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Important Properties of the z-Transform 107
4. The sequence x 3 (n) inExample 4.3 is a two-sided sequence. The ROC
for two-sided sequences is always an open ring R x− < |z| n 2 are called
finite-duration sequences. The ROC for such sequences is the entire
z-plane. Ifn 1 < 0, then z = ∞ is not in the ROC. If n 2 > 0, then
z =0isnot in the ROC.
6. The ROC cannot include a pole since X(z) converges uniformly in
there.
7. There is at least one pole on the boundary of a ROC of a rational X(z).
8. The ROC is one contiguous region; that is, the ROC does not come in
pieces.
In digital signal processing, signals are assumed to be causal since
almost every digital data is acquired in real time. Therefore the only
ROC ofinterest to us is the one given in statement 2.
4.2 IMPORTANT PROPERTIES OF THE z-TRANSFORM
The properties of the z-transform are generalizations of the properties
of the discrete-time Fourier transform that we studied in Chapter 3. We
state the following important properties of the z-transform without proof.
1. Linearity:
Z [a 1 x 1 (n)+a 2 x 2 (n)] = a 1 X 1 (z)+a 2 X 2 (z);
ROC: ROC x1 ∩ ROC x2
(4.4)
2. Sample shifting:
3. Frequency shifting:
4. Folding:
Z [x (n − n 0 )] = z −n0 X(z); ROC: ROC x (4.5)
Z [a n x(n)] = X
( z
a)
; ROC: ROC x scaled by |a| (4.6)
Z [x (−n)] = X (1/z); ROC: Inverted ROC x (4.7)
5. Complex conjugation:
Z [x ∗ (n)] = X ∗ (z ∗ ); ROC: ROC x (4.8)
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