Proceedings of the 23 rd International Conference on Digital Audio Effects (DAFx-20), Vienna, Austria, September 8–12, 2020 MOOG LADDER FILTER GENERALIZATIONS BASED ON STATE VARIABLE FILTERS Kurt James Werner and Russell McClellan iZotope, Inc., Cambridge, MA {kwerner|rmcclellan}@izotope.com ABSTRACT We propose a new style of continuous-time filter design composed of a cascade of 2nd-order state variable filters (SVFs) and a global feedback path. This family of filters is parameterized by the SVF cutoff frequencies and resonances, as well as the global feedback amount. For the case of two identical SVFs in cascade and a spe- cific value of the SVF resonance, the proposed design reduces to the well-known Moog ladder filter. For another resonance value, it approximates the Octave CAT filter. The resonance parameter can be used to create new filters as well. We study the pole loci and transfer functions of the SVF building block and entire filter. We focus in particular on the effect of the proposed parameteri- zation on important aspects of the filter’s response, including the passband gain and cutoff frequency error. We also present the first in-depth study of the Octave CAT filter circuit. 1. INTRODUCTION The Moog ladder filter [1, 2] was a landmark electronic music de- sign. It is a cascade of four identical 1st-order low-pass filters with global feedback. It has been studied extensively in the vir- tual analog literature, e.g. [3–9]. It also inspired a number of other circuit designs including higher-order generalizations of the Moog filter [8, 9] and “polygon filters” [4, 10–13]. Zavalishin [13] studied some modifications to the Moog ladder filter, including a “true high-pass” mode, “true band-pass” mode, and adding damp- ing controls to 2nd-order band-pass blocks. The Moog ladder filter is not the only classic filter design built from identical 1st-order filters. Filters made up of four cas- caded sections of integrator-based low-pass filters with variable global feedback are common in commercial synthesizer designs. The datasheets for the CEM3320 [14] and SSM2040 [15] voltage- controlled filter integrated circuits recommended this topology. Many synthesizers that contained these integrated circuits used this topology, notably including the Sequential Circuits Prophet 5 [16] and Oberheim OB-Xa [17], both early commercial poly- phonic synthesizers. Roland Corporation commercialized many designs based on this topology, including in their first polyphonic synthesizer, the Jupiter 4 [18]. These 1st-order filter blocks are implemented using op-amps, Moog’s original discrete-transistor design [1, 2], or diodes [19]. Transconductance-amplifier-based designs [20–26], where the am- plifier controls the cutoff frequency, are common. Another standard, 2nd-order, filter in electronic music is the state variable filter (SVF). Filters based on cascaded SVF sections Copyright: © 2020 Kurt James Werner et al. This is an open-access article dis- tributed under the terms of the Creative Commons Attribution 3.0 Unported License, which permits unrestricted use, distribution, and reproduction in any medium, pro- vided the original author and source are credited. ωc s ωc s ωc s ωc s −4 ˆ k −1 −1 −1 −1 + + + + + Xin(s) Yout(s) H Mg 1:4 (s) H Mg 2:4 (s) H Mg 3:4 (s) H Mg 4:4 (s) Figure 1: Full Moog ladder filter block diagram. are also common in commercial designs. Notable examples in- clude the Yamaha CS series [27], a line of early polyphonic syn- thesizers, and later Roland Corp. polyphonic models including the Jupiter 6 [28]. The Octave CAT [25,29] uses two SVFs in cascade, surrounded by a global negative feedback, like the Moog. In this paper, we introduce a novel continuous-time filter de- sign which replaces each pair of low-pass filters in the Moog lad- der filter with an SVF. This has a new degree of freedom: the SVF damping. This can be seen as filling the gaps between Moog’s original design, the Octave CAT, and Zavalishin’s proposed vari- ants, to a fully-parameteric SVF-core 4-pole filter. We study the pole loci, magnitude responses, and some time-domain behavior of this new filter, its stability bounds, and the error of its leading pole frequency. In the following, we first review the Moog ladder filter (§2) and analyze the circuit of the Octave CAT filter (§3). We propose the new generalization (§4) of these filters, study its continuous- time state space and time-varying behavior (§5), and show a discrete- time implementation (§6). 2. MOOG LADDER FILTER First, we review the Moog low-pass ladder filter. Its circuit analy- sis is well-represented in the literature (e.g. [3, 8, 9, 30]). We just review its pole parameterization and conditions on its stability. The Moog ladder filter is composed of four identical blocks, indexed by i ∈{1, 2, 3, 4}, with transfer functions H Mg i:4 (s)= Y Mg i:4 (s)/X Mg i:4 (s)= ωc /(s + ωc ) , (1) where ωc is the cutoff frequency in radians. ωc depends on the electronic circuit parameters and an applied control signal [8]. The input to the first block is X Mg 1:4 (s)= Xin(s) − 4 ˆ kY Mg 4:4 (s), where 0 ≤ ˆ k ≤ 1 is a “normalized” feedback gain 1 , ˆ k =0 is no feedback and ˆ k =1 is the edge of stability. The inputs to the other three, i ∈{2, 3, 4}, are X Mg i:4 (s)= Y Mg i−1:4 (s). The output is Yout (s)= Y Mg 4:4 (s). 1 Typically, an “unnormalized” coefficient k =4 ˆ k is used [8]. DAFx.1 DAF 2 x 21 in Proceedings of the 23 rd International Conference on Digital Audio Effects (DAFx2020), Vienna, Austria, September 2020-21 70