# Measurements and Modeling of the Nonlinear Behavior of a Guitar Pickup at Low Frequencies +
## Appendix A  
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 Figure 11 in which the first nonlinear system (NLS) (NL\({}_{1}\)) represents the shaker and the second NLS (NL\({}_{2}\)) represents the pickup under test.  
The excitation swept-sine signal \(s(t)\) (Equation (2)) is the input of the first NLS. The displacement \(x(t)\) is the output of the first NLS and the input of the second NLS. The voltage \(u(t)\) is the output of the second NLS. Consequently, \(u(t)\) may be seen as the output of the whole system for the input signal \(s(t)\).  
The method presented in [41] then allows the identification of the second NLS. The identification of the system is equivalent to estimating the linear filters \(G_{n}(f)\), \(n=1,N\), from the measured signals \(x(t)\) and \(u(t)\). The identification process is based on the off-line estimation of both the HHFRs \(\mathcal{H}_{m}^{(x,s)}(f)\) between \(s(t)\) and \(x(t)\), and the HHFRs \(\mathcal{H}_{m}^{(u,s)}(f)\) between \(s(t)\) and \(u(t)\), for \(m=1,M\), \(M\) being the total number of harmonics taken into account. We recall that, given an input signal \(a(t)\) and an output signal \(b(t)\) of an NLS, the HHFR \(\mathcal{H}_{m}^{(b,a)}(f)\) may be seen as the contribution, in both amplitude and phase, of the \(m\)-th harmonic at the output, for a sine at frequency \(f\) at the input, as  
\[\mathcal{H}_{m}^{(b,a)}(f)=|\mathcal{H}_{m}^{(b,a)}(f)|e^{ip_{m}^{(b,a)}(f)}. \tag{1}\]  
First, the HHFRs \(\mathcal{H}_{m}^{(x,s)}(f)\) and \(\mathcal{H}_{m}^{(u,s)}(f)\) are estimated using the synchronized swept sine method described in Section 4. Next, the displacement signal \(x(t)\), already distorted by the shaker (NL\({}_{1}\)), is taken to the powers of \(n\) and HHFRs \(\mathcal{H}_{m}^{(x^{n},s)}(f)\) for \(n=1,N\) are calculated.  
The HHFRs \(\mathcal{H}_{m}^{(u,s)}(f)\) of the output signal \(u(t)\) result in the combination of all HHFRs \(\mathcal{H}_{m}^{(x^{n},s)}(f)\) after filtering by filters \(G_{n}(f)\). The relationship between the HHFRs \(\mathcal{H}_{m}^{(x^{n},s)}(f)\), \(\mathcal{H}_{m}^{(u,s)}(f)\) and the linear filters \(G_{n}(f)\) can indeed be written in a matrix form as [41]  
\[\begin{bmatrix}\mathcal{H}_{1}^{(y,x)}(f)\\ \mathcal{H}_{2}^{(y,x)}(f)\\ \vdots\\ \mathcal{H}_{M}^{(y,x)}(f)\end{bmatrix} =\begin{bmatrix}\mathcal{H}_{1}^{(u,x)}(f)&\mathcal{H}_{1}^{(u^ {2},x)}(f)&\cdots&\mathcal{H}_{1}^{(u^{N},x)}(f)\\ \mathcal{H}_{2}^{(u,x)}(f)&\mathcal{H}_{2}^{(u^{2},x)}(f)&\cdots&\mathcal{H}_ {2}^{(u^{N},x)}(f)\\ \vdots&\vdots&\ddots&\vdots\\ \mathcal{H}_{M}^{(u,x)}(f)&\mathcal{H}_{M}^{(u^{2},x)}(f)&\cdots&\mathcal{H}_ {M}^{(u^{N},x)}(f)\end{bmatrix}\] \[\times\begin{bmatrix}G_{1}(f)\\ G_{2}(f)\\ \vdots\\ G_{N}(f)\end{bmatrix}. \tag{2}\]  
Figure 10: 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.  
Equation (A2) can then be solved for unknown \(G_{n}(f)\) by using a square matrix inversion in the case \(M=N\), or by using a pseudo-inversion in the case \(M>N\). The number of harmonics \(M\) chosen for the estimation of HHFRs must be equal to or greater than the number of the branches \(N\) of the Generalized Hammerstein model. The matrix (pseudo)-inversion must be computed for each frequency \(f\) separately.