Part A
Consider the system:
$ y(t)=\int_{-\infty}^{\infty}3x(t-1)dt $
The unit impulse response is then $ h(t) =3u(t-1) $
Using $ H(s) = \int_{-\infty}^{\infty}h(t)e^{-st}dt $
we find that
$ H(s) = \int_{-\infty}^{\infty}3u(t-1)e^{-st}dt $
$ =\int_{1}^{\infty}3e^{-st}dt $
$ =(\frac{-3}{s}e^{-st})|_{1}^{\infty} $
$ =\frac{3}{s} $
Part B
The Fourier series coefficients in $ x(t)=cos(3 \pi t) + sin(8 \pi t) $ are:
$ a_{3} = \frac{1}{2} $
$ a_{-3} = \frac{1}{2} $
$ a_{8} = \frac{1}{2j} $
$ -a_{-8} = \frac{1}{2j} $
For $ k \neq [3,-3,8,-8], a_{k} $ = 0
The response of the input $ x(t) $ to the system $ y(t) $ using $ H(s) $ and the above coefficients is:
$ y(t)=\sum_{k=-\infty}^{\infty}a_{k}H(s) $
$ =\frac{1}{s} + \frac{1}{s} + \frac{1}{sj} - \frac{1}{sj} $
$ =\frac{2}{s} $
