# A Review of Neural Network-Based Emulation of Guitar Amplifiers
## 2 Traditional Methods
### Gray-Box  
Gray-box methods seek to alleviate some of the labour required for the modelling task using white-box approaches. They do this by incorporating internal knowledge of the device in the form of blocks in order to simplify the model. Indeed, modelling each of the individual circuit components is not necessary in gray-box; only the behaviour of the device as a whole needs to be modelled. Block-Oriented structures are a subclass of Volterra series [11] that aim to separate the Device-Under-Test's (DUT) effect on the input signal into \"blocks\": one set dealing with the dynamic, linear behaviour of the system, and one to introduce the nonlinearity. The common configurations of these structures are the following:  
* Hammerstein (and their subclass cascade Hammerstein) models: Static nonlinearity followed by linear filter. * Wiener models: Linear filter followed by static nonlinearity. * Wiener-Hammerstein models: Static nonlinearity between two linear filters. Block-Oriented structures can be considered as gray-box methods as they can take into account the high-level structure of the DUT for their configuration and most examples of block-oriented structures applied to distortion approximation fall under the realm of gray-box modelling. One of these configurations (Wiener-Hammerstein) has even been described as the \"fundamental paradigm of electric guitar tone\" in [12] and this family of methods is used in commercial products and patents such as Fractal Audio's Axe-Fx [12] and a 2008 patent from Kemper [13]. This patent describes a Wiener-Hammerstein topology and Fractal's white paper extends this model to include extra linear and nonlinear blocks in their MIMIC technology. Most notable in the field of gray-box guitar amplifier modelling using block-oriented structures are the works of [PERSON] and [PERSON], in which system identification using input-output signals and optimization of the block topology using iterative methods to minimize the error between model output and the measured target are used. These approaches will be detailed in this section. In [14], a parametric Wiener-Hammerstein model for distortion effects is presented along with the iterative algorithm used for the optimization portion of the problem. The Levenberg-Marquardt algorithm was chosen for the minimization and the DUT was a Hughes and Kettner Tube Factor amplifier. This method produced satisfying results that were nearly indistinguishable from the target, but the model becomes more inaccurate for frequencies outside of the range [60 Hz, 18 kHz]. In a following work, an extended Wiener model was tested for the emulation of three different distortion devices: a diode clipper with pre-amplification of the input signal, a BIT distortion stage of an Electro-Harmonix Big Muff Pi, and the op-amp stage of an Ibanez Tube Screamer (all simulated using a SPICE circuit simulator). This simple topology had a harder time modelling the desired effects and only the diode clipper circuit was modeled successfully. [PERSON] and [PERSON] [15] presented a gray-box approach in which the modelling process is automatic using input-output measurements and iterative optimization. Its structure is the following: three linear filters inter-spaced with nonlinear blocks, the first consisting of a polynomial waveshaping function and the second consisting of a concatenation of hyperbolic tangents. An Exponential Sine Sweep (ESS) is used for the system identification process along with the Levenberg-Marquardt algorithm for the optimization problem. This method was used to model a Fender Bassman 100 and an Ampeg VT-22 but struggled at modelling distortion. Finally, in [16], analog amplifiers including a Bassman 100 and a JCM 900 were modeled with an automated procedure and an extended Wiener-Hammerstein configuration consisting of a nonlinear mapping between two Linear Time-Invariant (LTI) blocks connected in series.