Determine the Z-transform and region-of-convergence (ROC) of the following signal:
x[n]=−(1/​3)^n u[n]−(1/2​)^n u[-n−1]

We are given the signal:

\[
x[n] = -\left(\frac{1}{3}\right)^n u[n] - \left(\frac{1}{2}\right)^n u[-n-1]
\]

where \( u[n] \) is the unit step function. We need to determine the Z-transform \( X(z) \) of this signal and its region of convergence (ROC).

### Step 1: Split the signal
We can split the signal into two parts:
- The first part is \( x_1[n] = -\left(\frac{1}{3}\right)^n u[n] \).
- The second part is \( x_2[n] = -\left(\frac{1}{2}\right)^n u[-n-1] \).

### Step 2: Z-transform of \( x_1[n] \)
The signal \( x_1[n] = -\left(\frac{1}{3}\right)^n u[n] \) is a right-sided signal (defined for \( n \geq 0 \)). The Z-transform of a general right-sided signal \( a^n u[n] \) is given by:

\[
\mathcal{Z}\left\{a^n u[n]\right\} = \frac{1}{1 - az^{-1}}, \quad |z| > |a|
\]

For \( x_1[n] = -\left(\frac{1}{3}\right)^n u[n] \), the Z-transform is:

\[
X_1(z) = -\frac{1}{1 - \frac{1}{3}z^{-1}} = -\frac{z}{z - \frac{1}{3}}, \quad |z| > \frac{1}{3}
\]

### Step 3: Z-transform of \( x_2[n] \)
The signal \( x_2[n] = -\left(\frac{1}{2}\right)^n u[-n-1] \) is a left-sided signal (defined for \( n \leq -1 \)). The Z-transform of a general left-sided signal \( a^n u[-n-1] \) is given by:

\[
\mathcal{Z}\left\{a^n u[-n-1]\right\} = \frac{1}{1 - az^{-1}}, \quad |z| < |a|
\]

For \( x_2[n] = -\left(\frac{1}{2}\right)^n u[-n-1] \), the Z-transform is:

\[
X_2(z) = -\frac{1}{1 - \frac{1}{2}z^{-1}} = -\frac{z}{z - \frac{1}{2}}, \quad |z| < \frac{1}{2}
\]

### Step 4: Total Z-transform
The total Z-transform \( X(z) \) is the sum of the Z-transforms of \( x_1[n] \) and \( x_2[n] \):

\[
X(z) = X_1(z) + X_2(z) = -\frac{z}{z - \frac{1}{3}} - \frac{z}{z - \frac{1}{2}}
\]

### Step 5: Region of Convergence (ROC)
The ROC of the signal is determined by the individual ROCs of the two components:

- For \( X_1(z) \), the ROC is \( |z| > \frac{1}{3} \).
- For \( X_2(z) \), the ROC is \( |z| < \frac{1}{2} \).

Therefore, the total ROC is the intersection of these two regions:

\[
\frac{1}{3} < |z| < \frac{1}{2}
\]

### Final Answer:
The Z-transform of the signal is:

\[
X(z) = -\frac{z}{z - \frac{1}{3}} - \frac{z}{z - \frac{1}{2}}
\]

The region of convergence (ROC) is:

\[
\frac{1}{3} < |z| < \frac{1}{2}
\]

Would you like further details or explanations on any step?

---

Here are 5 related questions you might explore next:
1. How does the ROC affect the stability of a system?
2. How would the ROC change if \( u[-n-1] \) were replaced with \( u[n+1] \)?
3. What is the inverse Z-transform of \( X(z) \)?
4. How does the Z-transform relate to the Fourier transform?
5. How can we determine the poles and zeros from the Z-transform?

**Tip:** The region of convergence is always crucial for determining the behavior and causality of the system.

Determine the Z-transform and region-of-convergence (ROC) of the following signal: x[n]=−(1/3)^n u[n]−(1/2)^n u[-n−1]

Explore the Z-transform and region of convergence (ROC) for the signal x[n] = -(1/3)^n u[n] - (1/2)^n u[-n−1]. This solution involves key concepts from digital signal processing, including the Z-transform of right-sided and left-sided signals. Detailed steps provided for college-level students.

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