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THE Z-TRANSFORM AND DISCRETE-TIME LTI SYSTEMS
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1 in Eqs. (4.8) to (4.101, we obtain
z-Transform Pairs:
The z-transforms of some common sequences are tabulated in Table 4-1.
Table 4-1. Some Common z-Transform Pairs
All z
lzl > 1 Izl< 1
Z-"'
All z except 0 if ( m > 0) or m if ( m < 0)
1 Z 1-az-''2-a
Izl > lal
(COS Ron)u[nl (sin R,n)u[n] ( r n cos R,n)u[n]
( r nsin R,n)u[nI
n z 2 - (COS o ) z z 2 - (2cos R o )t + 1 (sin n o ) z z 2 - (2cos R,)z + 1 z2- (rcosR0)z z 2 - (2r cos R o ) z + r 2 ( r sin R,)z z 2 - (2r cos R,)z
lzl> 1 Izl> 1 Izl> r Izl> r lzl> 0
O<nsN-1 otherwise
1-~ " ' z - ~ 1- az-'
CHAP. 41
THE z-TRANSFORM AND DISCRETE-TIME LTI SYSTEMS
4.4 PROPERTIES OF THE 2-TRANSFORM Basic properties of the z-transform are presented in the following discussion. Verification of these properties is given in Probs. 4.8 to 4.14.
A. Linearity:
If x l b ] ++X1(z)
~ 2 b
ROC = R, R O C = R, R r ~ Rn lR 2 (4.17)
-Xz(z) I
then ++alXl(z) + a2XAz) where a , and a, are arbitrary constants.
Q I X I [ ~ ]+ a,xz[n]
B. Time Shifting:
then
Special Cases:
++X(z)
ROC = R R' = R n {O < (21< m} (4.18)
x [ n - n,] -z-"oX(z)
Because of these relationships [Eqs. (4.19) and (4.20)1, z-' is often called the unit-delay operator and z is called the unit-advance operator. Note that in the Laplace transform the operators s - = 1/s and s correspond to time-domain integration and differentiation, respectively [Eqs. (3.22) and (3.2011.
C. Multiplication by z,":
If then
In particular, a pole (or zero) at z = z , in X(z) moves to z 2," and .the ROC expands or contracts by the factor (z,(.
Special Case:
= zoz,
after multiplication by
THE Z-TRANSFORMAND DISCRETE-TIME LTI SYSTEMS
[CHAP. 4
In this special case, all poles and zeros are simply rotated by the angle R, and the ROC is unchanged. D. Time Reversal: If then
Therefore, a pole (or zero) in X ( z ) at z = z , moves to l / z , after time reversal. The relationship R' = 1 / R indicates the inversion of R , reflecting the fact that a right-sided sequence becomes left-sided if time-reversed, and vice versa.
E. Multiplication by n (or Differentiation in
If ~ [ n +l + X ( Z ) then
ROC = R
F. Accumulation: If x[nI + + X ( z ) then
ROC = R
Note that C z , _ , x [ k ] is the discrete-time counterpart to integration in the time domain and is called the accumulation. The comparable Laplace transform operator for integration is l / ~ .
G. Convolution:
If %[n] + + X I ( Z ) ~ 2 [ n+]+ X 2 ( 4 then R t 3 R 1n R 2 (4.26) This relationship plays a central role in the analysis and design of discrete-time LTI systems, in analogy with the continuous-time case.
XI[.]
ROC = R 1 ROC = R 2
* x2bI + + X I ( Z ) X Z ( Z )
CHAP. 41
T H E Z-TRANSFORM AND DISCRETE-TIME LTI SYSTEMS
Table 4-2. Some Properties of the z-Transform
Property Sequence Transform ROC
Linearity Time shifting Multiplication by z," Multiplication by einon Time reversal Multiplication by n Accumulation Convolution
&(z) d.
Summary of Some z-transform Properties
For convenient reference, the properties of the z-transform presented above are summarized in Table 4-2. 4.5 THE INVERSE Z-TRANSFORM Inversion of the z-transform to find the sequence x [ n ] from its z-transform X ( z ) is called the inverse z-transform, symbolically denoted as
~ [ n =]s - ' { X ( z > }
A. Inversion Formula:
(4.27)
As in the case of the Laplace transform, there is a formal expression for the inverse z-transform in terms of an integration in the z-plane; that is,
where C is a counterclockwise contour of integration enclosing the origin. Formal evaluation of Eq. (4.28) requires an understanding of complex variable theory.
B. Use of Tables of z-Transform Pairs:
In the second method for the inversion of X(z), we attempt to express X(z) as a sum X(z) =X,(z)
+ . . . +X,(z)
(4.29)
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