Signal Decomposition for ROC

The given signal is $x[n] = (1/3)^{|n|} -(1/2)^n u[n]$.

We need to find the Region of Convergence (ROC) for its Z-transform. Let's analyze the Z-transform and ROC for each component.

ROC Calculation for (1/3)^|n|

The signal $(1/3)^{|n|}$ consists of $(1/3)^n$ for $n \ge 0$ and $(1/3)^{-n}$ for $n < 0$. The Z-transform sum converges for two separate ranges:

• For the causal part ($n \ge 0$): $\sum_{n=0}^{\infty} (1/3)^n z^{-n}$ converges for $|z| > 1/3$.
• For the anti-causal part ($n < 0$): $\sum_{n=-\infty}^{-1} (1/3)^{-n} z^{-n}$ converges for $|z| < 3$.

The ROC for the sum $(1/3)^{|n|}$ is the intersection of these two conditions.

ROC for $(1/3)^{|n|}$ is $\frac{1}{3} < |z| < 3$.

ROC Calculation for -(1/2)^n u[n]

The Z-transform of the causal signal $(1/2)^n u[n]$ is $\frac{1}{1 - (1/2)z^{-1}}$.

The associated ROC is determined by the condition $|(1/2)z^{-1}| < 1$, which simplifies to $|z| > 1/2$.

Therefore, the Z-transform of $-(1/2)^n u[n]$ has the ROC $|z| > 1/2$.

Combined ROC for Sum of Signals

The ROC of the sum of two signals ($x[n] = x_1[n] + x_2[n]$) is the intersection of their individual ROCs ($\textrm{ROC}_1 \cap \textrm{ROC}_2$).

Combining the ROCs:

For the intersection to exist, $|z|$ must satisfy both conditions simultaneously. This means $|z|$ must be greater than $1/2$ and less than $3$.

Final ROC: $\frac{1}{2} < |z| < 3$.