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Consider the following statements about the z-transform, X(z) of the sequence
x(n) = 0 for n < 0 = 2n for n ≥ 0
(ROC denotes region of convergence in the z-plane)1. X(z) = 1 ROC: |z| > 2 1 – 2z– 1 2. X(z) = 1 + 2z– 1 ROC: |z| > 2 1 – 2z– 1
3. X(z) = 1 ROC: |z| > 2 1 – 2z– 1
Which of these statements are correct?
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- 1 and 3
- 1 and 2
- 2 and 3
- 1, 2 and 3
Correct Option: B
Given,
| x(t) = |  | 0, | for n < 0 |
then
| X(z) = | = | |z| > 2 | ||
| z - 2 | 1 - 2z-1 |
| (1) X(z) = | ROC, |z| > 2 → is true | |
| 1 - 2z-1 |
| (2) X(z) = 1 + | ROC : |z| > 2 | |
| 1 - 2z-1 |
| = | ; ROC : |z| > 2 | |
| 1 - 2z-1 |
| = | ;ROC: |z| > 2 → Also true | |
| 1 - 2z-1 |
(3) is not true since x(n) = 0 for n < 0.
Hence, alternative (B) is the correct choice.
