What are the necessary conditions for stability in $s$ domain, especially in regards to ROC.

I am able to understand that there are only two conditions

  1. First condition: all poles must be on left half plane.

  2. Second condition: ROC must include imaginary axis.

Are these the correct conditions? Are there other conditions?


One piece you are missing is the idea of causality.

Take, for example, a system with impulse response

$$ h(t) = e^{-at}u(t)+e^{at}u(-t) $$

This has poles at $+a$ and $-a$, but has a region of convergence that includes the imaginary axis and so is BIBO stable.

BIBO stability does not imply causality. Causality does not imply BIBO stability.

The only condition for BIBO stability is for the region of convergence to include the imaginary axis.

Your first condition all poles must be on left half plane is an indication that you require the stable system to also be causal.

  • 1
    $\begingroup$ Usually when you take a second-year undergraduate Signals and Systems course you're only presented with causal systems. You don't get into the whole field of two-sided Laplace transforms and noncausal-but-stable systems until you revisit signal processing at the graduate level. So -- it's OK if what Peter is saying is new to you. But it's still true, and has practical use (in that you can approximate a non-causal response with delay plus a truncated-from-the-left impulse response). $\endgroup$
    – TimWescott
    Sep 30 '21 at 16:50

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