Regarding the concepts of causality and stability in continuous-time systems, consider the following statements:1.A causal system produces output that depends only on present and past input values, not future inputs.2.A system is said to be BIBO stable if every bounded input produces a bounded output.3.All causal systems are inherently stable due to their dependence on past inputs only.4.For an LTI system, inclusion of the imaginary axis in the ROC of its Laplace transform implies stability.Which of the above statements is/are correct?

Correct Answer: C - 1, 2 and 4 only

Quick Concept Explanation

The Laplace transform must be considered together with its region of convergence. For a rational LTI system, BIBO stability requires the ROC to include the jω axis; causality imposes a right-sided ROC for a causal rational system. Key relation: X(s)=∫x(t)e^(−st)dt; stability ⇒ jω axis lies in ROC.

Exam focus: Transform expression + ROC identifies the time signal. Common trap: Never place a pole inside the ROC.

Statement-wise Verification

• Statement 1 - Correct.
  A causal system produces output that depends only on present and past input values, not future inputs.
  Causality imposes a right-sided ROC for a causal rational system.
• Statement 2 - Correct.
  A system is said to be BIBO stable if every bounded input produces a bounded output.
  For a rational LTI system, BIBO stability requires the ROC to include the jω axis
• Statement 3 - Incorrect.
  All causal systems are inherently stable due to their dependence on past inputs only.
  Causality and stability are independent properties; a causal system can be unstable
• Statement 4 - Correct.
  For an LTI system, inclusion of the imaginary axis in the ROC of its Laplace transform implies stability.
  For a rational LTI system, BIBO stability requires the ROC to include the jω axis

Core Concept

The Laplace transform must be considered together with its region of convergence. The ROC excludes poles, and the same algebraic transform can represent different time signals with different ROCs. For a rational LTI system, BIBO stability requires the ROC to include the jω axis; causality imposes a right-sided ROC for a causal rational system.

Formula / Key Relationship

X(s)=∫x(t)e^(−st)dt; stability ⇒ jω axis lies in ROC.

Why the Other Options Are Wrong

Option A is incorrect because it omits correct statement(s) 4. Option B is incorrect because it includes incorrect statement(s) 3 and omits correct statement(s) 1, 4. Option D is incorrect because it includes incorrect statement(s) 3.

Exam Shortcut / Approach

Transform expression + ROC identifies the time signal.

Common Mistake

Never place a pole inside the ROC.

Quick Revision

BIBO Stability and Laplace ROC is linked with Causality and Stability, System Properties and Signals and Systems. Transform expression + ROC identifies the time signal.