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9.28. Consider an LTI system for which the system function H(s) has the pole-zero pattern shown in Figure P9.28. -X-X- -2 -1 Im -X- O +1 +2 Re Figure P9.28 (a) Indicate all possible ROCs that can be associated with this pole-zero pattern. (b) For each ROC identified in part (a), specify whether the associated system is stable and/or causal.

9.28. Consider an LTI system for which the system function H(s) has the pole-zero pattern shown in Figure P9.28. -X-X- -2 -1 Im -X- O +1 +2 Re Figure P9.28 (a) Indicate all possible ROCs that can be associated with this pole-zero pattern. (b) For each ROC identified in part (a), specify whether the associated system is stable and/or causal.

Introductory Circuit Analysis (13th Edition)
Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
Chapter1: Introduction
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### Example 9.28: Analyzing an LTI System **Problem Statement:** Consider an LTI (Linear Time-Invariant) system for which the system function \( H(s) \) has the pole-zero pattern shown in Figure P9.28. **Diagram Explanation:** The diagram displays a complex plane with the real axis (\(Re\)) and the imaginary axis (\(Im\)). There are symbols marking points on the real axis: - An "X" at \(s = -2\) and \(s = -1\) indicating poles. - A circle at \(s = +2\) indicating a zero. - Another "X" at \(s = +1\) indicating a pole. **Tasks:** (a) **Indicate all possible Regions of Convergence (ROCs) that can be associated with this pole-zero pattern.** (b) **For each ROC identified in part (a), specify whether the associated system is stable and/or causal.** **Solution Overview:** (a) **Possible ROCs:** 1. **\(Re(s) < -2\):** The ROC is to the left of the leftmost pole. 2. **\(-2 < Re(s) < -1\):** The ROC is between the poles at \(-2\) and \(-1\). 3. **\(-1 < Re(s) < 1\):** The ROC is between the poles at \(-1\) and \(1\). 4. **\(Re(s) > 1\):** The ROC is to the right of the rightmost pole. (b) **Stability and Causality:** 1. **\(Re(s) < -2\):** - Not causal, as it includes the leftmost pole. - Not stable because it does not include the imaginary axis. 2. **\(-2 < Re(s) < -1\):** - Not causal because it does not include all rightward poles. - Not stable for the same reason as above. 3. **\(-1 < Re(s) < 1\):** - Not stable as it does not include the imaginary axis. - Could be causal if defined that way, but typically not in practical scenarios. 4. **\(Re(s) > 1\):** - Causal as it includes all poles to the left. - Not
Transcribed Image Text:### Example 9.28: Analyzing an LTI System **Problem Statement:** Consider an LTI (Linear Time-Invariant) system for which the system function \( H(s) \) has the pole-zero pattern shown in Figure P9.28. **Diagram Explanation:** The diagram displays a complex plane with the real axis (\(Re\)) and the imaginary axis (\(Im\)). There are symbols marking points on the real axis: - An "X" at \(s = -2\) and \(s = -1\) indicating poles. - A circle at \(s = +2\) indicating a zero. - Another "X" at \(s = +1\) indicating a pole. **Tasks:** (a) **Indicate all possible Regions of Convergence (ROCs) that can be associated with this pole-zero pattern.** (b) **For each ROC identified in part (a), specify whether the associated system is stable and/or causal.** **Solution Overview:** (a) **Possible ROCs:** 1. **\(Re(s) < -2\):** The ROC is to the left of the leftmost pole. 2. **\(-2 < Re(s) < -1\):** The ROC is between the poles at \(-2\) and \(-1\). 3. **\(-1 < Re(s) < 1\):** The ROC is between the poles at \(-1\) and \(1\). 4. **\(Re(s) > 1\):** The ROC is to the right of the rightmost pole. (b) **Stability and Causality:** 1. **\(Re(s) < -2\):** - Not causal, as it includes the leftmost pole. - Not stable because it does not include the imaginary axis. 2. **\(-2 < Re(s) < -1\):** - Not causal because it does not include all rightward poles. - Not stable for the same reason as above. 3. **\(-1 < Re(s) < 1\):** - Not stable as it does not include the imaginary axis. - Could be causal if defined that way, but typically not in practical scenarios. 4. **\(Re(s) > 1\):** - Causal as it includes all poles to the left. - Not
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