## 7 Discussion and Conclusions
However, the fifth-order binaural Ambisonics is only able to accurately synthesize binaural signals up to a frequency limit of 3.1 kHz. Above the upper frequency limit, the error in binaural signals results in timbre coloration in reproduction. Therefore, the efficiency of the proposed algorithm is higher than the binaural Ambisonics algorithm. In conclusion, the proposed algorithm can generate virtual sources in various directions and at different distances in dynamic VAD and yields localization performance similar to conventional algorithm. Moreover, it has the following advantages:  
1. It can control the direction and distance independently. 2. To simulate the variation of binaural pressures caused by head turning and create moving virtual source, it only requires changing the weights \(c_{n^{\prime}}\) rather than updating the filters or convolvers, avoiding the possibly audible artifacts caused by updating the HRTF-based filters in conventional dynamic VAD. 3. It only requires a set (eight) of common convolvers or filters, the number of convolvers is fixed (does not increases with the number of virtual source). Therefore, the efficiency of algorithm is improved when multiple(more than eight) virtual sources are synthesized simultaneously. Future work may include generating various complex auditory scenes composed of multiple sources by using the proposed method. Conceptualization, T.Z., B.X. and J.Z. ; methodology, T.Z., B.X. and J.Z. ; formal analysis, T.Z. and J.Z. ; validation, T.Z. ; writing--original draft preparation, T.Z. and B.X. ; writing--review and editing, T.Z. and B.X. ; visualization, T.Z. ; supervision, B.X. ; project administration, B.X. ; funding acquisition, B.X. All authors have read and agreed to the published version of the manuscript. This research was funded by the National Natural Science Foundation of China (No. 12174118). Not applicable. The authors declare no conflict of interest. The three matrices \(\mathbf{u}_{D\times D^{\prime}}^{d}\), \(\mathbf{u}_{M\times M^{\prime}}^{m}\) and \(\mathbf{u}_{N\times N}^{m}\) in Equation (5) can be derived using a procedure similar to that used in PCA or SVD calculation. For instance, to derive the matrix \(\mathbf{u}_{N\times N^{\prime\prime}}^{n}\), an \(N\times(DM)\) matrix \(\mathbf{h}_{N\times(DM)}\) is constructed. The rows of \(\mathbf{h}_{N\times(DM)}\)represent the HRIRs at different times, and the columns of \(\mathbf{h}_{N\times(DM)}\) represent the HRIRs at various directions and distances. An \(N\times N\) Hermitian matrix \(\mathbf{R}_{N\times N}\) is constructed from \(\mathbf{h}_{N\times(DM)}\) as:  
\[\mathbf{R}_{N\times N}=\frac{1}{DM}\mathbf{h}_{N\times(DM)}\mathbf{h}_{(DM)\times N}^{T}\, \tag{10}\]  
where superscript \"\(T\)\" denotes the transpose of the matrix. The \(N^{\prime}\) columns of matrix \(\mathbf{u}_{N\times N}^{n}\) are constructed from the orthonormal eigenvectors of matrix \(\mathbf{R}_{N\times N}\) associated with the \(N^{\prime}\leq N\) positive eigenvalues in a descending order:  
\[\lambda_{1}^{n}>\lambda_{2}^{n}>\dots\lambda_{N^{\prime}}^{n}>0. \tag{11}\]  
The other two matrixes \(\mathbf{u}_{D\times D^{\prime}}^{d}\) and \(\mathbf{u}_{M\times M^{\prime}}^{m}\) can be derived similarly. Once the matrixes \(\mathbf{u}_{D\times D^{\prime}}^{d}\), \(\mathbf{u}_{M\times M^{\prime}}^{m}\) and \(\mathbf{u}_{N\times N}^{n}\) are found, the core tensor \(\mathbf{w}_{D^{\prime}\times M^{\prime}\times N^{\prime}}\) in Equation (5) can be calculated using the orthonormal properties of Equation (6), as:  
\[\mathbf{w}_{D^{\prime}\times M^{\prime}\times N^{\prime}}=\mathbf{h}_{D\times M\times N }\times_{D}(\mathbf{u}^{d})_{D^{\prime}\times D}^{\mathrm{T}}\times_{M}(\mathbf{u}^{m })_{M^{\prime}\times M}^{\mathrm{T}}\times_{N}(\mathbf{u}^{n})_{N^{\prime}\times N }^{\mathrm{T}}. \tag{12}\]