#
Measurements and Modeling of the Nonlinear Behavior of a Guitar Pickup at Low Frequencies †^{ †}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Nonlinear Models

## 3. Synchronized Swept-Sine Technique

## 4. Measurement of the Pickup Nonlinearities

## 5. Nonlinear Parametric Model of the Pickup

## 6. Model vs. Real Guitar-String Signal

## 7. Discussion

## 8. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A

**Figure A1.**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.

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**Figure 1.**Nonlinear system usually used to model nonlinearities of a guitar pickup [12].

**Figure 2.**Generalized Hammerstein model for identifying the nonlinearities of the pickup; $x\left(t\right)$ and $u\left(t\right)$ represent the displacement of the guitar string and the output voltage of the pickup, respectively.

**Figure 3.**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.

**Figure 4.**First three Higher Harmonic Frequency Responses (HHFRs) of (

**a**) displacement of the string excited by the shaker and (

**b**) the output voltage of the pickup.

**Figure 5.**Magnitude values of the estimated filters ${G}_{n}\left(f\right)$ of the Generalized Hammerstein model (Figure 2) of the pickup.

**Figure 6.**Modulus (

**a**) and phase (

**b**) of the first filter ${G}_{1}\left(f\right)$ of the identified Generalized Hammerstein model of the pickup under test (blue solid line) and the equivalent frequency response of the simplified Hammerstein model followed by a differentiator (green dashed line).

**Figure 7.**Modulus (

**a**) and phase (

**b**) of the first filter ${G}_{2}\left(f\right)$ of the identified Generalized Hammerstein model of the pickup under test (blue solid line) and the equivalent frequency response of the simplified Hammerstein model followed by a differentiator (green dashed line).

**Figure 8.**Input–output graph of the static nonlinearity (power series development with coefficients ${\alpha}_{n}$ given in Table 1).

${\alpha}_{1}$ | $7.50\times {10}^{-2}$ |

${\alpha}_{2}$ | $6.75\times {10}^{-3}$ |

${\alpha}_{3}$ | $2.11\times {10}^{-3}$ |

${\alpha}_{4}$ | $4.75\times {10}^{-4}$ |

${\alpha}_{5}$ | $8.31\times {10}^{-4}$ |

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**MDPI and ACS Style**

Novak, A.; Guadagnin, L.; Lihoreau, B.; Lotton, P.; Brasseur, E.; Simon, L.
Measurements and Modeling of the Nonlinear Behavior of a Guitar Pickup at Low Frequencies †. *Appl. Sci.* **2017**, *7*, 50.
https://doi.org/10.3390/app7010050

**AMA Style**

Novak A, Guadagnin L, Lihoreau B, Lotton P, Brasseur E, Simon L.
Measurements and Modeling of the Nonlinear Behavior of a Guitar Pickup at Low Frequencies †. *Applied Sciences*. 2017; 7(1):50.
https://doi.org/10.3390/app7010050

**Chicago/Turabian Style**

Novak, Antonin, Leo Guadagnin, Bertrand Lihoreau, Pierrick Lotton, Emmanuel Brasseur, and Laurent Simon.
2017. "Measurements and Modeling of the Nonlinear Behavior of a Guitar Pickup at Low Frequencies †" *Applied Sciences* 7, no. 1: 50.
https://doi.org/10.3390/app7010050