How to isolate coefficiets from complex fourier series?
For LTI systems, use the linearity property.
$$
\begin{align}
H(s) \cdot \left(\sum X_i (s)\right) \mapsto{} &amp; \sum H(s)\cdot X_i (s)
&amp; {}={} &amp;
\sum Y_i(s)
\\
h(t) \circledast \cdot \left(\sum x_i (t)\right) \mapsto{} &amp; \sum h(t)\circledast x_i (t) 
&amp; {}={} &amp;
\sum y_i(t)
\end{align}
$$
From the above, \$\frac{y_i}{x_i}\$ or \$\frac{Y_i}{X_i}\$ can be found by comparing like terms.