###### Abstract  
With the development of acoustic simulation methods in recent decades, it has become feasible to simulate the sound pressure distribution of loudspeakers before actually setting physical speakers and measuring the sound field. The parametric array loudspeaker (PAL) has attracted attention due to its sharp directivity and unique applications. However, the sound reproduced by PALs is generated by the nonlinear interactions of ultrasound in the air, which makes it difficult to simulate the reproduced sound of a PAL with low computational load. Focusing on the sharp directivity of ultrasound, we extended conventional acoustic ray-tracing methods to consider the self-demodulation phenomenon of PALs. In this study, we developed a visualization method for the demodulated sound of a PAL. Specifically, the demodulated sound pressure distribution can be simulated to estimate and visualize the area covered by the reproduced sound of PAL before setting a real PAL. In the proposed method, acoustic rays were generated sequentially to express the generation of demodulated sound. Therefore, the proposed method is expected to simulate the demodulated sound of a PAL with acceptable accuracy and low calculation complexity. Quantitative evaluation between simulation results and practical measurement has been carried out, and the results demonstrate the effectiveness of the proposed method. parametric array loudspeaker; acoustic simulation; sequential ray tracing; self-demodulation phenomenon; sound pressure distribution +
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Electro-dynamic loudspeakers (EDLs) are widely used in audio applications. A typical EDL holds wide directivity as it emits audible sound directly. In addition, parametric array loudspeakers (PALs) [24; 25] use ultrasound to achieve much sharper directivity than can conventional EDLs [26; 27]. In spite of the poorer sound quality compared with conventional EDLs [28], PALs have been attracting attention because of their sharp directivity and unique audio applications [29; 30; 31; 32]. In the practical use of PALs, the target audible signal is converted into ultrasound with amplitude modulation [25] or frequency modulation [33]. The modulated signal holds multiple frequency components and is emitted into the air intensely. Due to the nonlinearity of air, the target audible signal is \"demodulated\" as one of the difference frequencies among these components. Because this phenomenon occurs due to the nature of the medium [34] rather than by artificial processing, the demodulation process of a PAL can be seen as self-demodulation [25; 27]. To simulate the demodulated sound of a PAL, it is essential to consider both the propagation of emitted ultrasound and the self-demodulation phenomenon. When a wave-based simulation such as FDTD is used, the grid (or element) size should be set as very small because the ultrasound wavelength is short [35]. This will lead to heavy computational cost and a long simulation time to simulate the demodulated sound of a PAL in a wide space with a scale of thousands of wavelengths. In the practical applications of PALs, we are mainly concerned with the sound pressure distribution of the demodulated sound. For example, in [31], an ideal situation is that the whole space can be covered by multiple PALs to obtain a sufficient sound pressure above a certain level at any position inside the space. A quick and feasible simulation of demodulated sound with low computational cost would be helpful in this case. Therefore, we focused on the ray-tracing method [22; 23], which can be used to simulate the propagation of ultrasound emitted from PALs [36]. However, it is difficult to use the traditional ray-tracing method to express self-demodulation. To solve this problem, we introduced self-demodulation to traditional acoustic ray-tracing methods based on [PERSON]'s theory [24], enabling the simulation of the demodulated PAL sound with low computational cost and short simulation time.