Lti Systems

Step 1: Define impulse and step responses.
The impulse response of an LTI system is the output when the input is a unit impulse .
Step 2: Relationship between unit step and impulse.
The unit impulse is the derivative of the unit step function:

Step 3: System response relation.
If is the step response of the system, then:

Step 4: Eliminate incorrect options.
Integrating responses does not yield the impulse response directly. Differentiating the ramp response gives the step response, not impulse.
Step 5: Final conclusion.
Hence, the impulse response can be obtained by differentiating the unit step response.

Step 1: Relation between step and impulse response.
For an LTI system, the impulse response is the derivative of the step response .
Step 2: Given step response.

Step 3: Differentiate the step response.

Step 4: Final conclusion.
Thus, the unit impulse response is .

Step 1: Stability condition for continuous-time LTI systems.
For a continuous-time causal and stable LTI system, all poles must lie strictly in the left half of the -plane.
Step 2: Analyze each option.
Option (A): Correct — poles must lie on the left of the imaginary axis.
Option (B): Correct — zeros do not affect stability and can be anywhere.
Option (C): Incorrect — the condition applies to discrete-time systems, not continuous-time systems.
Option (D): Correct — roots of the characteristic equation are poles and must lie in the left half-plane.
Step 3: Final conclusion.
Hence, option (C) is NOT true for a continuous-time causal and stable LTI system.