Complex gain

The complex gain $G$ of a system represents how strongly it responds to inputs and with what phase shift.

G(\omega) = \frac{\text{complexified system response}}{\text{complexified system input}}.

The gain $g(\omega) = \lvert G(\omega) \rvert$ and phase lag $\phi = - \arg G(\omega)$ .

In the case of a linear ODE with characteristic polynomial $p(\lambda)$ driven by an exponential driving force, we can apply the exponential response formula to find the gain.

\begin{align*} P(D) x &= Q(D)e^{i\omega t} \\ x &= \frac{Q(i\omega)}{P(i\omega)} e^{i\omega t} \\ G(\omega) &= \frac{Q(i\omega)}{P(i\omega)}, \quad P(i\omega) \ne 0. \end{align*}