Page 198 - Schaum's Outline of Theory and Problems of Signals and Systems

CHAP. 41 THE Z-TRANSFORM AND DISCRETE-TIME LTI SYSTEMS 187

(b) Differentiating Eq. (4.76) with respect to a, we have

Note that dividing both sides of Eq. (4.77) by a, we obtain Eq. (4.78).

4.13. Verify the convolution property (4.26), that is,

By definition (2.35)

Thus, by definition (4.3)

Noting that the term in parentheses in the last expression is the z-transform of the shifted
signal x2[n - k], then by the time-shifting property (4.18) we have

with an ROC that contains the intersection of the ROC of X,(z) and X,(z). If a zero of one
transform cancels a pole of the other, the ROC of Y(z) may be larger. Thus, we conclude that

4.14. Verify the accumulation property (4.25), that is,

From Eq. (2.40) we have

Thus, using Eq. (4.16) and the convolution property (4.26), we obtain

with the ROC that includes the intersection of the ROC of X(z) and the ROC of the
z-transform of u[n]. Thus,