Generalized Moog Ladder Filter: Part II–Explicit Nonlinear Model through a Novel Delay-Free Loop Implementation Method

Abstract

One of the most critical aspects of virtual analog simulation of circuits for music production consists in accurate reproduction of their nonlinear behavior, yet this goal is in many cases difficult to achieve due to the presence of implicit differential equations in circuit models, since they naturally map to delay-free loops in the digital domain. This paper presents a novel and general method for non-iteratively implementing these loops in such a way that the linear response around a chosen operating point is preserved, the topology is minimally affected, and transformation of nonlinearities is not required. This technique is then applied to a generalized model of the Moog ladder filter, resulting in an implementation that outperforms its predecessors with only a modest computational load penalty. This digital version of the filter is shown to offer strong stability guarantees w.r.t. parameter variation, allows the extraction of different frequency response modes by simple mixing of individual ladder stage outputs, and is suitable for real-time sound synthesis and audio effects processing.

Dynamical Systems for Audio Synthesis: Embracing Nonlinearities and Delay-Free Loops

“…This compare favorably with the results presented in [32] where compensation for the zero introduced by unit delay in the feedback path results in unbounded amplitude growth at higher values of f c . The relationship between f c and the measured frequency peaks in Figure 9b compares favorably with a similar analysis of another delay-free implementation given in [36]. Here there is noticeable flattening that results from lowering the value of r. This suggests that some coupling between this parameter and f c persists.…”

“…Many adjustments have been suggested to ameliorate this artifact of digitization [33] or to avoid the addition of delay to the feedback part of the filter without resorting to iteration. Examples of this include solution through the use of Volterra series [34,35] and linear compensation filters [36].…”

Virtual Analog Models of the Lockhart and Serge Wavefolders

“…We have recently begun to fill this research gap in [17], which presents a VA model of the wavefolder circuit in the seminal Buchla 259 module. Previous work on circuit-based VA modeling has researched the filters found in vintage synthesizers such as those produced by Moog [19][20][21][22], Electronic Music Studios (EMS) [23,24], Korg [25,26] and Buchla [16]. Extensive work has also been done on modeling guitar distortion pedals [13,27], tube amplifiers [11,28,29], modulation effects [30][31][32][33] and the Roland TR-808 drum machine [34,35].…”

Modeling and measuring a Moog voltage-controlled filter

“…Huovilainen and Välimäki simplified the nonlinear Moog filter model so that it only uses one memoryless mapping function [6]. Other authors have derived digital variations of both the linear Moog filter [16] and the nonlinear one [17]- [21]. Recently, also other analog synthesizer filters have been the subject of digital modeling [22], [23].…”

“…In this paper we investigate how well a digital model of the Moog ladder filter, which incorporates the nonlinear behavior of transistors, corresponds to the analog ladder filter. The analytical analog model is based on the work of D'Angelo and Välimäki [20], [21]. Two digital implementations are then compared: a trivial (i.e.…”