Page 218 - Schaum's Outline of Theory and Problems of Signals and Systems

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

4.45. Show the following properties for the z-transform.

(a) If x[n] is even, then X(zL') =X(z).
(b) If x[n] is odd, then X(z-') = -X(z).
(c) If x[n] is odd, then there is a zero in X(z) at z = 1.
Hint: (a) Use Eqs. (1.2) and (4.23).
(b) Use Eqs. (1.3) and (4.23).
(c) Use the result from part (b).

4.46. Consider the continuous-time signal

Let the sequence x[n] be obtained by uniform sampling of x(t) such that x[n] = x(nT,), where
T, is the sampling interval. Find the z-transform of x[n].

Am.

4.47. Derive the following transform pairs:

(sin n0n)u[n] - (sin Ro)z

z2 - (2cos Ro)z + 1 lzl> I

Hint: Use Euler's formulas.

and use Eqs. (4.8) and (4.10) with a = e * ''0.

4.48. Find the z-transforms of the following x[n]:
(a) x[n] = (n - 3)u[n - 31
(b) x[nl = (n - 3)u[n]
(c) x[nl = u[n] - u[n - 31
(dl x[nl = n{u[nl- u[n - 33)
z-~
Am. (a) ,, lz1> 1
(2- 1)