## 1 Introduction  
Three-dimensional (3-D) sound field reproduction systems have become increasingly popular as video technology has advanced. The 3-D sound field reproduction systems are classified with a psychoacoustics-based system and physical acoustics-based system. The binaural system and transaural system are traditional psychoacoustics-based systems [1]. These systems represent the sound pressure at a listener's ears by using a head-related transfer function, which represents the reflection and diffraction by a user's head and torso. In other words, the psychoacoustics-based system represents the direction of the sound image. On the other hand, physical acoustics-based systems, such as wave field synthesis [1], are based on the Kirchhoff-Helmholtz integral and reproduce the sound field by using multiple loudspeakers. In [2], the researchers proposed the sound field reproduction system by using dodecahedron loudspeaker array and achieved the reproducing sound field outside of the loudspeaker array. In [3], higher-order Ambisonics (HOA) is used to reproduce a 2-D sound field in the surrounding area by a circular loudspeaker array and cylindrical loudspeaker array. These researches depict the effectiveness of using loudspeaker arrays to achieve accurate sound field reproduction. In this paper, we focus on a 22.2 multichannel audio [4] as multiple loudspeakers. 22.2 multichannel audio [4] is a 3-D sound field reproduction systems, which has been developed for ultra-high definition television (UHDTV). Figures 1 and 2 and Table 1 show the loudspeaker arrangement of 22.2 multichannel audio, the labels and installation intervals of loudspeakers, and the requirement of the loudspeaker arrangement, respectively. This system can be divided into three layers: upper, middle, and lower. It consists of nine loudspeakers in the upper layer, ten in the middle layer, and three in the lower layer along with two sub-woofers called low frequency effects (LFEs). One of the practical uses of the 22.2 multichannel audio is the theater bar for home use ([[https://www.nhk.or.jp/strl/open2018/tenji/t2_e.html](https://www.nhk.or.jp/strl/open2018/tenji/t2_e.html)]([https://www.nhk.or.jp/strl/open2018/tenji/t2_e.html](https://www.nhk.or.jp/strl/open2018/tenji/t2_e.html))) (accessed date 20 January 2022). The theater bar is the home reproduction system of 22.2 multichannel audio with a line-array loudspeaker. Recently, the theater bar as a consumer product has been studied in Japan ([[https://www.jas-audio.or.jp/journal_contents/journal202111_post16264](https://www.jas-audio.or.jp/journal_contents/journal202111_post16264)]([https://www.jas-audio.or.jp/journal_contents/journal202111_post16264](https://www.jas-audio.or.jp/journal_contents/journal202111_post16264))) (accessed date 20 January 2022). According to the ITU-R standards [5], 22.2 multichannel audio can achieve the following effects in sound field. 1. The arrival of sound from all directions surrounding a listening position;
2. High quality 3-D sound impression beyond the 5.1 multichannel audio;
3. High accuracy adjustment of the position between sound and video images. To achieve these effects, 3-D sound field reproduction systems are generally required to control signals emitted from each loudspeaker. To reproduce the 3-D sound image or field, physical acoustics model-based methods have been studied, for example, in [6; 7; 8]. Physical acoustics model-based methods represent the target sound field on the basis of the Kirchhoff-Helmholtz integral equation. In other words, these methods represent the sound field as the physical quantity. In addition, a number of systems adopt the method that represents the arrival direction of the sound [4; 9]. The simplest system is the two channels stereophonic system, which is based on the perception of the direction of arrival [10]. Many sound field reproduction systems of this type utilize the panning method to control a 3-D sound image. Figure 1: Loudspeaker arrangement of 22.2 multichannel audio.