## 1 Introduction  
As the development of audio and video technologies has progressed in recent decades, the demand for highly realistic immersive auditory experiences is increasing. To meet this demand, the construction of virtual sound sources (VSSs) is an essential task, and provides listeners with perceptions of a sound source without placing an actual sound source at the perceived position. Some methods realize VSSs based on human auditory characteristics. For example, stereepohonic methods [1] reproduce auditory cues, such as interaural level differences and interaural time differences, to give the listener a perception of the direction of a VSS. Nowadays, this mechanism has been further developed and applied in surround sound systems [2; 3; 4]. In addition, some methods realize VSS by synthesizing sound field in physical approaches. Wave field synthesis (WFS) [5; 6; 7; 8; 9] and high-order ambisonics [10; 11; 12] can be raised as well-known examples. High-order ambisonics aims to reproduce the sound field at a certain point with spherical expansion, so it can reproduce the arrival direction of sound with high precision. However, because the controlled space is quite narrow, this method is difficult to use in large spaces or for multiple users. In contrast, WFS aims to reproduce the sound field in a wider range of space suitable for multiple users. Electro-dynamic loudspeakers (EDLs) are widely used in the above VSS construction methods. An EDL converts an electrical signal to a sound wave and emits the sound wave in a wide range of directions. In addition, parametric array loudspeakers (PALs) [13; 14; 15] have been attracting attention in recent years due to their sharp directivity. A typical PAL consists of multiple piezoelectric ultrasonic transducers, and it emits an ultrasonic signal that is modulated by a target sound. Due to the nonlinearity of the air, the interactionsamong the emitted ultrasonic components result in an audible component with the same frequency as the target sound, which is called the self-demodulation process [15]. A self-demodulated audible sound inherits the straightness of emitted ultrasound; therefore, the directivity of PALs is much sharper than that of EDLs. Though PALs suffer from poor bass reproduction and significant harmonic distortions [16], they have been employed in sound systems because of their different sound perception. It was reported that PALs realize more accurate and stable left-right sound image localization than do EDLs in stereophonic reproduction, as the interaural time differences and interaural level differences are different between cases using PALs and those using EDLs [17]. Employing PALs in amplitude panning [2] makes it possible to construct a sharper VSS compared with that produced by EDLs, which has been explained by the difference in interaural level differences and interaural correlation coefficients [18]. In addition, PALs also show unique characteristics in up-down sound image localization [19]. Recently, it has been reported that the sound reproduced by PAL can effectively draw people's attention [20; 21]. To construct VSSs based on WFS, PALs can be used instead of EDLs. It has been reported that WFS with PALs achieves better left-right perception than WFS with EDLs [22], which is consistent with the results of other studies [17; 18]. However, WFS with PALs shows poorer far-near perception than WFS with EDLs. This can be explained by the sharp directivity of PALs, as the direct-to-reverberant ratio (DRR) influences the distance perception of auditory signals [23]. It has been further reported that WFS with PALs provides better far-near perception when a VSS is constructed near the listener, while WFS with EDLs provides better far-near perception when a VSS is constructed far from the listener [24].