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?

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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.