## 1 Introduction
Figure 2: Labels and installation intervals of loudspeakers. For 22.2 multichannel audio, the conventional panning method, vector base amplitude panning (VBAP) [11; 12], can control the direction of the 3-D sound image by vector synthesis. VBAP divides the reproduction space into a triangular area consisting of three loudspeakers and calculates the gains for the respective loudspeakers. However, a VBAP-based system cannot control the distance of 3-D sound images because VBAP considers \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Layer**} & \multirow{2}{*}{**Channel No.**} & \multirow{2}{*}{**Channel Name**} & \multicolumn{2}{c}{**Setting Range**} \\ \cline{3-5}  & & & **Azimuth** & **Elevation** \\  & & & **(\(\mathbf{\phi}\) [degs.])** & **(\(\mathbf{\theta}\) [degs.])** \\ \hline \multirow{4}{*}{Middle} & 1 & Front left (FL) & \(135\leq\phi\leq 150\) & \(85\leq\theta\leq 90\) \\ \cline{2-5}  & 2 & Front right (FR) & \(30\leq\phi\leq 45\) & \(85\leq\theta\leq 90\) \\ \cline{2-5}  & 3 & Front center (FC) & \(90\) & \(85\leq\theta\leq 90\) \\ \hline \multirow{2}{*}{Lower} & 4 & Low frequency effects-1 & \(120\leq\phi\leq 180\) & \(105\leq\theta\leq 120\) \\ \hline \multirow{4}{*}{Middle} & 5 & Back left (BL) & \(200\leq\phi\leq 225\) & \(75\leq\theta\leq 90\) \\ \cline{2-5}  & 6 & Back right (BR) & \(315\leq\phi\leq 340\) & \(75\leq\theta\leq 90\) \\ \cline{2-5}  & 7 & Front left center (FLc) & \(112.5\leq\phi\leq 120\) & \(85\leq\theta\leq 90\) \\ \cline{2-5}  & 8 & Front right center (FRc) & \(60\leq\phi\leq 67.5\) & \(85\leq\theta\leq 90\) \\ \cline{2-5}  & 9 & Back center (BC) & \(270\) & \(75\leq\theta\leq 90\) \\ \hline \multirow{2}{*}{Lower} & 10 & Low frequency effects-2 & \(0\leq\phi\leq 60\) & \(105\leq\theta\leq 120\) \\ \hline \multirow{2}{*}{Middle} & 11 & Side left (SiL) & \(180\) & \(75\leq\theta\leq 90\) \\ \cline{2-5}  & 12 & Side right (SiR) & \(0\) & \(75\leq\theta\leq 90\) \\ \hline \multirow{4}{*}{Upper} & 13 & Top front left (TpFL) & \(135\leq\phi\leq 150\) & \(45\leq\theta\leq 60\) \\ \cline{2-5}  & 14 & Top front right (TpFR) & \(30\leq\phi\leq 45\) & \(45\leq\theta\leq 60\) \\ \cline{2-5}  & 15 & Top front center (TpFC) & \(90\) & \(45\leq\theta\leq 60\) \\ \cline{2-5}  & 16 & Top center (TpC) & N/A & \(0\) \\ \hline \multirow{2}{*}{Upper} & 17 & Top back left (TpBL) & \(200\leq\phi\leq 225\) & \(45\leq\theta\leq 60\) \\ \cline{2-5}  & 18 & Top back right (TpBR) & \(315\leq\phi\leq 340\) & \(45\leq\theta\leq 60\) \\ \cline{2-5}  & 19 & Top side left (TpSiL) & \(180\) & \(45\leq\theta\leq 60\) \\ \cline{2-5}  & 20 & Top side right (TpSiR) & \(0\) & \(45\leq\theta\leq 60\) \\ \cline{2-5}  & 21 & Top back center (TpBC) & \(270\) & \(45\leq\theta\leq 60\) \\ \hline \multirow{4}{*}{Lower} & 22 & Bottom front center (BtFC) & \(90\) & \(105\leq\theta\leq 120\) \\ \cline{2-5}  & 23 & Bottom front left (BtFL) & \(135\leq\phi\leq 150\) & \(105\leq\theta\leq 120\) \\ \cline{2-5}  & 24 & Bottom front right (BtFR) & \(30\leq\phi\leq 45\) & \(105\leq\theta\leq 120\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Requirement of loudspeaker arrangement in a 22.2 multichannel audio. only the direction of the 3-D sound image. In [12], the sound intensity is considered to obtain the gain vector for three selected loudspeakers to represent the sound intensity of the sound image with the assumption of locating the real loudspeaker at the position of the sound image. However, both VBAC cannot represent the various directivity pattern because these methods only use a direction vector from the loudspeaker to the sound image. From hereafter, we focus on the original VBAP [11] to simplify the discussion in this paper. To solve the problem for the original VBAP, we propose a novel 3-D sound image reproduction method based on spherical harmonic (SH) expansion [13]. The proposed method can control both the direction and distance of the 3-D sound image by controlling the sound directivity on the basis of SHs and mode matching [14].