## 1 Introduction  
Three-dimensional sound field reproduction [1, 2, 3] has been extensively investigated. Many methods and systems for reproduction have been proposed. In recent years, applications of this reproduction have attracted significant attention for achieving a high sense of presence in theaters, virtual live stages, and telecommunications. One of the simplest systems is a binaural system [4, 5, 6, 7] that reproduces a three-dimensional (3D) sound field through headphones. This system synthesizes the sound field by convolving a dry source signal with head-related transfer functions (HRTFs) [8, 9, 10], which are measured in advance. Since an HRTF is an acoustical transfer function between a sound source and an ear, it is necessary to measure the functions for all of the source positions to be reproduced. This requirement is generally costly because the HRTFs are specific to each individual. Although the HRTF of a dummy head can be used to reduce costs, accuracy and reproducibility are compromised. One large-scale system is a transaural system [11; 12; 13; 14] which consists of numerous remote loudspeakers. This system reproduces a three-dimensional sound field by controlling a multi-channel inverse filter between remote loudspeakers and both ears. Wave field synthesis (WFS) [15; 16; 17; 18] is another type of large-scale system that consists of a loudspeaker array with numerous independently drivable loudspeakers. WFS can control the position of a virtual sound source (VSS) by synthesizing the wavefront. As reasonable-scale systems, various surround systems have been proposed. These include stereo systems, 5.1-channel surround systems [19], and 22.2-channel surround systems [20], also referred to as channel-based systems. Channel-based systems can represent sound sources at the positions of the loudspeakers and therefore require numerous loudspeakers. On the other hand, object-based systems [21; 22], such as Dolby Atmos, have also been proposed. These systems use multiple audio objects, positioning them as virtual sound sources (VSSs) based on vector-based amplitude panning (VBAP) [23]. In addition, high-order ambisonics (HOA) [24] has been proposed as a full-sphere surround sound technique. These surround sound systems can represent the localization of VSSs in terms of their direction. However, localization in terms of distance is also important to accurately representing a VSS. A conventional approach to distance localization is VBAP, which controls the sound pressure level [25]. However, the performance of this method is insufficient because the distance perception cues include not only sound pressure levels but also the Doppler effect [26] and the direct-to-reverberant ratio (DRR) [27; 28; 29; 30; 31]. Therefore, we previously proposed a 3D sound field construction method using parametric-array loudspeakers (PALs) [32; 33; 34; 35]. A PAL [36; 37; 38; 39; 40; 41] achieves sharper directivity by utilizing ultrasonic waves and can form a narrow audible field called an \"audio spot\". A PAL has a significantly sharper directivity than that of a conventional loudspeaker (hereafter referred to simply as a loudspeaker). By using a PAL, we can easily construct a virtual sound source (VSS) in the air. We employ multiple PALs to form a focal point where the emitted sounds from each PAL converge. By doing so, we can reproduce the radiation characteristics of the acoustic waves emitted from a sound source and construct a VSS at the focal point. With this method, the listener perceives the position of the sound source at the focal point by listening to the sounds emitted from the PALs and the reverberation in the room. In addition, we have proposed a virtual sound source construction method based on wave field synthesis using a PAL array and a conventional loudspeaker array [42]. This method utilizes the PAL array and the loudspeaker array to improve the sound quality and focal point control, but it is insufficient for direct-to-reverberant ratio (DRR) control.