GATE EE 2022 Set 1 — Question 48
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Signals & Systems → LTI Systems → LTI, Causal & Stable Systems
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Question
Consider the system as shown below
where . The system is
where . The system is
Correct answer
(B) linear and non-causal.
Solution
The given system is .1. Linearity Test:
Let produce .
Let produce .
Consider a linear combination of inputs: .
The output for this input would be:
.
Since the output is a linear combination of the individual outputs, the system is linear.2. Causality Test:
A system is causal if its output at any time depends only on the input at the present time or past times ().
For the given system, .
Consider . The output .
Since , which is greater than , the output at depends on the input at a future time .
Therefore, the system is non-causal.Combining both properties, the system is linear and non-causal.Thus, the correct option is (B).
Let produce .
Let produce .
Consider a linear combination of inputs: .
The output for this input would be:
.
Since the output is a linear combination of the individual outputs, the system is linear.2. Causality Test:
A system is causal if its output at any time depends only on the input at the present time or past times ().
For the given system, .
Consider . The output .
Since , which is greater than , the output at depends on the input at a future time .
Therefore, the system is non-causal.Combining both properties, the system is linear and non-causal.Thus, the correct option is (B).
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