## 2 Proposed Virtual Sound Source Construction Method  
### Overview of Proposed Method  
In this paper, we propose a VSS construction based on WFS using crossfade processing with EDL and PAL arrays. The combination of EDL and PAL arrays is expected to provide a high-precision distance perception for VSSs, and the crossfade processing is expected to smooth the sound pressure transition and to compensate for the frequency characteristics.  
An overview of our VSS construction is given in Figure 1. In the proposed method, a linear EDL array and a linear PAL array are employed as secondary sound sources for WFS. These two loudspeaker arrays are set close together along the \(x\)-axis, and a VSS can be constructed in the \(xy\)-plane. In the following, all notations regarding positions are given in \(xy\)-coordinates, with \(z=0\) by default. As shown in Figure 1, a focused sound source [8; 22] is constructed at the desired position \(r_{\text{S}}=(x_{\text{S}},y_{\text{S}})\), emitting an audio signal \(s[n]\). The position of the VSS is used in the design of crossfade filters, which is described in Section 2.3. The position of the VSS is also used in the generation of driving functions, which is described in Section 2.2.  
The target audible signal of VSS \(s[n]\) is first processed with pre-designed crossfade filters, and the processed signals are given as  
\[s_{\text{EDL}}[n] =s[n]*w_{\text{EDL}}[n], \tag{1}\] \[s_{\text{PAL}}[n] =s[n]*w_{\text{PAL}}[n], \tag{2}\]  
where \(*\) denotes the convolution operator, \(w_{\text{EDL}}[n]\) denotes the pre-designed crossfade filter for the EDL array, and \(w_{\text{PAL}}[n]\) denotes the pre-designed crossfade filter for the PAL array.  
To construct a VSS at position \(r_{\text{S}}\), driving functions are generated based on the positions of the secondary sound sources. Defining \(i\) (\(i=1,2,\cdots I\)) as the index of the secondary sound sources, it should be noted that the \(i\)-th EDL and the \(i\)-th PAL share the same position in the \(xy\)-plane, so they share the same driving function. The driving functions for WFS are applied to the crossfade-filtered signals in order to generate the signals to be emitted from loudspeakers. The signals for EDLs can be given as  
\[u_{\text{EDL},i}[n]=s_{\text{EDL}}[n]*d_{i}[n], \tag{3}\]  
where \(d_{i}[n]\) denotes the driving filter for the \(i\)-th EDL. Similarly, the signals for PALs can be given as  
\[u_{\text{PAL},i}[n]=s_{\text{PAL}}[n]*d_{i}[n]. \tag{4}\]  
It should be noted that because PALs emit ultrasonic signals [15; 16; 26], the amplitude modulation is carried out after the driving functions are applied [22; 24; 25]. When single sideband modulation [15; 26] is utilized, the signal for the \(i\)-th PAL is given as  
\[v_{\text{PAL},i}[n]=v_{\text{C}}[n]+m_{\text{AM}}u_{\text{PAL},i}[n]v_{\text{C }}[n]+m_{\text{AM}}u^{\prime}_{\text{PAL},i}[n]v^{\prime}_{\text{C}}[n], \tag{5}\]  
where \(m_{\text{AM}}\) denotes the modulation factor, \(v_{\text{C}}[n]\) denotes the carrier wave in amplitude modulation, \(u^{\prime}_{\text{PAL},i}[n]\) denotes the signal \(u_{\text{PAL},i}[n]\) after a phase shift of \(\pi/2\), and \(v^{\prime}_{\text{C}}[n]\) denotes the signal \(v_{\text{C}}[n]\) after a phase shift of \(\pi/2\).