# A Review of Neural Network-Based Emulation of Guitar Amplifiers
## 2 Traditional Methods
### Black-Box  
Black-box approaches are system identification methods that seek to remedy some of the problems of gray- and white-box approaches by relying on only input-output measurements to describe the overall behaviour of the amplifier. These methods enable emulation without the need for prior knowledge of the internal circuitry of the amplifier. They also tend to be more computationally efficient than white-box models and allow for the replication of idiosyncrasies that can exist in the analogue devices (due to imperfect electrical components for example). The main methods in this category are Volterra series, dynamic convolution, block-oriented structures (which are a subclass of Volterra series), and kernel regression. Dynamic convolution, proposed in [17], is a variant of the system identification method used for linear systems in which instead of a single impulse response used to derive the transfer function of the DUT, multiple impulses are used as input at different amplitudes in order to obtain an approximation of the nonlinear behaviour. In this work, the standard convolution operation \(y(n)=\sum_{k=0}^{L-1}x(n-k)\cdot h(k)\) where \(h\) is the impulse response of length \(L,y\) is the output signal, and \(x\) the input, is replaced with so-called dynamic convolution operation defined as follows:  
\[y(n)=\sum_{k=0}^{L-1}x(n-k)\cdot h_{S(x(n-k))}(k) \tag{1}\]  
where \(S(x(n))\) is a selector function that determines which impulse response should be used for the input sample \(x(n)\). This method was used in the Sintefex FX8000 Audio Effects Replicator. Certain Machine Learning (ML) methods have also been proposed for the emulation of guitar distortion circuits. A Support Vector Machine (SVM), which is a class of linear ML algorithm, was used to emulate a common-cathode tube amplifier via kernel regression [18]. In the article, the authors linearize the nonlinear system in order to be able to apply the SVM to the regression task and suggest methods for choosing the proper kernel to linearize the system. The linearization is accomplished by mapping the data into a higher-dimensional vector space in which the nonlinear functions of the system can be replaced by linear operators. Kernel methods are used to avoid the computationally intensive task of representing these functions in the higher-dimension space, requiring instead only the computation of the inner products between the vectors in this new space. While this method is theoretically solid, the choice of the necessary mappings and kernel function can be a difficult task. ANN are a class of machine learning, black-box methods that use neurons and nonlinear activation functions such as sigmoid or hyperbolic tangent, functions frequently used in VA modelling to model various nonlinearities. This enables ANN to approximate complex mappings and recent technological improvements have enabled their wide-spread, leading to a rise of artificial intelligence in almost all fields in recent years. Similarities between a number of ANN architectures and certain black-box modelling methods can be drawn and suggest that they would appear to be well suited to the task of vacuum tube amplifier emulation as we shall discuss in further detail in a future section. The Volterra series [19] is a functional expansion of multidimensional convolution kernels that enables the emulation of the \"memory\" effect of an amplifier as the output of the nonlinear system is dependent on the input at previous time steps. It is an extension of the common model of a linear system: \(y(t)=Hx(t)=\int_{\mathbb{R}}h^{(1)}(\tau)x(t-\tau)d\tau\), where \(H\) is the transfer function of the system and \(x(t)\) the input.