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Question 3.SGPYQ.43: The impulse response of a system is h(t) = t u(t). For an in......

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

(a) \frac{t^2}{2} u(t)                 (b) \frac{t(t-1)}{2} u(t-1)

(c) \frac{(t-1)^2}{2} u(t-1)         (d) \frac{t^2-1}{2} u(t-1)

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Given that impulse response is h(t) = tu(t). For input response u(t-1),

\begin{aligned} \delta(t) & \rightarrow t a(t) \\ u(t) & \rightarrow t u(t) d t=u \\ \int_0^t t d t & =\frac{t^2}{2} u(t) \\ u(t-1) & \rightarrow \frac{(t-1)^2}{2} u(t-1) \end{aligned}

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