Assume that we have a SISO transfer function model $G(s)$ and we want to look how the output looks like when we have a input signal $u(t)$ that looks like this:
$$u_1(t) = \left\{\begin{matrix} 50 + 5\sin(2\pi10 t) & u(t) > 0 \\ 0 & u(t) \leq 0 \end{matrix}\right.$$
$$u_2(t) = \left\{\begin{matrix} A_2\sin(2\pi\omega_2 t) & u(t) > 0 \\ 0 & u(t) \leq 0 \end{matrix}\right.$$
Is it possible that I can prove that $G(s)U_1(s) = G(s)U_2(s)$ or $G(s)U_1(s) \sim G(s)U_2(s)$ by changing $\omega_2$ and $A_2$?
$U_i(s), i = 1, 2$ is the frequency domain signal on the view of $u_i(t), i = 1, 2$.