## 3 Results
### Harmonic Distortion  
In the audio range, there are different parameters and methods to evaluate the non-linear response of a speaker. The most widespread measure for characterizing the non-linear response of a speaker is the total harmonic distortion (THD) obtained by sinusoidal signals of increasing frequency. Using consecutive increments of stationary tones (step-by-step method) is the most common procedure employed to measure distortion of parametric loudspeakers that can be found in the literature [33]. However, in the audio range it is preferable to use the technique based on sweeps proposed by [PERSON] [34] that offers benefits in terms of ease of use, signal-to-noise ratio and immunity against the temporal variation of the DUT. We will use this method.  
The advantage of measuring impulse responses using sweeps is that the artifacts generated by the harmonic distortion can be eliminated, since these appear at negative times in relation to direct sound, and can be separated from the desired \(h(t)\)[35]. In addition, when using a logarithmic sweep as excitation signal, the harmonics will have a constant group delay independent of the frequency when it is deconvoluted with the reference spectrum. They will therefore appear in predictable positions of \(h(t)\) and can be separated by windowing if the sweep is sufficiently slow. Hence, with a single excitation signal, both the linear transfer function of the system and the harmonic distortion decomposed into several orders can be obtained, eliminating the influence of non-linearities. We measured harmonics up to the 5 th order.  
In Supplementary Figure S3, we present the decomposed harmonic distortion for the different preprocessing techniques. It can be seen that beyond the 2 nd order harmonic, the distortion was negligible. Most of the distortion resides in the second harmonic (see Supplementary Figure S3). It is worth noting that some techniques (e.g., MAM1) reduced the 2 nd harmonic considerably at some frequencies, almost to the levels of the 3 rd harmonic.  
The level of total harmonic distortion (\(L_{THD}\)) in decibels was determined by the formula:  
\[L_{THD}(f)=20\text{lg}\left(\frac{\sqrt{\sum_{n=1}^{5}\overline{p}_{nf}^{2}(f) }}{\overline{p}_{f}(f)}\right)dB. \tag{10}\]  
where \(\overline{p}_{nf}(f)\) is the RMS value of the nth-order harmonic component and \(\overline{p}_{f}(f)\) is the RMS value of the first harmonic of the fundamental component.  
Figure 6 shows the \(L_{THD}\) obtained for all modulations with m = 1. \(L_{THD}\) smaller than 5% (i.e., \(-26\) dB) were generally acceptable levels for domestic audio equipment. MAM techniques  
Figure 5: Transfer functions measured with different preprocessing techniques (modulation index m = 1).  
provided the best results up to 2.6 kHz, and single sidebands (i.e., LSBAM and USBAM) provided the best overall result, staying below the distortion threshold throughout the working frequency of the speaker. For DSBAM, the high level of the second harmonic caused \(L_{THD}\) greater than \(-20\) dB in virtually the entire frequency range. MAM and SRAM modulations obtained good results up to 2 kHz but for higher frequencies. Its behavior was similar to the conventional DSBAM modulation. The smallest values for harmonic distortion were obtained with single sideband modulations. This is because these modulations used only one band, thus there was only one difference-signal. The other techniques used a double band and thus produced two difference-signals generating more harmonics and increasing the harmonic distortion.  
In Supplementary Figure S4, we show the \(L_{THD}\) for each predistortion technique and modulation index ranging from 0.1 to 1 at 0.1 steps. In general, the larger the modulation index, the more distortion appeared. However, this increment was more pronounced for DSBAM along all frequencies, for SRAM from 1 to 6 kHz, for MAM1 beyond 1 kHz, and for MAM2 beyond 1.5 kHz. Both LSBAM and USBAM were less affected by an increase of total harmonic distortion as the modulation index increased.