## 1 Introduction  
Parametric loudspeakers exploit the non-linear behavior of acoustic waves travelling through air to generate audible sound along a highly directive path due to the self-demodulation property of finite-amplitude ultrasonic waves [1]. The audible components are more directional than sounds produced by conventional loudspeakers, hence they can find application in contexts where audio must be targeted precisely in space. The directional nature of parametric speakers has been used for directing users towards specific objects [2], and a hand-held directional speaker was used to provide targeted information about the objects pointed by the user [3]. Additionally, sound landscapes in which the audience receives sound stimuli from specific locations can be created with directional speakers [4]. In general, directional speakers enable the targeted delivery of audio for applications in advertising, dual-language systems or notifications [5].  
In 1963, [PERSON] [6] theoretically described the generation of difference frequency waves from two high-frequency collimated beams referred to as primary waves. [PERSON] [7] extended this approach and evaluated some possible applications in underwater acoustic transmission. His analysis was not limited to two primary waves and could be applied to a single self-demodulated primary wave. If the primary wave \(p_{1}\) is a generic carrier modulated in amplitude such that:  
\[p_{1}(t)=E(t)P_{0}sin\omega_{c}t, \tag{1}\]where \(P_{0}\) and \(\omega_{c}\) are the amplitude and the angular frequency of the carrier, and \(E(t)\) is the envelope function, then [PERSON]'s farfield solution predicts a self-demodulated wave \(p_{2}\) proportional to the second time derivative of the square of the modulation envelope:  
\[p_{2}\left(t\right)\propto P_{0}^{2}\partial^{2}E^{2}(t)/\partial^{2}, \tag{2}\]  
This dependence implies that the self-demodulated wave (\(p_{2}\)) is not linear to \(E(t)\) and that it will suffer from high levels of distortion due to the generated harmonics and a strong low-pass equalization.  
Various preprocessing techniques have been developed to reduce distortion using different modulations of the envelope \(E(t)\). Existing experimental measurements [8; 9; 10] are mostly based on the total harmonic distortion or at certain representative frequencies, which may not completely capture the non-linear response of the speakers. Here, we present an extensive comparison of the different amplitude modulation techniques under various modulation indices in terms of frequency response, harmonic distortion and intermodulation distortion. This study analyzes the intermodulation distortion and the harmonics distortions of the different amplitude modulation techniques for various modulation indices. Furthermore, the analysis is split by order. To avoid the presence of spurious signal in the measurements, our method previously selects a dynamic microphone with limited frequency response.  
In the Section 2 we describe: the experimental setup, the measurement procedure, the method employed to measure audible sound in the presence of high-levels of ultrasonic carrier and the evaluated amplitude modulation techniques. In the Section 3, we report and analyze the response and distortion of the different modulation techniques and indices. We conclude by summarizing the results and their potential implications.