This paper is an extended version of our paper published in the 19th International Conference on Digital Audio Effects (DAFx-16), Brno, Czech Republic, 5–9 September 2016.

This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

Description of the physical behavior of electric guitars is still not very widespread in the scientific literature. In particular, the physical models describing a nonlinear behavior of pickups still requires some refinements. The study presented in this paper is focused on nonlinear modeling of the pickups. Two main issues are raised. First, is the currently most used nonlinear model (a Hammerstein model) sufficient for the complex nonlinear behavior of the pickup? In other words, would a more complex model, such as a Generalized Hammerstein that can deal better with the nonlinear memory, yield better results? The second troublesome issue is how to measure the nonlinear behavior of a pickup correctly. A specific experimental set-up allowing for driving the pickup in a controlled way (string displacement perpendicular to the pickup) and to separate the nonlinear model of the pickup from other nonlinearities in the measurement chain is proposed. Thanks to this experimental set-up, a Generalized Hammerstein model of the pickup is estimated for frequency range 15–500 Hz and the results are compared with a simple Hammerstein model. A comparison with experimental results shows that both models succeed in describing the pickup when used in realistic conditions.

Physical modeling and synthesis of musical instruments has been an active research field for the last few decades [

A few models of pickup are available in the literature. Some of them are based on integral equations leading to the variation of magnetic flux at the coil location [

On the other hand, studies on nonlinear modeling have led to many nonparametric nonlinear models, discussed in

The goal of this paper is to proceed with the identification of pickup linearities based on a Generalized Hammerstein representation of the pickup. For this purpose, a specific experimental set-up is used to drive the pickup in a controlled way, and a technique is carried out to get rid of nonlinearities due to the driver. One of the aims of this study is to find out if it is meaningful, or not, to use a simple Hammerstein structure given in

Study of a nonlinear physical system, such as a guitar pickup, usually begins with a basic model based on a memory-less polynomial representation [

When dealing with modeling of nonlinear systems, two different methodological approaches are usually considered. The first approach studies the physical causes of the nonlinear phenomena and the model is derived purely from the physical laws governing the behavior of the system. These laws can then be used for the derivation of a theoretical nonlinear model of the system [

Probably the best known block-oriented nonlinear model is the Hammerstein model combining two elements in series: a static nonlinear system and a linear filter [

On the other hand, the Volterra series representation is usually considered to be more effective but less practical due to the highly complex calculation of multidimensional kernels [

To estimate the filters

The synchronized swept-sine is generated using [

The method, originally developed by Angelo Farina [

Finally, to get the Higher Harmonic Frequency Responses (HHFRs), the time-delayed higher harmonic impulse responses

The first goal of this paper is to identify the pickup in terms of the Generalized Hammerstein model (

To control the string displacement, we use the system shown in

The shaker is driven by a Synchronized Swept Sine signal [

To protect the shaker from a possible destruction due to excessive displacement or current, the frequency range is furthermore limited to the span 15–500 Hz. The excitation signal is pre-filtered using a linear filter so as to obtain a displacement whose amplitude is almost constant over the frequency span. The peak amplitude is set here to 1 mm. The displacement of the string portion (that is, the displacement of the shaker) is measured by means of a single-point vibrometer (OFV-503 and OFV-505, Polytec, Irvine, CA, USA), pointing at a piece of reflective tape glued to the string support. The electrical output of the pickup is then connected to an acquisition card that exhibits a high input impedance (470 kΩ). Consequently, the measured output voltage corresponds to the open-circuit voltage that does not take into account the effect of pickup output impedance.

The Higher Harmonic Frequency Responses (HHFRs) for both the string displacement

The fundamental harmonic of the string displacement (

To overcome this problem, we use a technique presented in [

For the pickup under test, a fifth order Generalized Hammerstein model, the magnitude values of which are depicted in

Observing the estimated filters of the Generalized Hammerstein model depicted in

It is thus tempting to fit all the filter responses

Fixing the phase of all of the estimated filters

This relation being a time derivative of a Taylor series, we can simplify the Generalized Hammerstein model to a Hammerstein model consisting of a static nonlinearity followed by a linear filter (

To test the validity of the identified Hammerstein model, we set up a different experiment corresponding to a realistic use of the pickup. For that purpose, we use a lab guitar prototype [

The string is struck using an impact hammer. A single-point vibrometer (OFV-503 and OFV-505, Polytec, Irvine, CA, USA) pointing at a piece of reflective tape glued to the string and located above the pickup allows the measurement of the string displacement in the vertical plane. Temporal evolution of both string displacement and pickup output voltage are recorded simultaneously and depicted in

The displacement signal measured with the vibrometer is then used as the input of estimated parametric Hammerstein model of the pickup and both the measured and the synthesized pickup outputs are compared on the same graph (

The results presented in this paper show that a simple Hammerstein model seems to be sufficient for the pickup modeling and that using a Generalized Hammerstein model is not necessary. However, several hypotheses have been put forward, simplifying the problem that might be at the origin of small differences between the measured and modeled pickup outputs compared in

First, the frequency range of the excitation signal is limited to 15–500 Hz due to the capacities of the shaker. Using a larger frequency range might have been beneficial. The nonlinearities of the pickup may differ at higher frequencies, and thus, in such a case, a complete Hammerstein Generalized model might be useful. A supplementary study would be necessary to draw a meaningful conclusion.

Next, in the first experimental setup used for the model identification, the movement of the rigid string attached to the shaker exhibits only

Even though these phenomena have been neglected, the results presented in this paper show a very good agreement between the output predicted by the model and the output obtained from the experimental measurement.

Physical modeling of musical instruments is still a challenge for not only physicists, acousticians, and instrument makers, but also for engineers from the digital audio community. An electric guitar is not an exception.

This paper focuses on nonlinear modeling of a pickup of an electric guitar capturing vibrations of ferromagnetic steel strings. Two main issues are raised: how we should proceed to measure the nonlinear properties of the pickup and what model we should use. An experimental set-up, in which a piece of steel string is attached to a shaker, is proposed. It allows for driving the pickup by shaker vibrations in a controlled way (string displacement perpendicular to the pickup). The shaker is driven by a Synchronized Swept Sine signal and a Generalized Hammerstein model of the pickup is estimated. Next, a simple Hammerstein model, usually used to represent pickup nonlinearities, is derived from the Generalized Hammerstein model and the validity of the Hammerstein model is verified for a pickup operating in a realistic way with a string excited by a hammer-like impact. Even if the experimental bench for model identification has a limited working frequency range (15–500 Hz), the estimated Hammerstein model precisely predicts the signal output.

In future work, the model can be used to synthesize different kinds of existing pickups (single coil pickups, humbuckers). However, two important questions need to be answered: is the Hammerstein model sufficient for all the single coil pickups on the market, and, if yes, would their corresponding coefficients

The results of this study shows that the Generalized Hammerstein model does not bring much more complementary information in a low-frequency range (15–500 Hz) and that the simple Hammerstein model is a very good compromise between precision and complexity.

This research was funded by the Region Pays de la Loire within the Le Mans Acoustic Project. The paper is an extended version of [

Antonin Novak, Leo Guadagnin, Bertrand Lihoreau, Pierrick Lotton, Emmanuel Brasseur, and Laurent Simon conceived and designed the experiments; Antonin Novak performed the experiments, Antonin Novak, Leo Guadagnin, Bertrand Lihoreau, Pierrick Lotton, Emmanuel Brasseur, and Laurent Simon analyzed the data; and Antonin Novak wrote the paper.

The authors declare no conflict of interest.

Since the measurement technique described in this paper uses a shaker as the excitation device to create the displacement of the guitar string, the nonlinearities caused by the shaker must be taken into account. The problem is depicted in

Two dynamic nonlinear systems in series; the first one represents the shaker, and the second one, represented by a Generalized Hammerstein model, is the pickup under test.

The excitation swept-sine signal

The method presented in [

First, the HHFRs

The HHFRs

Equation (

Nonlinear system usually used to model nonlinearities of a guitar pickup [

Generalized Hammerstein model for identifying the nonlinearities of the pickup;

Measurement device used to characterize the nonlinearities of the pickup. A sample of a guitar string is glued on a non-magnetic rigid support, itself fixed to a shaker. An mu-metal shielding covers the shaker in order to limit its electromagnetic radiation.

First three Higher Harmonic Frequency Responses (HHFRs) of (

Magnitude values of the estimated filters

Modulus (

Modulus (

Input–output graph of the static nonlinearity (power series development with coefficients

Picture of the second experiment in which the pickup is placed under a vibrating string.

Recorded signals of the vibrating string.

Recorded and synthesized signals of the voltage from the pickup.

Coefficients