  It is common in the accelerator community to use the impedance of accelerator
components to describe wake interactions in the frequency domain. However, it
is often desirable to understand such wake interactions in the time domain in a
general manner for excitations that are not necessarily Gaussian in nature. The
conventional method for doing this involves taking the inverse Fourier
Transform of the component impedance, obtaining the Green's Function, and then
convolving it with the desired excitation distribution. This method can prove
numerically cumbersome, for a convolution integral must be evaluated for each
individual point in time when the wake function is desired. An alternative to
this method would be to compute the wake function analytically, which would
sidestep the need for repetitive integration. Only a handful of cases, however,
are simple enough for this method to be tenable. One of these cases is the case
where the component in question is an RLC resonator, which has a closed-form
analytical wake function solution. This means that a component which can be
represented in terms of resonators can leverage this solution. As it happens,
common network synthesis techniques may be used to map arbitrary impedance
profiles to RLC resonator networks in a manner the accelerator community has
yet to take advantage of. In this work, we will use Foster Canonical Resonator
Networks and partial derivative descent optimization to develop a technique for
synthesizing resonator networks that well approximate the impedances of
real-world accelerator components. We will link this synthesis to the
closed-form resonator wake function solution, giving rise to a powerful
workflow that may be used to streamline beam dynamics simulations.
