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Let x[n] = (-1/9)ⁿ u(n) - (-1/3)ⁿ u(-n-1). The Region of Convergence (ROC) of the z-transform of x[n]

1. is |z| > 1/9.
2. is |z| < 1/3.
3. is 1/3 > |z| > 1/9.
4. does not exist.

Correct answer: is 1/3 > |z| > 1/9.

Solution

The ROC is determined by the behavior of the two components of the signal: the first term converges for |z| > 1/9 and the second term converges for |z| < 1/3. Therefore, the overall ROC is the intersection of these two regions, which results in the range 1/3 > |z| > 1/9.

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