If I consider a LTI system whose response to the signal $x_1(t) = u(t) − u(t − 1)$ is the signal $y_1(t)$. How would I determine the response $y_2(t)$ of the system to the input $x_2(t)$ shown in the figure below in terms of $y_1(t)$?
I should add, I know I just have to find $x_2(t)$ in terms of $x_1(t)$ and I can get the output from that. I'm just having some problems with the "writing in terms" part.
You should realize that $$x_1(t)=u(t)-u(t-1)=\mathrm{rect}(t)$$ So $x_1(t)$ is a rectangular pulse whose value is one between $[0,1]$, and zero elsewhere.
Now define $x_2(t)$ in terms of four rectangular pulses:
$$x_2(t)=2\mathrm{rect}(t+2)+\mathrm{rect}(t+1)+\mathrm{rect}(t)-2\mathrm{rect}(t-1)$$ Since the system is LTI, the the response to $x_2(t)$ would be $$y_2(t)=2y_1(t+2)+y_1(t+1)+y_1(t)-2y_1(t-1)$$