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The impulse response of a system is h(t) = t u(t). For an input u(t-1), the output is

1. t²/2 u(t)
2. t(t-1)/2 u(t)
3. (t-1)²/2 u(t)
4. t²/2 u(t-1)

Correct answer: (t-1)²/2 u(t)

Solution

The output of a linear time-invariant system can be found by convolving the input with the impulse response. In this case, the convolution of the input u(t-1) with the impulse response h(t) = t u(t) results in the output (t-1)²/2 u(t), which reflects the shifted nature of the input.

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