Question

2. For each of the Z transforms below: a. Sketch the Pole-Zero plot. b. Indicate if there is a ROC for which it is stable. c. Indicate if there is a ROC for which it is causal. d. Indicate if there is a ROC for which it is both causal and stable. $X_1(z) = \frac{1 - \frac{1}{2}z^{-1}}{1 + 2z^{-1}}$ $X_2(z) = \frac{1 - \frac{1}{3}z^{-1}}{(1 + \frac{1}{2}z^{-1})(1 - \frac{2}{3}z^{-1})}$ $X_3(z) = \frac{1 + z^{-1} - 2z^{-2}}{1 - \frac{11}{6}z^{-1} + z^{-2}}$

          2. For each of the Z transforms below:
a. Sketch the Pole-Zero plot.
b. Indicate if there is a ROC for which it is stable.
c. Indicate if there is a ROC for which it is causal.
d. Indicate if there is a ROC for which it is both causal and stable.
$X_1(z) = \frac{1 - \frac{1}{2}z^{-1}}{1 + 2z^{-1}}$
$X_2(z) = \frac{1 - \frac{1}{3}z^{-1}}{(1 + \frac{1}{2}z^{-1})(1 - \frac{2}{3}z^{-1})}$
$X_3(z) = \frac{1 + z^{-1} - 2z^{-2}}{1 - \frac{11}{6}z^{-1} + z^{-2}}$

2. For each of the Z transforms below:
a. Sketch the Pole-Zero plot.
b. Indicate if there is a ROC for which it is stable.
c. Indicate if there is a ROC for which it is causal.
d. Indicate if there is a ROC for which it is both causal and stable.
X1(z) = (1 - (1)/(2)z^-1)/(1 + 2z^-1)
X2(z) = (1 - (1)/(3)z^-1)/((1 + (1)/(2)z^-1)(1 - (2)/(3)z^-1))
X3(z) = (1 + z^-1 - 2z^-2)/(1 - (11)/(6)z^-1 + z^-2)

Added by Maria S.

University Physics with Modern Physics

Hugh D. Young 14th Edition

Instant Answer

Solved by Expert Ronald Prasad

Step 1

There is a ROC for which it is causal because the pole is inside the unit circle. There is no ROC for which it is both causal and stable. For the second Z transform, 1 + 2 - 11z^(-1), the pole-zero plot is as follows: Pole: z = 11 Zero: None For this Z Show more…

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For each of the Z transforms below: a. Sketch the Pole-Zero plot. b. Indicate if there is a ROC for which it is stable. c. Indicate if there is a ROC for which it is causal. d. Indicate if there is a ROC for which it is both causal and stable. 1 - z^(-1) 1 - z^(-1) X^2z = 1 + 2 - 11z^(-1) 1 + z^(-1) - 2z^(-2) X^3z = 1 - 1z^(-1) + z^(-2)