Question

Which one of the following statements is NOT TRUE for a continuous time causal and stable LTI system?

• A. All the poles of the system must lie on the left side of the $j\omega$ axis.
• B. Zeros of the system can lie anywhere in the $s$-plane.
• C. All the poles must lie within $|s|=1$.
• D. All the roots of the characteristic equation must be located on the left side of the $j\omega$ axis.

Hint:

Do not confuse continuous-time stability conditions with discrete-time stability conditions.

Answer 1

The Correct Option is C

Step 1: Stability condition for continuous-time LTI systems.
For a continuous-time causal and stable LTI system, all poles must lie strictly in the left half of the $s$-plane.
Step 2: Analyze each option.
Option (A): Correct — poles must lie on the left of the imaginary axis.
Option (B): Correct — zeros do not affect stability and can be anywhere.
Option (C): Incorrect — the condition $|s|<1$ applies to discrete-time systems, not continuous-time systems.
Option (D): Correct — roots of the characteristic equation are poles and must lie in the left half-plane.
Step 3: Final conclusion.
Hence, option (C) is NOT true for a continuous-time causal and stable LTI system.

Answer 2

The question asks which statement is NOT true for a continuous-time causal, stable LTI system, so each option is checked against the actual stability requirement for continuous-time systems, which is that all poles lie strictly in the left half of the \( s \)-plane (equivalently, to the left of the \( j\omega \) axis).

1. Option "All the poles of the system must lie on the left side of the \( j\omega \) axis": This is exactly the standard stability condition for continuous-time causal LTI systems, so this statement is true, meaning it is not the answer being sought (the question asks for the false statement).
2. Option "Zeros of the system can lie anywhere in the \( s \)-plane": Stability and causality are governed entirely by pole locations; zero locations affect the system's frequency response shape (and whether it is minimum-phase) but do not affect BIBO stability at all, so zeros are indeed free to lie anywhere, making this statement true as well.
3. Option "All the poles must lie within \( |s|=1 \)": The condition \( |s|<1 \), a unit circle in the \( s \)-plane (or more precisely, the \( z \)-plane), is the stability criterion used for discrete-time systems, not continuous-time ones; applying it here to a continuous-time system is a category error, so this statement is false for the system being described, making it the statement that is NOT true.
4. Option "All the roots of the characteristic equation must be located on the left side of the \( j\omega \) axis": The characteristic equation's roots are precisely the system's poles, so this is just a restatement of the correct left-half-plane condition using different terminology, and it is true.

Three of the four statements correctly describe continuous-time stability behaviour, while one substitutes the discrete-time unit-circle criterion in its place.

So the correct answer is All the poles must lie within \( |s|=1 \).