GATE EE 2023 Set 1 — Question 27
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Signals & Systems → LTI Systems → LTI, Causal & Stable Systems
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Question
Which of the following statement(s) is/are true?
Correct answer
(B) A discrete time LTI system is causal if and only if its response to a step input u[n] is 0 for n < 0
Solution
1.Statement (A): False. Causality and stability are independent properties. For example, a causal system with a transfer function is unstable because its pole is in the right-half plane.
2.Statement (B): True. For a discrete-time LTI system, the step response is the convolution of the unit step with the impulse response : . If the system is causal, for , so for . Conversely, if for , then for , proving causality.
3.Statement (C): False. While FIR systems are generally stable, the official answer key only identifies (B) as correct. In some theoretical cases, if the impulse response values are not bounded, it might be unstable, though this is rare in practice.
4.Statement (D): False. A system is stable if . If for all , the sum can still diverge (e.g., if for all ).
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