## 1 Introduction
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]. Measurement-based VA modeling, commonly known as \"black-box modeling\", has also been thoroughly studied within the context of guitar amplifiers and pedals [36; 37; 38]. This approach is particularly useful when the original circuit schematics are not available. A major challenge in VA modeling of nonlinear circuits, and digital waveshaping in general, is aliasing suppression. Early research on waveshaping synthesis addressed this issue by using low-order polynomial transfer functions, which not only allowed full parametric control of the produced spectrum but also ensured that the output waveform was bandlimited [4]. In VA modeling, high oversampling factors are usually necessary to prevent harmonics introduced by nonlinearities from reflecting into the baseband as aliases [13]. Oversampling increases the computational requirements of the model, by introducing additional filtering stages and scaling the number operations required to compute each output sample. For VA models that require evaluating transcendental functions, as is the case with the proposed Lockhart and Serge models, these added costs could compromise the integration of the system within a larger, real-time computer music system. A sizable portion of VA research has concentrated on designing efficient algorithms to generate alias-free geometric waveforms like those used in analog subtractive synthesizers, the so-called classic analog waveforms. Well-known techniques include the bandlimited impulse train (BLIT) family of methods, which involves the use of bandlimited basis functions and their integrated forms [39; 40; 41], and the use of differentiated polynomial waveforms (DPW) [42; 43; 44]. Moreover, [PERSON] and [PERSON] have applied the antialasing principle behind the DPW algorithm to tackle aliasing in wavetable oscillators [45]. Recent work on antialiasing techniques has extended the use of the bandlimited ramp (BLAMP) method, originally proposed to antialias triangular oscillators in [25; 41], to special cases of linear piecewise nonlinearities such as signal rectification, and inverse/hard clipping [17; 46; 47]. In this work, we propose the use of the antiderivative antialiasing method introduced in [48; 49]. This approach can be used to reduce the aliasing caused by arbitrary nonlinear waveshaping functions and is applicable to the proposed wavefolder models. In its first-order form, the method can be derived by analytically convolving a linear continuous-time representation of the input signal with a rectangular lowpass kernel [48]. As shown in this work, the use of the antiderivative method reduces the oversampling requirements of the proposed wavefolder models. A VA model of the Lockhart wavefolder was originally presented in [50]. This paper extends that work by introducing a second wavefolding circuit and studying the similarities between both systems. Additionally, we present a different treatment of the required Lambert-W function and an extended evaluation of the proposed antialiasing method in terms of computational costs. This paper is organized as follows. Sections 2 and 3 describe the model derivation of the Lockhart and Serge wavefolders, respectively. Time-domain simulations of the circuits are also presented in these two sections. Section 4 deals with two implications of VA wavefolding in the digital domain, namely aliasing suppression and evaluation of the Lambert-W function. Section 5 presents frequency-domain results of the Lockhart and Serge wavefolders, as well as an evaluation of the proposed antialiasing method in terms of perceived sound quality and computational costs. Section 6 discusses the practical synthesis usage of both circuits and compares the behavior of the middle Serge Wave Multiplier with a recommended four-stage topology built around the Lockhart wavefolder. Concluding remarks and perspectives appear in Section 7.