Optimizing differentiated discretization for audio circuits beyond driving point transfer functions

One goal of Virtual Analog modeling of audio circuits is to produce digital models whose behavior matches analog prototypes as closely as possible. Discretization methods provide a systematic approach to generate such models but they introduce frequency response error, such as frequency warping for the trapezoidal method. Recent work showed how using different discretization methods for each reactive element could reduce such error for driving point transfer functions. It further provided a procedure to optimize that error according to a chosen metric through joint selection of the discretization parameters. Here, we extend that approach to the general case of transfer functions with one input and an arbitrary number of outputs expressed as linear combinations of the network variables, and we consider error metrics based on the L
²
and the L
¹
norms. To demonstrate the validity of our approach, we apply the optimization procedure for the response of a Hammond organ vibrato/chorus ladder filter, a 19-output, 36th order filter, where each output frequency response presents many features spread across its passband.