Question

S(t) is step response and h(t) is impulse response of a system. Its response y(t) for any input u(t) is given by

Answer 1

$\text{y(t) = u(t) * h(t) (always)}$
$\text{where h(t) is impulse response}$
$\because \text{s(t) step response (given)}$

$\Rightarrow \frac{d}{dt}\text{s(t) = h(t)}$
$\therefore \text{y(t) = u(t) *}\frac{d}{dt}\text{s(t)}$
$\text{=}\frac{d}{dt}\text{[u(t) * s(t)] (using property)}$

$\text{=}\frac{d}{dt}{\int }_{-\infty }^{\infty }\text{u(}\tau \text{) s(t-}\tau \text{) d}\tau$

$\text{(by definition of convolution)}$
$\text{=}\frac{d}{dt}{\int }_0^t\text{u(}\tau \text{) s(t-}\tau \text{)d}\tau$

$\text{(}\because \text{u(t) and s(t) are causal functions)}$