## 1 Introduction  
Modeling of virtual analog audio effects is the process of emulating an audio effect unit and seeks to recreate the sound, behaviour and main perceptual features of an analog reference device [1]. Audio effect units are analog or digital signal processing systems that transform certain characteristics of the sound source. These transformations can be linear or nonlinear, time-invariant or time-varying and with short-term and long-term memory. Most typical audio effect transformations are based on dynamics, such as compression, tone such as distortion, frequency such as equalization, and time such as artificial reverberation or modulation based audio effects. Nonlinear audio effects: These effects are widely used by musicians and sound engineers and can be classified into two main types of effects: dynamic processors such as compressors or limiters; and distortion effects such as tube amplifiers [2]. Distortion effects are mainly used for aesthetic reasons and are usually applied to electric musical instruments such as electric guitar, bass guitar, electric piano or synthesizers. The main sonic characteristic of these effects is due to their nonlinearities and the most common processors are overdrive, distortion pedals, tube amplifiers and guitar pickup emulators. Dynamic range processors are nonlinear time-invariant audio effects with long temporal dependencies, and their main purpose is to alter the variation in volume of the incoming audio. This is achieved with a varying amplification gain factor, which depends on an envelope follower along with a waveshaping nonlinearity. These effects tend to introduce a low amount of harmonic distortion, while for tube amplifiers a strong distortion is desired [2]. Thus, distortion effects and dynamic range processors are based on the alteration of the waveform which leads to various degrees of amplitude and harmonic distortion. The nonlinear behavior of certain components of the effects' circuit performs this alteration, which can be seen as a waveshaping nonlinearity applied to the amplitude of the incoming audio signal in order to add harmonic and inharmonic overtones. For example, a waveshaping transformation depends on the amplitude of the input signal and consists in using a nonlinear function, such as an hyperbolic tangent, to distort the shape of the incoming waveform [3]. Modulation based audio effects: Modulation based or time-varying audio effects involve audio processors that include a modulator signal within their analog or digital implementation [4]. These modulator signals are in the low frequency range (usually below 20 Hz). Their waveforms are based on common periodic signals such as sinusoidal, squarewave or sawtooth oscillators and are often referred to as a Low Frequency Oscillator (LFO). The LFO periodically modulates certain parameters of the audio processors altering the timbre, frequency, loudness or spatialization characteristics of the audio. Based on how the LFO is employed and the underlying signal processing techniques used when designing the effect units, we can classify modulation based audio effects into _time-varying filters_ such as phaser or wah-wah; _delay-line_ based effects such as flanger or chorus; and _amplitude modulation_ effects such as tremolo or ring modulator [2]. The Leslie speaker cabinet is a type of modulation based effect that combines amplitude, frequency and spatial modulation. It consists of a vacuum-tube amplifier and crossover filter followed by a rotating _horn_ and rotating _woofer_ inside a wooden cabinet. This effect can be interpreted as a combination of tremolo, Doppler effect and reverberation [5]. Audio effects modeling: Modeling these types of effect units or analog circuits has been heavily researched and remains an active field, see Section 2 for more details. Virtual analog methods for modeling nonlinear and time-varying audio effects mainly involve circuit modeling and optimization for specific analog components such as vacuum-tubes, operational amplifiers or transistors.