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THE Z-TRANSFORM AND DISCRETE-TIME LTI SYSTEMS
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one-sided) z-transform, which is defined as
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Clearly the bilateral and unilateral z-transforms are equivalent only if x[n] = 0 for n < 0. The unilateral z-transform is discussed in Sec. 4.8. We will omit the word "bilateral" except where it is needed to avoid ambiguity. As in the case of the Laplace transform, Eq. (4.3) is sometimes considered an operator that transforms a sequence x[n] into a function X ( z ) , symbolically represented by
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The x[n] and X ( z ) are said to form a z-transform pair denoted as
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B. The Region of Convergence: As in the case of the Laplace transform, the range of values of the complex variable z for which the z-transform converges is called the region of convergence. T o illustrate the z-transform and the associated R O C let us consider some examples.
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EXAMPLE 4.1. Consider the sequence
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x [ n] =a"u[n]
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Then by Eq. (4.3) the z-transform of x [ n ] is
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For the convergence of X(z) we require that
Thus, the ROC is the range of values of z for which laz -'I < 1 or, equivalently, lzl > lal. Then
Alternatively, by multiplying the numerator and denominator of Eq. (4.9) by z, we may write X(z) as X(z) = z-a
Izl > la1
Both forms of X ( z ) in Eqs. ( 4 . 9 ) and (4.10) are useful depending upon the application. From Eq. (4.10) we see that X ( z ) is a rational function of z. Consequently, just as with rational Laplace transforms, it can be characterized by its zeros (the roots of the numerator polynomial) and its poles (the roots of the denominator polynomial). From Eq. (4.10) we see that there is one zero at z = 0 and one pole at z = a . The ROC and the pole-zero plot for this example are shown in Fig. 4-1. In z-transform applications, the complex plane is commonly referred to as the z-plane.
CHAP. 4 1
THE z-TRANSFORM AND DISCRETE-TIME LTI SYSTEMS
Unit circle
-I ca<O
a<-I
Fig.4-1 ROCofthe form lzl>lal.
EXAMPLE 4.2. Consider the sequence
x [ n ] = - a n u [ - n - 11
Its z-transform X(z) is given by (Prob. 4.1)
THE Z-TRANSFORMAND DISCRETE-TIME LTI SYSTEMS
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Again, as before, X ( z ) may be written as X(z)
Izl < la1
Thus, the ROC and the pole-zero plot for this example are shown in Fig. 4-2. Comparing Eqs. (4.9) and (4.12) [or Eqs. (4.10) and (4.13)], we see that the algebraic expressions of X ( z ) for two different sequences are identical except for the ROCs. Thus, as in the Laplace
Fig. 4-2 ROC of the form I z I < lal.
CHAP. 41
THE Z-TRANSFORM AND DISCRETE-TIME LTI SYSTEMS
transform, specification of the z-transform requires both the algebraic expression and the ROC.
C. Properties of the ROC: As we saw in Examples 4.1 and 4.2, the ROC of X ( z ) depends on the nature of x [ n ] . The properties of the ROC are summarized below. We assume that X ( Z ) is a rational function of z.
Property 1 : Property 2:
The ROC does not contain any poles. If x [ n ] is a finite sequence (that is, x [ n ] = 0 except in a finite interval N l ~ n s N,, where N , and N , are finite) and X(z) converges for some value of z, then the ROC is the entire z-plane except possibly z = 0 or z = co. If x [ n ] is a right-sided sequence (that is, x [ n ] = 0 for n < N, < 03) and X(z) converges for some value of z, then the ROC is of the form
Property 3:
where r,, equals the largest magnitude of any of the poles of X(z). Thus, the ROC is the exterior of the circle lzl= r,, in the z-plane with the possible exception of z = m.
Property 4:
If x [ n ] is a left-sided sequence (that is, x [ n l = 0 for n > N , > for some value of z, then the ROC is of the form
- 03) and
X(z) converges
where r,, is the smallest magnitude of any of the poles of X(z). Thus, the ROC is the interior of the circle lzlE rminin the z-plane with the possible exception of z = 0.
Property 5:
If x [ n ] is a two-sided sequence (that is, x [ n ] is an infinite-duration sequence that is neither right-sided nor left-sided) and X ( z ) converges for some value of z, then the ROC is of the form
where r , and r, are the magnitudes of the two poles of X(z). Thus, the ROC is an annular ring in the z-plane between the circles lzl= r , and lzl = r2 not containing any poles. Note that Property 1 follows immediately from the definition of poles; that is, X(z) is infinite at a pole. For verification of the other properties, see Probs. 4.2 and 4.5.
z-TRANSFORMS OF SOME COMMON SEQUENCES From definition (1.45) and ( 4 . 3 )
A. Unit Impulse Sequence 61 nl:
X ( z )=
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