###### Abstract
Another advantage of basis function decomposition-based algorithms is that multiple virtual sources can share a common set of filters; therefore, the required number of filters is fixed and independent from the number of virtual sources. This feature improves the efficiency of generating multiple virtual sources simultaneously. Spherical harmonic function (SHF) decomposition is a well-known example of spatial basis function decomposition, by which HRTFs are decomposed by a set of (source-direction-dependent) spherical harmonic functions associated with (frequency- and distance-dependent) weights [3; 4; 5]. The corresponding basis function decomposition-based algorithm is known as binaural Ambisonics [2; 6; 7; 8]. However, binaural Ambisonics requires a large number of fixed filters to synthesize the binaural signals accurately up to a frequency of 20 kHz. Therefore, its computational efficiency is not high. Binaural Ambisonics with limited order is usually used in practice, which causes errors in the resultant binaural signals. Accordingly, appropriate psychoacoustic principles should be utilized to generate virtual sources in various directions and at various distances. Principal component analysis (PCA), which is basically equivalent to singular value decomposition (SVD), is often used to derive the spectral shape basis function decomposition of HRTFs. By using PCA, the correlation among the HRTFs at different positions are removed so that HRTFs can be decomposed as a combination of spectral shape basis functions associated with source position-dependent weight. Previous studies have indicated that a small set (usually ranging from 5 to 20) of spectral shape basis functions is enough to represent HRTFs at different positions with appropriate accuracy [9; 10; 11]. Therefore, PCA yields a highly efficient spectral shape basis function decomposition of HRTFs. Moreover, PCA is also applicable to derive the spatial basis function decomposition of HRTFs. Accordingly, PCA-based algorithms for synthesizing binaural signals in dynamic VAD have been suggested [6; 12]. However, HRTFs/HRIRs are multivariable functions, including the frequency (or time), source direction, and distances. PCA cannot separate the variations in HRTFs/HRIRs caused by each variation, and the corresponding algorithm cannot manipulate the directions and distance of virtual sources independently. Therefore, the efficiency of the PCA-based algorithm in synthesizing multiple virtual sources in different directions and at different distances should be improved. Tensor decomposition is a multi-linear modelling technique and can be regarded as an extension of PCA and SVD [13]. Tensor decomposition removes the correlation among the HRTFs/HRIRs at different frequencies (or times), directions, distances, and even for different individuals, and represents the HRTFs with variations in multiple independent modes. Tensor methods have been used to decompose the HRTFs at a fixed (far field not less than 1.0 m) source distance and even the HRTFs for different individuals [14; 15; 16]. In the present work, we further applied tensor decomposition to near-field HRTFs in various sources directions and at different distances, and then propose an algorithm for generating multiple virtual sources at various directions and distances. The proposed algorithm enables the manipulation of the directions and distance of virtual sources independently with high efficiency. The paper is organized as follows. Section 2 outlines the auditory localization cues and the conventional algorithm of synthesizing binaural signals of virtual sources. Section 3 discusses the basic principle of tensor decomposition of near-field HRIRs. Section 4 presents the algorithm of generating multiple virtual sources at various directions and distances. Section 5 presents an example of analysis. Section 6 presents the psychoacoustic experiment and results. Section 7 includes some discussion and conclusions.