## 2 Materials and Methods
### Preprocessing Techniques  
Amplitude modulation (AM) is the most common preprocessing technique in parametric loudspeakers. It consists of changing the amplitude of a relatively high-frequency carrier of angular frequency \(\omega_{c}\) according to the amplitude of a modulating signal. The envelope of the carrier wave is weighted with the desired audible sound signal \(s(t)\).  
Double side band amplitude modulation (DSBAM) was the original method used by [PERSON] [27] to produce wideband audio with parametric loudspeakers. The modulated ultrasonic wave \(p(t)\) is expressed as  
\[p_{DSBAM}(t)=[1+m\;s(t)]sin(\omega_{c}t), \tag{4}\]  
where \(m\) is the modulation index and has values ranging from 0 to 1. DSBAM is simple to implement and requires low computational resources. In addition, it generates a louder audible signal since it uses both sidebands. However, due to the tone difference between the upper (USB) and lower band (LSB), the second harmonic is also louder.  
A simple solution to reduce the second harmonic is to transfer the energy to a single band as in the single side band amplitude modulations (SSBAM) [28], which are named LSBAM for the lower or USBAM for the upper sideband. This modulation can be obtained by using high-pass, low-pass filter or adding a quadrature term to the conventional AM. For the latter, a Hilbert filter is needed to convert the modulating signal \(s(t)\) into its orthogonal counterpart \(\hat{s}(t)\). The modulations are expressed as:  
\[p_{LSBAM}(t)=[1+ms(t)]sin(\omega_{c}t)-m\hat{s}(t)cos(\omega_{c}t), \tag{5}\]  
Figure 4: Frequency responses of the difference-frequency signal obtained with the tested microphones. The ultrasonic array was used as an emitter with a conventional DSBAM modulation.  
\[p_{USBAM}(t)=[1+ms(t)]sin(\omega_{c}t)+m\bar{s}(t)cos(\omega_{c}t). \tag{6}\]  
Inspired by [PERSON]'s solution, a square root operation was added to the AM modulation, leading to the so-called square root amplitude modulation (SRAM):  
\[p_{SRAM}(t)=\sqrt{[1+ms(t)]}sin(\omega_{c}t). \tag{7}\]  
This procedure theoretically removes distortion but introduces infinite harmonics of the original signal \(s(t)\). Therefore, the resulting reduction in the distortion is limited by the bandwidth of the ultrasonic emitters [29].  
A set of preprocessing techniques based on quadrature amplitude modulation has been proposed to deal with the bandwidth limitation of the emitters [30]. This class of preprocessing techniques referred to as modified amplitude modulation (MAM) can be adapted to the available bandwidth using different orders of the Taylor expansion for the distortion term modulated by the orthogonal carrier. The modulated ultrasonic waves for the first two orders are described as:  
\[p_{MAM1}(t)=[1+ms(t)]sin(\omega_{c}t)+\bigg{[}1-\frac{1}{2}m^{2}s^{2}(t)\bigg{]} cos(\omega_{c}t), \tag{8}\]  
\[p_{MAM2}(t)=[1+ms(t)]sin(\omega_{c}t)+\bigg{[}1-\frac{1}{2}m^{2}s^{2}(t)-\frac{ 1}{8}m^{4}s^{4}(t)\bigg{]}cos(\omega_{c}t). \tag{9}\]  
Due to the low-cost of its implementation, FM-based methods have also been analyzed empirically [31]. However, demodulating FM waves causes higher harmonic distortions at conventional sound pressure levels. There are combined methods that achieve higher sound pressure levels at lower frequencies [32]. We kept the scope of the paper to amplitude modulation techniques, but frequency modulation is a promising technique that could be included in future studies.  
Double side band amplitude modulation (DSBAM), lower side band amplitude modulation (LSBAM), upper side band amplitude modulation (USBAM), square root amplitude modulation (SRAM), modified amplitude modulation 1 (MAM1) and 2 (MAM2) can be found as block implementations in Supplementary Figure S1. These preprocessing techniques were implemented in Matlab.