Page 183 - Schaum's Outline of Theory and Problems of Signals and Systems

172 THE Z-TRANSFORM AND DISCRETE-TIME LTI SYSTEMS [CHAP. 4

In this special case, all poles and zeros are simply rotated by the angle R, and the ROC is
unchanged.

D. Time Reversal:

If

then

Therefore, a pole (or zero) in X(z) at z =z, moves to l/z, after time reversal. The
relationship R' = 1/R indicates the inversion of R, reflecting the fact that a right-sided
sequence becomes left-sided if time-reversed, and vice versa.

E. Multiplication by n (or Differentiation in 2):
If
~[nl ++X(Z) ROC = R
then

F. Accumulation:

If
x[nI ++X(z) ROC = R
then

Note that Cz, _,x[k] is the discrete-time counterpart to integration in the time domain
and is called the accumulation. The comparable Laplace transform operator for integra-
tion is l/~.

G. Convolution:
If
%[n] ++XI(Z) ROC =R1
~2[n] ++X2(4 ROC = R2

then
XI[.] * x2bI ++XI(Z)XZ(Z) Rt3R1 nR2 (4.26)
This relationship plays a central role in the analysis and design of discrete-time LTI
systems, in analogy with the continuous-time case.