# How does this digital signal controlling a switch in the circuit affect the output voltage?

Suppose you have a circuit which has the input signal $x(t)=2\sin (ω_ot + \pi/6)$. The switch in the circuit is controlled with a digital signal of the form: $s(t)=\sum_{k=-\infty}^{+\infty} (u(t+ε-kT_s) - u(t-ε-kT_s))$, $\frac{2\pi}{T_s}=800\pi$, $ε\to 0$, so that when the signal equals 1 the switch is closed and the output signal equals the input signal and when it is open the output signal is 0. Note that $u(t)$ is a step function evaluated as $u(0)=1/2$.

The task is finding the Laplace Transform of the output signal. The digital signal form to me looks like a very thin rectangular impulse series of width $2ε$ and a period of $T_s$.

My question is: does the output signal in time domain equal the product of these two signals so it translates to the convolution of their respective laplace transforms or is something else going on that i'm not seeing?

• Hint: the signal $x(t)$ is being sampled by the switch. You're correct that the output is the product of the two signals. – MBaz Jun 30 '18 at 16:03
• I have a feeling that you are very close to the answer. When you are ready, could you have a go at answering the question yourself and accepting it? That would stop it from circulating the board. – A_A Jun 30 '18 at 20:47
• @A_A Im still working on it, i could post what i calculated some time tommorow, because i have a test tommorow afternoon. Is it a problem to leave it as it is until i have time to add to it? – edward_d Jun 30 '18 at 21:02
• It's just a suggestion. Not urgent anyway. Good luck with your test. – A_A Jun 30 '18 at 21:05