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Solve this z transform problem and explain the steps throughly 

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?

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### 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.

Solve this Z-transform problem and explain the steps thoroughly

This page provides a step-by-step solution for finding the inverse Z-transform of X(z) = (1 + 1/2 z^{-1}) / (1 - 1/2 z^{-1}). The solution involves breaking down the expression into simpler terms and recognizing standard Z-transform pairs to derive the time-domain sequence x(n).

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