### Determining the Impulse Response of a Noncausal LTI System**Task:** Determine the impulse response of a noncausal Linear Time-Invariant (LTI) system.#### Given System Function: \[ H(z) = \frac{1 + 0.5z^{-1}}{1 - 0.25z^{-1}} \]#### Region of Convergence (ROC):\[ |z| < 0.25 \]### Explanation of the Given Information:**1. System Function \( H(z) \):** - This is a rational function representing the Z-transform of the system's impulse response. - The numerator \( 1 + 0.5z^{-1} \) includes the direct term and one delayed term. - The denominator \( 1 - 0.25z^{-1} \) also includes one term that suggests a pole at \( z = 0.25 \).**2. Region of Convergence (ROC) \( |z| < 0.25 \):** - The ROC is the region where the Z-transform converges. - For this system, the ROC being \( |z| < 0.25 \) indicates a noncausal system.### Understanding Noncausal Systems:Noncausal systems have impulse responses that are not zero for \( n < 0 \), meaning the system's output depends on future input values.**Interactive Steps:**1. **Find the Inverse Z-Transform:** - Express the Z-transfer function in partial fractions if necessary. - Use Z-transform tables or properties to find the corresponding time-domain impulse response.2. **Analyze the System in Time Domain:** - Determine the nature of the system (causal, noncausal, or both). - Check the ROC to fully understand the system behavior.**Graphs and Diagrams:**There are no specific graphs or diagrams given in this context. However, if provided, they might illustrate poles and zeroes on the Z-plane, system responses, and ROC shapes to help visualize the problem and solution.### Conclusion:In this exercise, we determined the impulse response and analyzed the ROC condition for a given noncausal LTI system represented by its Z-transform. Understanding these fundamentals is crucial for signal processing and control systems.

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Question

### Determining the Impulse Response of a Noncausal LTI System

**Task:**
Determine the impulse response of a noncausal Linear Time-Invariant (LTI) system.

#### Given System Function:
\[ H(z) = \frac{1 + 0.5z^{-1}}{1 - 0.25z^{-1}} \]

#### Region of Convergence (ROC):
\[ |z| < 0.25 \]

### Explanation of the Given Information:

**1. System Function \( H(z) \):**
- This is a rational function representing the Z-transform of the system's impulse response.
- The numerator \( 1 + 0.5z^{-1} \) includes the direct term and one delayed term.
- The denominator \( 1 - 0.25z^{-1} \) also includes one term that suggests a pole at \( z = 0.25 \).

**2. Region of Convergence (ROC) \( |z| < 0.25 \):**
- The ROC is the region where the Z-transform converges.
- For this system, the ROC being \( |z| < 0.25 \) indicates a noncausal system.

### Understanding Noncausal Systems:
Noncausal systems have impulse responses that are not zero for \( n < 0 \), meaning the system's output depends on future input values.

**Interactive Steps:**
1. **Find the Inverse Z-Transform:**
- Express the Z-transfer function in partial fractions if necessary.
- Use Z-transform tables or properties to find the corresponding time-domain impulse response.

2. **Analyze the System in Time Domain:**
- Determine the nature of the system (causal, noncausal, or both).
- Check the ROC to fully understand the system behavior.

**Graphs and Diagrams:**
There are no specific graphs or diagrams given in this context. However, if provided, they might illustrate poles and zeroes on the Z-plane, system responses, and ROC shapes to help visualize the problem and solution.

### Conclusion:
In this exercise, we determined the impulse response and analyzed the ROC condition for a given noncausal LTI system represented by its Z-transform. Understanding these fundamentals is crucial for signal processing and control systems.

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