Question

Which of the following is not true for a continuous-time causal and stable LTI system?

• A. All poles lie on the left half of s-plane
• B. Zeros can lie anywhere in s-plane
• C. All poles lie within \( |s|=1 \)
• D. Roots of characteristic eqn lie left of jω-axis

Hint:

For CT LTI stability: poles must lie in the left half of s-plane.

Answer 1

The Correct Option is C

A continuous-time causal and stable Linear Time-Invariant (LTI) system is characterized by specific properties related to its poles and zeros. To analyze which statement is not true for such a system, let's break down the concepts:

• All poles lie on the left half of s-plane: For a system to be stable, all poles must be located on the left half of the complex s-plane. This ensures that the system's response does not grow unbounded over time.
• Zeros can lie anywhere in s-plane: The location of zeros does not affect the stability of the system. They can be anywhere in the s-plane without influencing the system's causal and stable nature.
• All poles lie within \(|s|=1\): This condition is relevant for discrete-time systems involving the unit circle in the z-plane, not continuous-time systems in the s-plane. Thus, this statement does not apply to continuous-time causal and stable systems.
• Roots of characteristic eqn lie left of jω-axis: This statement is akin to saying the poles must be in the left half-plane, which, as previously stated, is true for stability in continuous-time systems.

Therefore, the statement "All poles lie within \(|s|=1\)" is not true for a continuous-time causal and stable LTI system. This condition applies to discrete-time systems and reflects a misunderstanding in the context of continuous-time systems. As such, it is the correct answer to the given question.