# Real-Time Guitar Amplifier Emulation with Deep Learning +
## 1 Introduction  
Many popular guitar amplifiers and distortion effects are based on analog circuitry. To achieve the desired distortion of the guitar signal, these circuits use nonlinear components, such as vacuum tubes, diodes, or transistors. As music production becomes increasingly digitised, the demand for faithful digital emulations of analog audio effects is increasing [1; 2; 3]. The main objective of the field of Virtual Analog (VA) modelling is to create digital emulations of these analog systems. This allows bulky, expensive and fragile analog equipment to be replaced by software plugins that can be used on a modern desktop computer. A common approach for VA modelling of distortion effects is \"white-box\" modelling [4; 5; 6; 7]. White-box modelling is based on analysis and discrete-time simulation of the analog circuitry. If the circuit and the characteristics of its nonlinear components are known, white-box modelling can be very accurate. However, circuit simulation can be computationally demanding when there are manyreactive components and nonlinear elements in the circuit, and the involved design process can be labor intensive. An alternative approach for VA modelling is \"black-box\" modelling. Black-box modelling is based on measuring the circuit's response to some input signals, and creating a model which replicates the observed input-output mapping. Black-box models for VA modelling include block-oriented models, which are based on assumptions about the design of the modelled circuit [8; 9; 10; 11; 12]. As an example, a Wiener model [8; 11] emulates the circuit as a linear filter followed by a static nonlinearity. Other black-box modelling methods include Volterra series models [13; 14] and dynamical convolution [15]. Approaches which fall somewhere in between white and black box are known as \"grey-box\" models. These use knowledge of the circuit to create the model structure, but also require input/output data measured from the device being modelled. In recent years, more machine learning based models have been proposed for nonlinear circuit modelling. One approach has been to use a state space representation of the circuit, and train a machine learning algorithm to predict both the circuit's states and output, using, for example, kernel regression [16] or a neural network [17]. These are examples of grey-box models, as analysis was required to determine the number and location of the circuit states. In the past year, a number of neural network models have been suggested for black-box vacuum tube amplifier emulation. [PERSON] et al. [18] proposed a long short-term memory (LSTM) model, with many layers but a small hidden size in each layer, although the authors reported clearly audible differences between the resulting model and the target device. [PERSON] and [PERSON] have proposed a hybrid convolutional and recurrent model [19], as well as number of other recurrent, dense and convolutional networks [20]. In [21] we presented a single layer recurrent model along with a real-time C++ implementation. Additionally, in [22] the authors presented a convolutional neural network which was used for modelling several digital nonlinear audio effects. This paper focuses on a model we proposed [23] for nonlinear circuit black-box modelling, that was based on the WaveNet convolutional neural network [24]. The neural network model is made up of a series of convolutional layers, with each layer consisting of a dilated filter followed by a dynamically gated nonlinear activation function. In [23] listening tests were conducted to validate the performance of the model when emulating the Fender Bassman tube amplifier. Following from this, we presented models, with a focus on real-time performance, of three guitar distortion effects: the Ibanez Tube Streamer, the Boss DS-1, and the Electro-Harmonia Big Muff Pi [25]. This paper is an extended version of our previous work [25].