## 7 Discussion and Conclusions  
The experiment in Section 6 validates that the proposed algorithm can generate directional localization and distance perception consistent with that of the conventional algorithm in dynamic VAD. To evaluate the efficiency of the proposed algorithm, some comparisons to the existing algorithms, including the conventional algorithm and binaural Ambisonics algorithm, are beneficial. The required memory and computational cost for synthesizing the signal of each ear in the conventional and proposed algorithms are estimated. The required resource for synthesizing binaural signals is twice that for each ear. The conventional algorithm requires storing a full set of minimum-phase HRTFs at different distances and directions for binaural synthesis. In the example in Section 5.1, the dimensionality of minimum-phase HRIRs for each ear is 7 distances \(\times\) 64,440 directions \(\times\) 128 discrete frequencies = 57,738,240. Moreover, the conventional algorithm requires a 128-point HRIR convolver or finite-impulse-response HRTF filter for each virtual source. Dynamic binaural synthesis requires updating and interpolating on the 128 coefficients of the HRTF filter. The proposed algorithm requires storing the data of tensor presentation of HRIRs. In the example in Section 5.1, the size of the tensor presentation of minimum-phase HRTFs is \(7\times 13\times 8\) (for core tensor) \(+\)\(7\times 7\) (for \(\mathbf{u}_{D\times D^{\prime}}^{d}\)) \(+\)\(64440\times 13\) (for \(\mathbf{u}_{M\times M^{\prime\prime}}^{m}\)) \(+\)\(128\times 8\) (for \(\mathbf{u}_{N\times N^{\prime\prime}}^{n}\)) = 839,521, which is about 1.5% of that of the conventional algorithm. Moreover, the proposed algorithm requires eight 128-point common HRIR convolvers or HRTF filters for synthesizing the signal of each ear. Dynamic binaural synthesis requires updating the 20 weights (weights of 7 distance-related eigen modes \(+\) weights of 13 direction-related eigen modes) for each virtual source, which is about 15.6% of that of the conventional algorithm. Therefore, the required memory and computational cost for the proposed algorithm is less than, or even much less than, those of the conventional algorithm when  
Figure 4: Results of distance perception for virtual sources on the horizontal plane. more than eight virtual sources are generated simultaneously. Moreover, the proposed algorithm can synthesize a binaural signal with high accuracy, with a mean relative error less than \(-20\) dB (with respect to the conventional method), which is enough to accurately reconstruct various information for auditory perception. Binaural Ambisonics is another algorithm for generating a virtual source at various distances and directions in dynamic VAD. According to the Shannon-Nyquist spatial sampling theorem [1], the upper frequency limit for accurately synthesizing binaural signals is related to the order \(L\) of binaural Ambisonics by \(f_{\text{max}}=(Lc/2\pi a)\), and the \(L\)th-order binaural Ambisonics requires \((L+1)^{2}\) virtual loudspeakers or HRTF-based filters for synthesizing the signal of each ear, where \(c=343\) m/s and \(a=0.0875\) m are the speed of sound and mean head radius, respectively. Therefore, both the upper frequency limit and complexity of binaural Ambisonics increases with the order. A 33 rd-order binaural Ambisonics with 1156 HRTF-based filters is required for accurately synthesizing the signal of each ear up to a frequency limit of 20 kHz. Binaural Ambisonics of appropriate order is usually used in practice, which causes error in binaural signals above the upper frequency limit. Although previous analyses and experiments indicated that, by using the redundancy among various auditory direction and distance perception cues, a fifth-order dynamic binaural Ambisonics is able to generate the auditory perception of virtual sources in various directions and at different distances [32].