## 5 The Results of Analysis
### Results of Analysis  
Tensor decomposition, as discussed in Section 3, is applied to the aforementioned HRIRs, yielding a core tensor of \(\mathbf{w}_{128\times 64440\times 7}\). Tables 1 and 2 list the relationship between the cumulative percentage variance of energy and the number of direction- and time-related modes for the left ear, respectively. The results indicate that the cumulative percentage variations in the energy of HRIRs in Equation (8) increase with the number of eigen modes in the decomposition. Moreover, the first mode accounts for nearly 69.9% and 70.7% of the direction- and time-related variance respectively, and the contributions of other modes descend with the order. This is a general characteristic of tensor decomposition of HRIRs/HRTFs and is consistent with some previous results [26]. For both left- and right-ear minimum-phase HRIRs, \(M^{\prime\prime}=13\) direction-related eigen modes and \(N^{\prime\prime}=8\) time-related eigen modes account for over 99.0% of energy variance. Thus, the order of direction- and time-related modes were truncated to \(M^{\prime\prime}=13\) and \(N^{\prime\prime}=8\), yielding a truncated core tensor of \(\mathbf{w}_{8\times 13\times 7}\) and two truncated unitary matrices of \(\mathbf{u}_{64440\times 13}^{m}\) and \(\mathbf{u}_{128\times 8}^{n}\) respectively. The minimum-phase HRIRs are approximately reconstructed with \(M^{\prime\prime}=13\) direction-related eigen modes and \(N^{\prime\prime}=8\) time-related eigen modes according to Equation (7), and the mean relative error across all frequencies and distances calculated from Equation (9) is \(-21.9\) dB for the left ear and \(-22.0\) dB for the right ear.  
Figure 1: The block diagram of algorithm.  
Figure 2 illustrates the mean relative error calculated from Equation (9) across all distances on the horizontal plane for the left ear. The error varies with the direction (azimuth) and frequency. Actually, reconstruction errors on the horizontal plane are larger than other elevations. As seen in Figure 2b, the mean relative error across distances is less than \(-10\) dB for most cases, which is basically sufficient for the HRTF reconstruction. Calculation from the results indicate that the mean relative error across distances is \(-27.7\) dB at the ipsilateral direction (\(\theta=265^{\circ}\), \(\phi=0^{\circ}\)) and \(-17.5\) dB at the contralateral direction (\(\theta=85^{\circ}\), \(\phi=0^{\circ}\)). Generally, the errors are relatively large at high frequencies above 10 kHz and for source azimuth contralateral to the left ear. These errors are due to the complex nature of contralateral HRTFs at high frequency caused by head scattering and shadow effects. However, contralateral HRTFs at high frequencies contribute less to auditory perception because their magnitudes are attenuated by head shadows. Overall, the results of analysis indicate that \(M^{\prime\prime}=13\) direction-related eigen modes and \(N^{\prime\prime}=8\) time-related eigen modes are enough to reconstruct the minimum HRIRs with high accuracy.  
The reconstructed results for the right-ear HRTF are quite similar to those for the left ear; thus, these results are omitted here.