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

This is a conceptual question distinguishing continuous-time from discrete-time stability criteria, rather than a calculation, so here is the reasoning for each option.
For continuous-time causal LTI systems, BIBO stability strictly requires every pole to lie in the open left half of the \( s \)-plane, i.e. to the left of the \( j\omega \) axis; this is true both when phrased directly (poles left of \( j\omega \)) and when phrased via the characteristic equation's roots, since those roots are the poles.
Zero locations never affect stability, so "zeros can lie anywhere" is also a true statement about such systems.
The condition \( |s|<1 \) (inside a unit circle) is the discrete-time stability criterion applied to the \( z \)-plane, and mistakenly applying it to a continuous-time \( s \)-plane system makes that particular statement false. \[ \boxed{\text{False statement: poles confined to } |s|=1} \]

Answer 2

Another way to isolate the false statement is to recall precisely which plane and which region each stability criterion belongs to, then match that against every option.

1. Option "All the poles of the system must lie on the left side of the \( j\omega \) axis": This correctly identifies the \( s \)-plane left-half region used for continuous-time stability, matching the standard criterion exactly, so it is a true statement.
2. Option "Zeros of the system can lie anywhere in the \( s \)-plane": Since stability is a pole-only property (it concerns whether bounded inputs produce bounded outputs, which depends on the system's natural response modes, i.e. its poles), zero placement has no bearing on it, so this statement holds true regardless of where zeros sit.
3. Option "All the poles must lie within \( |s|=1 \)": This describes a bounded disk region in the complex plane, which is the correct picture for a discrete-time system's \( z \)-plane stability (poles inside the unit circle), not for a continuous-time system's \( s \)-plane, where the relevant region is a half-plane, not a disk; using the wrong plane's criterion here makes the statement false.
4. Option "All the roots of the characteristic equation must be located on the left side of the \( j\omega \) axis": Because a system's characteristic equation roots are identical to its poles, this statement is mathematically equivalent to the first option and is therefore also true.

Recognising which geometric region (half-plane vs. unit disk) belongs to which domain (continuous vs. discrete time) reveals exactly one mismatched statement.

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