# Measurements and Modeling of the Nonlinear Behavior of a Guitar Pickup at Low Frequencies +
## 4 Measurement of the Pickup Nonlinearities  
The first goal of this paper is to identify the pickup in terms of the Generalized Hammerstein model (Figure 2). Since the pickup is an electromagnetic transducer that converts string vibration into an electrical output signal, its experimental characterization is not straightforward. Usually, when measuring a linear or a nonlinear device, excitation signal is a controlled one (impulses, swept-sine, pseudo-random sequences) so that output signal can be used to identify the system in terms of a frequency response function (FRF) for a linear system, or in terms of a set of describing functions when dealing with a nonlinear system. For a pickup, the excitation signal is the displacement of a plucked string exhibiting a multi-modal and non stationary behavior. Such an excitation is useful for a study in real conditions [39] but can hardly be used to get an FRF or to identify nonlinearities. To control the string displacement, we use the system shown in Figure 3. A sample of a guitar string (steel, low E-string, diameter 1.42 mm) is glued on a non-magnetic rigid support, itself fixed to a shaker which imposes a string displacement perpendicular to the pickup [40]. The rigid connection between the string sample, rigid support and the shaker ensures a piston-like motion. An mu-metal shielding covers the shaker in order to limit its electromagnetic radiation. The pickup under test (SSL-5 single-Coil pickup, [PERSON], Santa Barbara, CA, USA) is set on a precision movement device, which allows for adjusting the distance at rest \(d_{0}\) between the string and the magnet. For this experiment, \(d_{0}\) is set to \(d_{0}=5\) mm, corresponding approximately to the distance set on a real guitar. Figure 2: Generalized Hammerstein model for identifying the nonlinearities of the pickup; \(x(t)\) and \(u(t)\) represent the displacement of the guitar string and the output voltage of the pickup, respectively. The shaker is driven by a Synchronized Swept Sine signal [37] so that the nonlinearities of the whole system, that is, the shaker and the pickup in series can be easily identified using a Generalized Hammerstein model [16]. 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 k\(\Omega\)). 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 \(\mathcal{H}_{m}^{(x,s)}(f)\) and for the pickup output voltage \(\mathcal{H}_{m}^{(u,s)}(f)\) calculated using the Synchronized Swept Sine method [37] are given in Figure 4. The noise level of the measurement is visible on the first HHFR (blue line) at frequencies higher than the highest excitation frequency (500 Hz), giving at least 60 dB signal-to-noise ratio (SNR) for the displacement measurement and 80 dB SNR for the voltage measurement. The fundamental harmonic of the string displacement (Figure 4a) is not flat despite the pre-filtering and the second harmonic reaches \(-40\) dB relative to fundamental harmonic. It is thus rather difficult, or almost impossible, to estimate which part of the HHFRs of the pickup output voltage (Figure 4b) is due to the pickup behavior and which part is due to the shaker behavior. To overcome this problem, we use a technique presented in [41] and detailed in the Appendix A.