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THE z-TRANSFORM AND DISCRETE-TIME LTI SYSTEMS
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Applying the z-transform and using the time-shift property (4.18) and the linearity property (4.17) of the z-transform, we obtain
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Hence, H ( z ) is always rational. Note that the ROC of H ( z ) is not specified by Eq. (4.44) but must be inferred with additional requirements on the system such as the causality or the stability.
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D. Systems Interconnection:
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For two LTI systems (with h,[n] and h2[n], respectively) in cascade, the overall impulse response h[n] is given by h[nl = h , [ n l * h 2 b l Thus, the corresponding system functions are related by the product (4.45)
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Similarly, the impulse response of a parallel combination of two LTI systems is given by
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=h,[nl +h*lnl
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(4.47)
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THE UNILATERAL Z-TRANSFORM Definition:
The unilateral (or one-sided) z-transform X,(z) of a sequence x[n] is defined as [Eq. (4.511 X,(z) =
x[n]z-"
(4.49)
and differs from the bilateral transform in that the summation is carried over only n 2 0. Thus, the unilateral z-transform of x[n] can be thought of as the bilateral transform of x[n]u[n]. Since x[n]u[n] is a right-sided sequence, the ROC of X,(z) is always outside a circle in the z-plane.
THE 2-TRANSFORM AND DISCRETE-TIME LTI SYSTEMS
[CHAP. 4
B. Basic Properties:
Most of the properties of the unilateral z-transform are the same as for the bilateral z-transform. The unilateral z-transform is useful for calculating the response of a causal system to a causal input when the system is described by a linear constant-coefficient difference equation with nonzero initial conditions. The basic property of the unilateral z-transform that is useful in this application is the following time-shifting property which is different from that of the bilateral transform.
Time-Shifting Property:
If x[n] t X,( z ), then for m 2 0, , x[n - m ] - Z - ~ X , ( Z ) +z-"+'x[-11 +z-"+~x[-~+ ]
+x[-m]
x [ n + m] t,zmX,(z) -zmx[O] - z m - ' x [ l ] -
. . - - ~ [ m 11 -
The proofs of Eqs. (4.50) and (4.51) are given in Prob. 4.36.
D. System Function:
Similar to the case of the continuous-time LTI system, with the unilateral z-transform, the system function H(z) = Y(z)/X(z) is defined under the condition that the system is relaxed, that is, all initial conditions are zero.
Solved Problems
THE Z-TRANSFORM
Find the z-transform of
(a) From Eq. ( 4 . 3 )
By Eq. (1.91)
( a - ~ z =~ ) I n =O
- a-'z
if la-'zl< 1 or lz( < la1
CHAP. 41
THE Z-TRANSFORM AND DISCRETE-TIME LTI SYSTEMS
Thus, X(z) = 1( b ) Similarly,
1 1-a-'z
-a-'z -
1-a-'z
1 z z-a 1-az-'
Izl < I4
(4.52)
Again by Eq. (1.91)
Thus,
A finite sequence x [ n ] is defined as
=O N, I n I N , otherwise
where N, and N, are finite. Show that the ROC of X(z) is the entire z-plane except possibly z = 0 or z = m.
From Eq. (4.3)
For z not equal to zero or infinity, each term in Eq. (4.54) will be finite and thus X(z) will converge. If N, < 0 and N2 > 0, then Eq. (4.54) includes terms with both positive powers of z and negative powers of z. As lzl- 0, terms with negative powers of z become unbounded, and as lzl+ m, terms with positive powers of z become unbounded. Hence, the ROC is the entire z-plane except for z = 0 and z = co. If N, 2 0, Eq. (4.54) contains only negative powers of z, and hence the ROC includes z = m. If N, I 0, Eq. (4.54) contains only positive powers of z, and hence the ROC includes z = 0.
A finite sequence x [ n ] is defined as
Find X(z) and its ROC.
THE Z-TRANSFORM AND DISCRETE-TIME LTI SYSTEMS
[CHAP. 4
From Eq. (4.3) and given x [ n ] we have
=5~~+3~-2+4z-~-3z-~
For z not equal to zero or infinity, each term in X ( z ) will be finite and consequently X ( z ) will converge. Note that X ( z ) includes both positive powers of z and negative powers of z. Thus, from the result of Prob. 4.2 we conclude that the ROC of X ( z ) is 0 < lzl < m.
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