With reference to stability of continuous-time LTI systems, consider the following statements:1.A system is stable if all poles of its transfer function lie in the left half of the s-plane.2.Inclusion of the imaginary axis in the ROC ensures bounded output for bounded input.3.Absolute integrability of the impulse response guarantees system stability.4.Every causal system is automatically stable.Which of the above statements is/are correct?

Correct Answer: B - 1, 2 and 3 only

Quick Concept Explanation

The impulse response uniquely characterizes an LTI system for a specified input-output relationship because any input can be decomposed into shifted impulses and the output follows by convolution. This concept applies in both continuous and discrete time. Key relation: y(t)=x(t)*h(t); y[n]=x[n]*h[n].

Exam focus: For LTI questions, think 'impulse response + convolution'. Common trap: Impulse response is not a discrete-time-only concept.

Statement-wise Verification

• Statement 1 - Correct.
  A system is stable if all poles of its transfer function lie in the left half of the s-plane.
  Y(t)=x(t)*h(t); y[n]=x[n]*h[n].
• Statement 2 - Correct.
  Inclusion of the imaginary axis in the ROC ensures bounded output for bounded input.
  The impulse response uniquely characterizes an LTI system for a specified input-output relationship because any input can be decomposed into shifted impulses and the output follows by convolution.
• Statement 3 - Correct.
  Absolute integrability of the impulse response guarantees system stability.
  For LTI questions, think 'impulse response + convolution'.
• Statement 4 - Incorrect.
  Every causal system is automatically stable.
  A causal LTI system can have right-half-plane poles and be unstable

Core Concept

The impulse response uniquely characterizes an LTI system for a specified input-output relationship because any input can be decomposed into shifted impulses and the output follows by convolution. This concept applies in both continuous and discrete time.

Formula / Key Relationship

y(t)=x(t)*h(t); y[n]=x[n]*h[n].

Why the Other Options Are Wrong

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

Exam Shortcut / Approach

For LTI questions, think 'impulse response + convolution'.

Common Mistake

Impulse response is not a discrete-time-only concept.

Quick Revision

Pole Location ROC and Impulse Integrability is linked with Stability, LTI Systems and Signals and Systems. For LTI questions, think 'impulse response + convolution'.