Let's solve the given Z-transform problem step by step.

Problem:

We are given the function $X(z)$ as: $X(z) = \frac{1 + \frac{1}{2}z^{-1}}{1 - \frac{1}{2}z^{-1}}$ We need to find the corresponding $x(n)$, which is the inverse Z-transform of $X(z)$.

Step-by-step solution:

1. Given Expression: The given Z-transform is: $X(z) = \frac{1 + \frac{1}{2}z^{-1}}{1 - \frac{1}{2}z^{-1}}$ We need to express this in a form that allows us to find the inverse Z-transform.

2. Break the Expression into Simpler Terms: Let's split the fraction into two terms. We can rewrite $X(z)$ as: $X(z) = \frac{1}{1 - \frac{1}{2}z^{-1}} + \frac{\frac{1}{2}z^{-1}}{1 - \frac{1}{2}z^{-1}}$ Now we have two terms that we can work with separately.

3. Recognize the Inverse Z-transform Patterns: The two terms can now be interpreted using known inverse Z-transform pairs. Specifically:

   • The first term, $\frac{1}{1 - \frac{1}{2}z^{-1}}$, is of the form $\frac{1}{1 - az^{-1}}$, whose inverse Z-transform is $a^n u(n)$, where $u(n)$ is the unit step function. In this case, $a = \frac{1}{2}$, so the inverse Z-transform is: $x_1(n) = \left(\frac{1}{2}\right)^n u(n)$

   • The second term, $\frac{\frac{1}{2}z^{-1}}{1 - \frac{1}{2}z^{-1}}$, is of the form $\frac{bz^{-1}}{1 - az^{-1}}$, whose inverse Z-transform is $a^{n-1} u(n-1)$. Here, $b = \frac{1}{2}$ and $a = \frac{1}{2}$, so the inverse Z-transform is: $x_2(n) = \frac{1}{2} \left(\frac{1}{2}\right)^{n-1} u(n-1)$ Simplifying this gives: $x_2(n) = \frac{1}{2^n} u(n-1)$

4. Combine the Two Results: Now that we have the inverse Z-transforms of both parts, we can combine them to get the overall inverse Z-transform: $x(n) = \left(\frac{1}{2}\right)^n u(n) + \frac{1}{2^n} u(n-1)$

This is the solution for $x(n)$.

Summary of Steps:

1. Start with the given Z-transform function.
2. Break the expression into simpler terms.
3. Recognize each term as a known Z-transform pair and apply inverse Z-transform rules.
4. Combine the results to find the final time-domain sequence.

Would you like more details on any of these steps, or do you have any specific questions?

Relative Questions:

1. How can you identify standard Z-transform pairs when solving problems?
2. What is the significance of the unit step function $u(n)$ in Z-transform problems?
3. How does the splitting of fractions help in simplifying Z-transform problems?
4. What is the physical interpretation of the Z-transform in signal processing?
5. How do you handle Z-transforms when the denominator has higher powers of $z^{-1}$?

Tip:

Always look for standard Z-transform pairs (from tables) to simplify the process of finding inverse Z-transforms in complex expressions.