*3.3.2 Spatial average of SDR and SD*

In order to show the macroscopic tendencies of the relation between reconstruction accuracies and domains, accuracies in time and frequency domains with number

*Spatial Principal Component Analysis of Head-Related Transfer Functions and Its Domain… DOI: http://dx.doi.org/10.5772/intechopen.104449*

#### **Figure 6.**

*Change of SD in source azimuth at various elevation in each cases with number of PCs set to the least value achieving the cumulative proportion of variance of 0.999 in Table 2. (a) Domain I (No. of PCs = 39). b) Domain C (No. of PCs = 20). (c) Domain F (No. of PCs = 31). (d) Domain L (No. of PCs = 78). (e) Domain CL (No. of PCs = 39).*

of PCs set to *K* are computed in a certain domain X (X = I,C,F,L,CL), the overall average of SDR and SD were calculated by using the following equations:

$$\text{AvSDR}\left(X, K\right) = \mathbf{10} \log\_{10} \left\{ \frac{\mathbf{1}}{M} \sum\_{m=1}^{M} \mathbf{10}^{\text{SDR}\left[\mathbf{h}\_{m}, \left(\mathbf{h}\_{m}^{(\mathbf{X})}\right)\_{k}\right]/10} \right\},\tag{21}$$

$$\text{AvSD}\left(X, K\right) = \sqrt{\frac{1}{M} \sum\_{m=1}^{M} \left\{ \text{SD}\left[\mathbf{A}\_m, \left(\mathbf{A}\_m^{(\mathbf{X})}\right)\_K\right] \right\}^2} . \tag{22}$$

**Figure 7.**

*Changes of (a) average SDR and (b) average* SD *with number of component(s) in five domains. For SDR, five lines respectively corresponds to AvSDR I*ð Þ ,*K , AvSDR C*ð Þ ,*K , AvSDR F*ð Þ ,*K , AvSDR L*ð Þ ,*K and AvSDR CL* ð Þ ,*K , and for* SD*, they respectively corresponds to AvSD I*ð Þ ,*K , AvSD C*ð Þ ,*K , AvSD F*ð Þ , *K , AvSD L*ð Þ , *K and AvSD CL* ð Þ ,*K* ð Þ *K* ¼ 1, ⋯, 40 *. Note that average SDR is calculated with its reference to the minimum phase HRIRs in the domains* F *and* L*.*

Changes of the average SDR and SD with number of component(s) *K* in each case are plotted in **Figure 7** respectively. It is clearly found from these figures that the reconstruction accuracy improves (the larger SDR and the smaller SD) commonly in all domains as the number of PC(s) increases. This means that the CPV corresponds to the tendency of the average accuracies in both time and frequency domains. However, these values are different among five domains. Seeing **Figure 7(a)**, the largest average SDR is achieved with the domain C in most of number of PCs. The domains I and F have the similar tendency, and the domains L and CL are the lowest SDR values when the number of PC(s) is more than 5. On the other hand, as shown in **Figure 7(b)**, the domains L and CL have exceptionally the lowest SD in almost all number of PCs. The domains L and CL has the best accuracy in frequency domain. The third lowest average SD is obtained in the domain C, and the value is almost the same as in the domains L and CL when the number of PCs are relatively large (≥35).
