## 4 Tensor Decomposition-Based Algorithm of Generating Multiple Virtual Sources  
The proposed algorithm is derived from Equation (7). A HRTF can be approximated as its minimum-phase functions \(H_{min}\) associated with a linear phase or linear delay \(\tau_{2}\) which depends on the source position but is independent of frequency [23]. For discrete source distance \(d\), direction \(m\), and frequency \(k\), we have:  
\[H(d,m,k)=H_{\min}(d,m,k)\exp[-j\tau_{2}(d,m)]. \tag{10}\]  
Accordingly, the binaural signals in Equation (2) can be generated by filtering the input signal with minimum-phase HRTFs, scaling with an inverse distance gain \(1/r_{d}\) and supplementing a linear delay:  
\[ \begin{split} E(d,m,k)&=\frac{1}{r_{d}}H_{\min}(d,m,k)\exp\{-j[\tau_{1}(d)+\tau_{2}(d,m)]\}E_{0}(k)\\ &=\frac{1}{r_{d}}H_{\min}(d,m,k)\exp(-j\tau)E_{0}(k)\end{split} , \tag{11}\]  
where \(\tau=\tau_{1}(d)+\tau_{2}(d,m)\). In the time domain, Equation (11) becomes following convolution form:  
\[e(d,m,n)=\frac{1}{r_{d}}h_{\min}(d,m,n)\otimes_{n}e_{0}(n-\tau), \tag{12}\]  
where \(e(d,m,n)\), \(h_{\min}(d,m,n)\), and \(e_{0}(n-\tau)\) denote the binaural signals, minimum-phase HRIR, and input signal in the time domain, respectively. Minimum-phase approximation of HRTFs/HRIRs simplifies the filters of generating binaural signals. The minimum-phase HRIRs can be represented as the tensor decomposition form according to Equation (7). To derive the algorithm, we express Equation (7) in a different form. Let \(\mathbf{h}_{\min,d,m}=[h_{\min}(d,m,1),h_{\min}(d,m,2)\ldots h_{\min}(d,m,N)]^{T}\) be an \(N\times 1\) column vector or matrix, which represent the \(N\)-point minimum-phase HRIR of a given distance d and direction m, where superscript \"\(T\)\" denotes the transpose of the matrix. Additionally, let \(\mathbf{u}_{n^{\prime}}=[u_{1n^{\prime}},u_{2n^{\prime}}\ldots u_{nn^{\prime}}]^{T}\) with \(n^{\prime}=1,2\ldots N^{\prime\prime}\) as a set of \(N\times 1\) column vectors or matrices taken from matrix \(u_{N\times N^{\prime\prime}}^{n}\) in Equation (7), which represents \(N^{\prime\prime}\) impulse responses (coefficients of finite impulse response filters) corresponding to \(N^{\prime\prime}\) time-related eigen modes of minimum-phase HRIRs. Then, Equation (7) yields:  
\[ \begin{split}\mathbf{h}_{\min,d,m}&=\sum_{n^{\prime}=1 }^{N^{\prime\prime}}\left[\sum_{n^{\prime}=1}^{D^{\prime}}\sum_{n^{\prime}=1} ^{M^{\prime\prime}}w_{d^{\prime}n^{\prime}}u_{dd^{\prime}}^{d}u_{nn^{\prime}} ^{m}\right]\mathbf{u}_{n^{\prime}}\\ &=\sum_{n^{\prime}=1}^{N^{\prime\prime}}c_{n^{\prime}}(d,m)\mathbf{u} _{n^{\prime}},\end{split}  \tag{13}\]  
where  
\[c_{n^{\prime}}=\left[\sum_{d^{\prime}=1}^{D^{\prime}}\sum_{n^{\prime}=1}^{M^{ \prime\prime}}w_{d^{\prime}n^{\prime}n^{\prime}}u_{dd^{\prime}}^{d}u_{mm^{ \prime}}^{m}\right]. \tag{14}\]  
Equation (13) indicates that HRIR of an arbitrary distance and direction can be decomposed into a weight combination of \(N^{\prime\prime}\) time eigen-mode-related impulse responses,which are independent of distance and direction, and the weights depend on the direction and distance. Let \(e_{d,m}(n)\) be the binaural signals of distance \(d\) and direction \(m\), and \(e_{0}(n)\) be the input signal. Then, Equations (12) and (13) yield:  
\[e_{d,m}(n)=\left[\sum_{n=1}^{N^{\prime\prime}}c_{n^{\prime}}\mathbf{u}_{n^{\prime}} \right]\otimes\frac{1}{r_{d}}e_{0}(n-\tau). \tag{15}\]  
Equation (15) indicates that the algorithm of generating binaural signals can be implemented by the following step:  
1. Delay the input signal with a source distance- and direction-related \(\tau(d,m)\), and then scale with a distance-related factor or gain \(1/r_{d}\);
2. Multiply the signal with \(N^{\prime\prime}\) weights of \(c_{n^{\prime}}\) and then convolute or filter with \(\mathbf{u}_{n^{\prime}}\), respectively;
3. The outputs of \(N^{\prime\prime}\) convolvers are summed to form the signal of a given ear;
4. The signal of each ear is generated, respectively. Figure 1 is the block diagram of the algorithm given by Equation (15).