Question

The unit impulse response of a linear time invariant system is the unit step function U(t). For t >0, the response of the system to an excitation $e^{-at}u\left(t\right),a>0$ will be

Answer 1

Given that,
Impulse response = h(t) = u(t)
So, transfer function
$$H\left(s\right)=\frac{1}{s}$$$$H\left(s\right)=\frac{Y\left(s\right)}{X\left(s\right)}$$$$Y\left(s\right)=H\left(s\right)\times X\left(s\right)$$Given that$$x\left(t\right)=e^{-at}u\left(t\right)$$$$X\left(s\right)=\frac{1}{(s+a)}$$$$Y\left(s\right)=\frac{1}{s}\times \frac{1}{(s+a)}$$$$Y\left(s\right)=\frac{1}{a}[\frac{1}{s}-\frac{1}{s+a}]$$Taking inverse Laplace transform$$Y\left(t\right)=\frac{1}{a}[1-e^{-at}]$$