**Transcription:**For a continuous-time Linear Time-Invariant (LTI) system to be Bounded Input Bounded Output (BIBO) stable, demonstrate that the Region of Convergence (ROC) of its system function \( H(s) \) must include the imaginary axis, i.e., \( s = j\omega \).---**Explanation:**This text refers to the properties of LTI systems and their stability in the context of the Laplace Transform. Here's a breakdown:- **Continuous-time LTI System:** A system that is linear and time-invariant, meaning its behavior and characteristics do not change over time.- **BIBO Stability:** A stability criterion where a system's output remains bounded whenever the input is bounded.- **ROC (Region of Convergence):** In the context of the Laplace Transform, ROC is the range of values in the complex plane for which the Laplace Transform of a signal converges.- **System Function \( H(s) \):** Represents the system's response in the s-domain (Laplace domain).- **Imaginary Axis (\( s = j\omega \))**: Refers to the vertical line in the complex plane where the real part is zero and the imaginary part varies (common in frequency analysis).This concept is important for determining the stability and performance of signals and systems in control theory and signal processing. Understanding these principles helps in designing systems that behave predictably and remain stable under various input conditions.

A continuous-time system is called BIBO stable if and only if for any bounded input, the output of…

Answered: If a continuous-time LTI system is BIBO…

If a continuous-time LTI system is BIBO stable, then show that the ROC of its system function H(s) must contain the imaginary axis; that is, s = jo. %3D

Introductory Circuit Analysis (13th Edition)

13th Edition

ISBN:9780133923605

Author:Robert L. Boylestad

Publisher:Robert L. Boylestad

Chapter1: Introduction

Section: Chapter Questions

Problem 1P: Visit your local library (at school or home) and describe the extent to which it provides literature...

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Question

**Transcription:**

For a continuous-time Linear Time-Invariant (LTI) system to be Bounded Input Bounded Output (BIBO) stable, demonstrate that the Region of Convergence (ROC) of its system function \( H(s) \) must include the imaginary axis, i.e., \( s = j\omega \).

---

**Explanation:**

This text refers to the properties of LTI systems and their stability in the context of the Laplace Transform. Here's a breakdown:

- **Continuous-time LTI System:** A system that is linear and time-invariant, meaning its behavior and characteristics do not change over time.
- **BIBO Stability:** A stability criterion where a system's output remains bounded whenever the input is bounded.
- **ROC (Region of Convergence):** In the context of the Laplace Transform, ROC is the range of values in the complex plane for which the Laplace Transform of a signal converges.
- **System Function \( H(s) \):** Represents the system's response in the s-domain (Laplace domain).
- **Imaginary Axis (\( s = j\omega \))**: Refers to the vertical line in the complex plane where the real part is zero and the imaginary part varies (common in frequency analysis).

This concept is important for determining the stability and performance of signals and systems in control theory and signal processing. Understanding these principles helps in designing systems that behave predictably and remain stable under various input conditions.

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