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Determine the impulse response of a noncausal LTI system H(2) = It 0,5 -0.252

Determine the impulse response of a noncausal LTI system H(2) = It 0,5 -0.252

Introductory Circuit Analysis (13th Edition)
Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
Chapter1: Introduction
Section: Chapter Questions
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### 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.
Transcribed Image Text:### 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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