When we need to do a measurement of a nonlinear system, we usually use an excitation device (e.g., amplifier, loudspeaker, shaker) that can provide the desired energy (e.g., electrical, acoustical, mechanical). Unfortunately, these devices are very often nonlinear and can spoil the measurement by their nonlinearities. To overcome this problem, we propose a simple nonlinear technique, that can provide spectrally clean signals at the output of these devices.

[Novak et al. (2018)] Novak, A., Simon, L., & Lotton, P. (2018). A simple predistortion technique for suppression of nonlinear effects in periodic signals generated by nonlinear transducers. Journal of Sound and Vibration. Vol. 420(0), pp. 104-113.

[Novak et al. (2019)] Novak, A., Simon, L., & Lotton, P. (2019). Predistortion technique for generating spectrally clean excitation signals in audio and electro-acoustic nonlinear measurements. Audio Engineering Society Convention 146, Dublin, Ireland

The basic idea of the predistortion consists in modifying the amplitudes and the phases of the harmonic components of the signal $u(t)$ generated by the Data Acquisition System used for the measurement. In the specific case of a pure sine wave generation (see Figure below), the goal is to add higher harmonics to the input signal $u(t)$ so that the harmonics contained in the signal $x(t)$ at the output of the excitation device are suppressed, the only component present in this signal $x(t)$ being then the fundamental harmonic with desired target amplitude and phase.

### Algorithm for the predistortion technique:

1. procedure FRF Estimation (with low-amplitude signal)
2. $M \gets$ number of harmonics to be used
3. ${\bf H_{lin}} \gets$ {complex vector of estimated FRF at frequencies $m f_0$, $m \in (0,1,...,M)$}

4. procedure Initialization
5. ${\bf X^{\oplus}} \gets$ {complex target vector of $M$ harmonics}
6. ${\bf U}_0 \gets \dfrac{{\bf X^{\oplus}}}{{\bf H_{lin}}}$ estimate initial values of input harmonics

7. procedure Process $k$-th frame
8. generate signal $u$ containing harmonics ${\bf U}_k$ according to Eq. (1)
9. acquire signal $x$
10. ${\bf X}_{k} \gets$ complex values of $M$ harmonics of FFT of $x$
11. ${\bf E}_k \gets {\bf X}_{k} - {\bf X^{\oplus}}$ error values of each harmonic
12. ${\bf U}_{k+1} \gets {\bf U}_k - \dfrac{{\bf E_k}}{{\bf H_{lin}}}$ estimated new values of input harmonics

The following figure shows an example of the measurement on a saker: a) the time-domain waveforms of the uncorrected (red solid) and corrected (blue dashed) acceleration measured at the output of the shaker; b) spectra of the uncorrected acceleration (red) with many higher harmonics; c) time-evolution of first five harmonics after the pre-distortion procedure is started; d) spectra of the corrected acceleration (blue) with almost no higher harmonics (up to 100 dB).

The following videos show the recordings from the measurement on a shaker. An accelerometer, attached to the moving part of the shaker is used as a reference signal for the linearization.

Two graphs are shown in the video. The left one shows the waveshape of the acceleration, the right one the power spectral density of the acceleration. The red color represents the state (not varying with time) of the shaker's acceleration without any predistortion; the blue color is the actual state. When the predistortion technique is activated, the blue curves (waveshape and spectra) converges quickly to the desired target signals such as sine multitone, triangular, or even rectangular one.

The first video show the shaker loaded by a vibrating beam.

### Sine Acceleration:

Next videos show the shaker loaded by a metalic rod.

### Rectangular Acceleration:

The following videos show the recordings from the measurement on an electrodynamic loudspeaker. The first loudspeaker (on the left) is used to excite the volume of the sealed box in which a second loudspeaker (on the right) is also placed. The role of the first loudspeaker is to excite the second loudspeaker pneumatically. A microphone placed flash mounted inside the box measures the acoustic pressure created by the first loudspeaker and is used as the reference signal for the predistortion algorithm.

Two graphs are shown in the video. The left one shows the waveshape of the acoustic pressure, the right one its spectrum. The red color represents the state (not varying with time) of the acoustic pressure created by the loudspeaker without any predistortion; the blue color is the actual state. When the predistortion technique is activated, the blue curves (waveshape and spectrum) converges quickly to the desired target with spectrally clean (without unwanted spectral components) signals.