Page 195 - Schaum's Outline of Theory and Problems of Signals and Systems
P. 195
THE Z-TRANSFORM AND DISCRETE-TIME LTI SYSTEMS [CHAP. 4
Fig. 4-7
PROPERTIES OF THE Z-TRANSFORM
4.8. Verify the time-shifting property (4.18), that is,
By definition (4.3)
By the change of variables m = n - no, we obtain
Because of the multiplication by 2-"0, for no > 0, additional poles are introduced at r = 0 and
will be deleted at z = w. Similarly, if no < 0, additional zeros are introduced at z = 0 and will
be deleted at z = m. Therefore, the points z = 0 and z = oo can be either added to or deleted
from the ROC by time shifting. Thus, we have
where R and R' are the ROCs before and after the time-shift operation.
