# A Review of Neural Network-Based Emulation of Guitar Amplifiers
## 2 Traditional Methods  
### White-Box  
The first class of methods used for the emulation of vacuum-tube distortion aims to create a model of the physical device using internal information of the amplifier to establish a system of differential equations. The main methods of this approach are state-space models, Modified Nodal Analysis (MNA), Port-Hamiltonian formalism, and the use of Wave Digital Filters (WDF). Here we present a non-exhaustive list of white-box methods in order to illustrate the limitations that can occur. More detail on the landscape of white-box methods can be found, for example, in the works of [PERSON] and [PERSON] [5,6]. Nodal analysis is a method used in circuit simulation techniques in order to establish the system of equations in matrix form to be solved. This is accomplished using the Kirchhoff Circuit Laws (KCL) at each node. Modified Nodal Analysis extends this to incorporate auxiliary equations into the system using the circuit's Branch Constitutive Equations (BCE) in order to solve the resulting system, which has the following form: MV = I where M is the matrix of conductance coefficients for KCL, I is a vector of independent current sources in the circuit, and V is the vector of unknowns (which are typically voltages). The most well-known electronic circuit simulator is SPICE (Simulation Program with Integrated Circuit Emphasis), which uses a combination of component-wise discretization of the circuit and MNA to create the system to be solved. This approach, while accurate, is computationally heavy and therefore poses a problem for real-time use [6]. Moreover, if an element within the circuit is changed, the MNA process needs to be repeated to take into account this change. The WDF approach constructs digital filters based on the traveling-wave formulation of the physical elements of the device, obtained via a change of variables from the Kirchhoff to the wave domain which describes circuit dynamics in terms of incident waves \(a=v+Ri\) and reflected waves \(b=v-Ri\). Unlike nodal analysis, this formalism allows for modularity without delay-free loops (i.e., loops that cannot be computed as the input and resulting output are needed at the same time step in order to accomplish this). Discrete-time models of each of the individual components (resistors, capacitors, and inductors) are connected to construct the filter. An interesting benefit of this approach is that the finite difference scheme resulting from the use of the bilinear transform for the discretization remains explicit for any number of models. WDF was used to model a number of distortion effects including a triode amplifier in [7]. State-space models rely on the principle that the equations of motion for any physical system may be formulated in terms of the state of the system: \(x^{\prime}(t)=f_{t}[x(t),u(t)]\) where \(x(t)\) is the state of the system at time \(t\), \(u(t)\) is a vector of external inputs, and the general vector function \(f_{t}\) specifies how the current state \(x(t)\) and inputs \(u(t)\) cause a change in the state at time \(t\). \(f_{t}\) may be time-varying itself. All past states and the entire input history are \"summarized\" by the current state \(x(t)\). Thus, \(x(t)\) must include all the \"memory\" of the system. To solve the nonlinear state space system, two notable approaches are the K-method and the Discrete K-method (DK-method). The K-method was originally proposed to avoid noncomputable delay-free loops in the system. It does this by transforming the system to highlight all loops involving nonlinear maps. The delay-free paths are isolated in order to be eliminated via geometrical transformations [8]. The DK-method establishes a system of Ordinary Differential Equations (ODE) based on MNA which is then solved to find the state transition matrix and the nonlinear relationships between variables in order to produce a nonlinear filter similar to that of the K-method. However, in the K-method, the ODE are discretized after derivation whereas in the DK-method, discretization is applied first [9].