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The symbols, a and T, represent positive quantities, and u(t) is the unit step function. Which one of the following impulse responses is NOT the output of a causal linear time-invariant system?

1. e^(−at)u(t)
2. e^(−a(t+T))u(t)
3. 1 + e^(−at)u(t)
4. e^(−a(t−T))u(t)

Correct answer: 1 + e^(−at)u(t)

Solution

The option 1 + e^(−at)u(t) is not the output of a causal linear time-invariant system because it includes a constant term (1) that does not decay over time, violating the principle that the impulse response must be zero for negative time values, which is a requirement for causality.

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