**1.3 Head-related transfer functions and principal component analysis**

Another promising method for the compact representation of HRTFs is the Principal Component Analysis (PCA) [16, 17]. In some studies, the PCA has an alternative name, the spatial feature extraction method [18–20]. Both have their theoretical basis on the PCA or the Singular Value Decomposition (SVD). In these researches, the spatial variation of HRTFs is modeled by using small number of principal components or eigenvectors. Xie called the PCA adopted to the dataset(s) of HRTFs the Spatial PCA (SPCA) of HRTFs [19]. The author uses this name after Xie in this chapter. As a result of the SPCA, a HRTF in a certain direction is represented as the linear combination of relatively small number of fixed Principal Components (PCs), meaning that these components do not change according to the sound source positions against the listener. The coefficients for the PCs represent such variation. This property has a potential for effective real-time processing concerning their spatial variation due to dynamic factors. The VAD that can synthesize the HRTFs from multiple sound sources in real-time is currently available, for example by using the computational power of the Graphics Processing Units (GPUs) [21].

Many researches have been carried out on the SPCA of HRTFs [16–20], but there are some differences among these studies. One of the obvious differences is the domain to execute the SPCA. Kistler *et al.* applied the log-amplitude of the HRTF to the SPCA [17], Chen *et al.* applied the complex-valued frequency spectrum [18], and Xie applied the amplitude of the HRTF with the assumption of the minimum phase approximation [19]. On the other hand, Wu *et al.* applied the HRIRs [20]. Xie surveyed and summarized those results in his book [22]. These studies indicate that the SPCA can be successively and commonly adopted by using each domain. In contrast, the use of different domains may bring about the different properties in the results of the SPCA. If the HRTF/HRIR can be reconstructed by using the smallest number of PCs in a certain domain, the SPCA in that domain may bring about the most compact representations. There exists a study with the similar purpose. Liang *et al.* compared between the SPCA of the linear and logarithmic magnitudes of the HRTFs [23]. The conclusion of this research was that the SPCA on the linear magnitudes of the HRTFs was better than that on their logarithmic magnitudes in the reconstruction accuracy of their monaural loudness spectra. However, their used HRTFs were limited only in horizontal plane, and they only dealt with two domains with the assumption of the minimum phase approximation. Furthermore, Takane proposed the new domain for the SPCA, the complex logarithm of the HRTFs [24].

In this chapter, all domains dealt with the previous researches are picked up together and the compactness brought by the SPCA using each domain is compared.
