Periodic input, frequency response

Let a system \(H\) have a periodic input \(u\) represented by a Fourier series. For reals \(a_0\), \(\omega_1\) (fundamental frequency), \(\mathcal{A}_n\), and \(\phi_n\), let \[\begin{aligned} u(t) = \frac{a_0} {2} + \sum_{n=1}^\infty \mathcal{A}_n \sin(n\omega_1 t + \phi_n). \end{aligned}\] The \(n\)th harmonic is \[\begin{aligned} u_n(t) = \mathcal{A}_n \sin(n\omega_1 t + \phi_n),\end{aligned}\] which, from yields forced response \[\begin{aligned} y_n(t) = \mathcal{A}_n |H(j n \omega_1)| \sin(n\omega_1 t + \phi_n + \angle H(j n \omega_1)).\end{aligned}\]

Applying the principle of superposition, the forced response of the system to periodic input \(u\) is \[\begin{aligned} \label{eq:periodic_response_trig} y(t) = \frac{a_0} {2} H(j 0) + \sum_{n=1}^\infty \mathcal{A}_n |H(j n \omega_1)| \sin(n\omega_1 t + \phi_n + \angle H(j n \omega_1)). \end{aligned}\]

Similarly, for inputs expressed as a complex Fourier series with components \[\begin{aligned} u_n(t) = c_n e^{j n \omega_1 t}, \end{aligned}\] each of which has output \[\begin{aligned} y_n(t) = c_n H(j n \omega_1) e^{j n \omega_1 t}, \end{aligned}\] the principle of superposition yields \[\begin{aligned} \label{eq:periodic_response_exp} y(t) = \sum_{n=-\infty}^\infty c_n H(j n \omega_1) e^{j n \omega_1 t}. \end{aligned}\]

, tell us that, for a periodic input, we obtain a periodic output with each harmonic \(\omega_n\) amplitude scaled by \(|H(j\omega_n)|\) and phase offset by \(\angle H(j\omega_n)\). As a result, the response will usually undergo significant distortion, called phase distortion. The system \(H\) can be considered to filter the input by amplifying and suppressing different harmonics. This is why systems not intended to be used as such are still sometimes called “filters.” This way of thinking about systems is very useful to the study of vibrations, acoustics, measurement, and electronics.

All this can be visualized via a Bode plot, which is a significant aspect of its analytic power. An example of such a visualization is illustrated in figure 13.6.