## 1 Introduction
(\(-45\leq\theta\leq 45\)) is the direction of the VSS, and \(h_{\text{L}_{\theta}}\) and \(h_{\text{R}_{\theta}}\) are the weighting coefficients for the left- and right-side loudspeakers, respectively. Figure 3: Overview of the proposed method with four pairs of PALs and loudspeakers. The weighting coefficients for the distance of the VSS are calculated using a linear ratio for the loudspeaker and an energy ratio for the PAL as follows:  
\[d_{\mathrm{E_{r}}} = \sigma_{r}^{-1}\cdot\frac{r}{d}, \tag{3}\] \[d_{\mathrm{P_{r}}} = \sigma_{r}^{-1}\cdot\sqrt{\frac{d-r}{d}},\] (4) \[\sigma_{r} = \frac{r}{d}+\sqrt{\frac{d-r}{d}}, \tag{5}\]  
where \(r\) [m] (\(0\leq r\leq d\)) is the distance to the VSS, \(\sigma_{r}^{-1}\) is the amplitude normalization factor for \(r\) m, and \(d_{\mathrm{E_{r}}}\) and \(d_{\mathrm{P_{r}}}\) are the weighting functions for the distance \(r\) m of the loudspeakers and PALs, respectively. The distance attenuation function is calculated as follows:  
\[a_{r}=e^{-\eta r}, \tag{6}\]  
where \(\eta\) is the attenuation factor. \(\eta\) is determined using a regression analysis based on the room impulse responses measured at 0.1 m, 0.2 m, and 0.3 m. Additionally, the proposed method must normalize the sound pressure level of the audible sound from both the loudspeakers and PALs at the listener's head position. The output signals for the front area are calculated as follows:  
Figure 2:0 Figure 2:1 (8) Figure 2:2 (9) Figure 2:3  
where \(\xi_{r}\) is the weighting function for DRR correction at the position \((r,\theta)\), \(c(t)\) is the ultrasonic carrier wave, \(s(t)\) is the normalized input signal of the audible sound, \(a_{r}\) is the  
Figure 2:4 Overview of front area output signal design in the proposed method. weighting function for distance attenuation at the distance \(r\), \(\hat{x}_{\text{L}_{r,\theta}}(t)\) is the output signal for the loudspeaker in the FL direction, \(\hat{x}_{\text{R}_{r,\theta}}(t)\) is the output signal for the loudspeaker in the FR direction, \(\hat{y}_{\text{L}_{r,\theta}}(t)\) is the output signal for the PAL in the FL direction, and \(\hat{y}_{\text{R}_{r,\theta}}(t)\) is the output signal for the PAL in the FR direction. These outputs simply achieve DRR control for distance and amplitude panning for direction. To apply this method to another area, the output channels should be selected based on back-projection to the VSS coordinates for each area. In the right area (45\({}^{\circ}\) to 135\({}^{\circ}\)) in Figure 3, the corresponding relationship between the output signals and loudspeakers is as follows:  
* The output signal for the front R-ch loudspeaker: \(\hat{x}_{\text{L}_{r,\theta}}(t)\);
* The output signal for the front R-ch PAL: \(\hat{y}_{\text{L}_{r,\theta}}(t)\);
* The output signal for the rear R-ch loudspeaker: \(\hat{x}_{\text{R}_{r,\theta}}(t)\);
* The output signal for the rear R-ch PAL: \(\hat{y}_{\text{R}_{r,\theta}}(t)\). In the rear area (135\({}^{\circ}\) to \(-\)135\({}^{\circ}\)) in Figure 3, the corresponding relationship between the output signals and loudspeakers is as follows:  
* The output signal for the rear R-ch loudspeaker: \(\hat{x}_{\text{L}_{r,\theta}}(t)\);
* The output signal for the rear R-ch PAL: \(\hat{y}_{\text{L}_{r,\theta}}(t)\);
* The output signal for the rear L-ch loudspeaker: \(\hat{x}_{\text{R}_{r,\theta}}(t)\);
* The output signal for the rear L-ch PAL: \(\hat{y}_{\text{R}_{r,\theta}}(t)\). In the left area (\(-\)135\({}^{\circ}\) to \(-\)45\({}^{\circ}\)) in Figure 3, the corresponding relationship between the output signals and loudspeakers is as follows:  
* The output signal for the rear L-ch loudspeaker: \(\hat{x}_{\text{L}_{r,\theta}}(t)\);
* The output signal for the rear L-ch PAL: \(\hat{y}_{\text{L}_{r,\theta}}(t)\);
* The output signal for the front L-ch loudspeaker: \(\hat{x}_{\text{R}_{r,\theta}}(t)\);
* The output signal for the front L-ch PAL: \(\hat{y}_{\text{R}_{r,\theta}}(t)\). By switching the above outputs, a VSS can be constructed in all directions on the horizontal plane.