 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 kF(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 1 0 1 1 1 1 1 1 1 2 1 3 2 2 z a z a a u n a a z a z a a u n a a a z a z a n n - > - = { } - < - - = Ï --- Ì Ô Ó Ô ¸ ˝ Ô ˛ Ô - > - - , ( ) , , , , ( ) , , ( ) , ( * * * * * * * * * * * * n a u n a a z a z a n a u n aaa z a z a n n - - = { } - < - - - = Ï Ì Ô Ó Ô ¸ ˝ Ô ˛ Ô - > - - 1 1 0 1 2 3 1 1 3 2 1 1 2 2 2 2 4 3 2 3 ) ( ) , , , ( ) , ( ) ( ) , , ( ) , , . . . 2b. for . . . , 3a. for * * * * * * * * * * * * 1 2 1 2 1 0 0 1 3 6 1 1 2 1 2 6 3 1 3 2 3 3 5 4 3 ( )( ) ( ) , , , , ( ) , ( )( ) ( ) , , n n a u n a a z a z a n n a u n a a a n n -- - = { } - < - - - - = Ï - - - Ì Ô Ó Ô ¸ ˝ Ô ˛ Ô - - , . . . 3b. for . . . , ( ) , ( )! ( ) ( ) ( ) , ( )! ( ) ( ) , , 4a. for 4b. for 5a. for 1 1 1 1 1 1 1 0 1 1 1 1 z a z a m n k a u n z a z a m n k a u n z z m m n m k m m n m k m m - > - - - - < - - - - ¹ ³ - = - - = - - ’ ’ * * * * * * * * 0 0 0 1 0 0 0 0 0 1 0 0 d d ( ) , , , , , , ( ) , , , n m z z m n m m - = { } < • ³ + = { } + . . . , . . . , . . . , , . . . 5b. for * * . . . , . . . , . . . , , . . ., , . .
for positive or negative integer k The region of convergence of z F(z)is the same as for F(z)for positive k nly 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, ff(n)*g(n))=F(z)G(z) Multiplication by This operation corresponds to a rescaling of the z-plane. For a>0, z"f(m)}= for aR, <z< aR where F(z)is defined for,<zl Time Reversal zff(n)=F(2-) for R=<|z|< R- where F()is defined for R, lz<R 8.3 Unilateral z-Transform The unilateral z-transform is defined as Z,x(n))=X(z)=>x(n)2-mn for z>R where it is called single-sided since n 20, just as if the sequence xn) 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 (n) 2+{f(n+1)}=zF(z)-zf(0) More generally, z,f(n+k)}=2F(z)-zf(0)-2=f(1)-…-zf(k-1) 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 e 2000 by CRC Press LLC
© 2000 by CRC Press LLC for positive or negative integer k. The region of convergence of z kF(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 R1 < ½z½ < R2. Time Reversal where F(z) is defined for R1 < ½z½ < R2. 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 { f n( )} F z a = aR z aR Ê Ë Á ˆ ¯ ˜ for 1 2 < * * < Z {f n (± )} = F(z ) for R < z < R - - 1 - 2 1 1 1 * * Z+ - = • {x n( )} = = X( )z  x(n)z z > R n n 0 for * * Z+ - {f(n + = k)} z F( )z - z f( ) - z f( ) - - zf k( - ) kkk 0 1 1 1 . .
Initial Signal Value f(0)=lim F(z) where F(z)=ZIf(n)) forz>R If f(n)=0 for n<0 and ZIf(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 f(oo)= lim f(n)= lim(1-z)F(z) n=00 8.4 z-Transform Inversion We operationally denote the inverse transform of F(z) in the form f(n)=z-{F(z)} There are three useful methods for inverting a transformed signal. They are Expansion into a series of terms in the variables z and tl 2. Complex integration by the method of residue 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, ie, an annulus centered at the origin of the z-plane.This first method is to obtain a series expression of the form F(z which is valid in the annulus of convergence. When F(z) has been expanded as in the previous equation, that is, when the coefficients n, n=0, +1, +2,... have been found, the corresponding sequence is specified by f(n)=c by uni 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 e 2000 by CRC Press LLC
© 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, 8.4 z-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 cn , n = 0, ±1, ±2, … have been found, the corresponding sequence is specified by f(n) = cn 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. f( )0 = F(z) fi • lim z f( ) • = f(n) = ( )F(z) fi • fi • lim lim 1 – z n z –1 F z c zn n n ( ) = - =-• • Â