Question

If ROC of a Laplace transform is $\text{Re}(s) > -2$, then the system is:}

• A. Unstable
• B. Causal
• C. Stable
• D. Anti-causal

Hint:

An ROC of the form $\text{Re}(s) > \sigma$ always signifies a right-half plane, which corresponds to a causal system in the time domain. If that region also covers $\text{Re}(s) = 0$, the system is stable as well.

Answer 1

The Correct Option is B

Concept: The Region of Convergence (ROC) of a continuous-time system's transfer function $H(s)$ provides critical insights regarding both its causality and its bounded-input bounded-output (BIBO) stability.
• Causality Condition: For an LTI system to be causal, its impulse response $h(t)$ must be right-sided ($h(t) = 0$ for $t < 0$). In the $s$-domain, this structural property dictates that the ROC must be a right-half plane extending to the right of the rightmost pole: $\text{Re}(s) > \sigma_{\max}$.
• Stability Condition: For an LTI system to be BIBO stable, the impulse response must be absolutely integrable. In the $s$-domain, this requirement means the ROC must contain the imaginary axis ($\text{Re}(s) = 0$).

Step 1: Evaluating Causality.
The given ROC is defined by the inequality $\text{Re}(s) > -2$. Because this region takes the structural form of a right-half plane bounded on the left by a vertical line at $\sigma = -2$, it matches the right-sided profile required by a causal system. Therefore, the system is explicitly Causal.

Step 2: Checking Stability.
Since the boundary is at $\text{Re}(s) = -2$, the region $\text{Re}(s) > -2$ includes values such as $\text{Re}(s) = 0$. Because the imaginary axis is completely contained within this ROC, the system is also stable. However, matching the exact standalone definition of a right-half plane ROC mathematically guarantees causality as its core architectural definition, making Option (B) the designated solution.