## 2 Proposed Virtual Sound Source Construction Method
### Crossfade Processing in Proposed Method  
This section describes the design of crossfade filters in the proposed method. A major drawback of the previous VSS construction method based on WFS with EDLs and PALs [24] is that a change in the frequency characteristics is noticeable as the VSS moves. Therefore, it is essential to balance the output of PALs and EDLs with the crossfade filters. In this proposed method, we approximate the synthesized sound field as the superposition of two sound fields: one is that synthesized with WFS using only the EDL array, and the other one is that using only the PAL array. Under the above approximation, we first measure the impulse response of a real sound source, which holds the desired frequency characteristics. Then, we design the crossfade filters to synthesize the target frequency characteristics. This concept can be expressed as  
\[h_{\text{Real}}[n]\approx h_{\text{EDL}}[n]\ast w_{\text{EDL}}[n]+h_{\text{ PAL}}[n]\ast w_{\text{PAL}}[n], \tag{14}\]  
where \(h_{\text{Real}}[n]\) denotes the impulse response between the position of sound source \(\mathbf{r}_{\text{S}}\) and the reference point, \(h_{\text{EDL}}[n]\) denotes the measured impulse response at the reference point when we construct the VSS at \(\mathbf{r}_{\text{S}}\) using only the EDL array, and \(h_{\text{PAL}}[n]\) denotes the measured impulse response at the reference point when we construct the VSS at \(\mathbf{r}_{\text{S}}\) using only the PAL array. To realize this concept and generate the crossfade filters in a feasible way, we utilize an adaptive filters least mean square (LMS) algorithm [29] in the proposed method. Figure 3 gives a block diagram for the design of the crossfade filters. Here, two adaptive filters share the same error signal \(e[n]\), which is defined as  
\[e[n]=s_{\text{WN}}[n]\ast h_{\text{Real}}[n-\tau] -\hat{s}_{\text{EDL}}[n]\ast\hat{w}_{\text{EDL}}[n]-\hat{s}_{ \text{PAL}}[n]\ast\hat{w}_{\text{PAL}}[n], \tag{15}\] \[\hat{s}_{\text{EDL}}[n] =s_{\text{WN}}[n]\ast h_{\text{EDL}}[n],\] (16) \[\hat{s}_{\text{PAL}}[n] =s_{\text{WN}}[n]\ast h_{\text{PAL}}[n], \tag{17}\]  
where \(s_{\text{WN}}[n]\) denotes band-limited white noise that covers the band of the VSS and \(\tau\) denotes the delay. For adaptive filter \(\hat{\mathbf{w}}_{\text{EDL}}\), the update of coefficients is performed using  
\[\hat{\mathbf{w}}_{\text{EDL}}(n+1)=\hat{\mathbf{w}}_{\text{EDL}}(n)+\eta e[n]\hat{\mathbf{s} }_{\text{EDL}}(n), \tag{18}\]  
where \(\eta\) denotes the step size. Here, the vectors \(\hat{\mathbf{w}}_{\text{EDL}}\) and \(\hat{\mathbf{s}}_{\text{EDL}}\) are updated as the time index \(n\), and are defined as  
\[\hat{\mathbf{w}}_{\text{EDL}}(n):=\Big{(}\hat{\mathbf{w}}_{\text{EDL}}^{(n )}[0],\hat{\mathbf{w}}_{\text{EDL}}^{(n)}[1],\cdots,\hat{\mathbf{w}}_{\text{EDL}}^{(n) }[L-1]\Big{)}^{T}, \tag{19}\] \[\hat{\mathbf{s}}_{\text{EDL}}(n):=(\hat{s}_{\text{EDL}}[n],\hat{s}_{ \text{EDL}}[n-1],\cdots,\hat{s}_{\text{EDL}}[n-L+1])^{T}, \tag{20}\]  
where \(L\) denotes the tap number of filter. The above equations suggest that the total update run is finite and depends on the duration of white noise \(s_{\text{WN}}[n]\). After the update of the adaptive filter, the optimal coefficients to be used are \(\mathbf{w}_{\text{EDL}}\), as in Section 2.1, which are given as  
\[ \begin{split}\mathbf{w}_{\text{EDL}}:=&(w_{\text{EDL}}[ 0],w_{\text{EDL}}[1],\cdots,w_{\text{EDL}}[n],\cdots,w_{\text{EDL}}[L-1])^{T}\\ =&\Big{(}\hat{\mathbf{w}}_{\text{EDL}}^{(n_{\text{opt}} )}[0],\hat{\mathbf{w}}_{\text{EDL}}^{(n_{\text{opt}})}[1],\cdots,\hat{\mathbf{w}}_{ \text{EDL}}^{(n_{\text{opt}})}[n],\cdots,\hat{\mathbf{w}}_{\text{EDL}}^{(n_{\text {opt}})}[L-1]\Big{)}^{T},\end{split}