Generalized Moog Ladder Filter: Part I–Linear Analysis and Parameterization

The Moog ladder filter, which consists of four cascaded first-order ladder stages in a feedback loop, falls within the class of devices that have attracted greatest interest in virtual analog research. On one hand, this work confirms that the presence of exactly four stages in the original analog circuit is motivated by specific filter control issues and, on the other, that such a limitation can be overcome in the digital domain with relative ease. First, a continuous-time large-signal model is defined for a version of the circuit that is generalized to an arbitrary number of ladder stages. Then, the linear behavior of the filter around its natural operating point and the effect of control parameters on the resulting frequency response are studied in depth, to obtain exact analytical expressions for the position of poles in the transfer function and for the dc gain of the filter, as well as a parameterization strategy that is consistent for any number of ladder stages. A previously-introduced linear digital model of the device suggested by Smith is eventually generalized based on these general results, which remain, however, relevant and similarly applicable to other discretizations of the filter. The proposed model faithfully reproduces the linear behavior of the generalized device while providing sensible parametric control for any number of ladder stages.