## 2 Auditory Localization Cues and Conventional Algorithm in Dynamic VAD  
In the present work, we use a spherical coordinate system with respect to the head center. The virtual source position is specified by distance \(0\leq r<\infty\), elevation \(-90^{\circ}\leq\phi\leq 90^{\circ}\), and azimuth \(0^{\circ}\leq\theta\leq 360^{\circ}\), where \(\phi=-90^{\circ}\), \(0^{\circ}\), and \(90^{\circ}\) represents the bottom, horizontal, and top direction, respectively; on the horizontal plane, \(\theta=0^{\circ}\), \(90^{\circ}\), and \(180^{\circ}\) correspond to the front, right, and back direction, respectively. Auditory localization includes directional localization and distance perception. Multiple cues contribute to directional localization for a real sound source in free-field [17]. The inter-aural time difference (ITD) and inter-aural level difference (ILD) are two cues for lateral localization; spectral cues caused by head and pinna scattering and dynamic cues caused by head turning contribute to vertical and front-back localization. Multiple cues also contribute to distance perceptions [18; 19; 20; 21]. For a real source in a free-field, distance perception cues include distance-dependent pressure (\(1/r\) law), distance-dependent ILD, and spectral cues at proximal source distances (\(r\leq 1.0\) m). High-frequency attenuation caused by air absorption is also a weak distance perception cue and is ignored in the presented work. Moreover, in a reflective environment, the direct-to-reverberation ratio is an effective distance perception cue [22], but it is also ignored here because the present work is focused on generating a free-field virtual source. Various auditory localization cues for the free-field source are encoded in HRTFs, which are defined as the normalized acoustic transfer functions from a point source to two ears [1]:  
\[H_{\alpha}(r,\theta,\phi,f)=\frac{P_{\alpha}(r,\theta,\phi,f)}{P_{0}(f)}, \tag{1}\]  
where \(\alpha=\) L or R denotes the left and right ear, respectively, and \(f\) is the frequency. \(P_{\alpha}\) is the pressure at the ear and \(P_{0}\) is the pressure at the position of the center of the head with the head absent. For simplicity, we omit the subscript \"\(\alpha\)\" in the following discussion. HRTFs generally vary with frequency, source distance, and direction. At far-field distances with \(r>1.0\) m, HRTFs are asymptotically distance independent. Otherwise, at near-field distances with \(r<1.0\) m, HRTFs depend on the source distance.  
In the conventional algorithm in VAD, binaural signals of a virtual source are synthesized by filtering the input stimulus \(E_{0}\) with a pair of HRTFs. If the distance-dependent pressure and delay is taken into account, the binaural signals are given by [1]:  
\[E(f)=\frac{1}{r}H(r,\theta,\phi,f)\exp(-j\tau_{1}(r))E_{0}(f). \tag{2}\]  
where \(\tau_{1}\) is the distance-dependent pure delay. In the time domain, Equation (2) can be written as the convolution of input signal with a pair of HRIRs:  
\[e(t)=\frac{1}{r}h(r,\theta,\phi,t)\otimes e_{0}[t-\tau_{1}(r)]. \tag{3}\]  
where notation \"\(\otimes\)\" denotes convolution over time. The time-domain functions in Equation (3) are related to the corresponding frequency-domain functions in Equation (2) by inverse Fourier transformation. In dynamic VAD, the HRTFs in Equation (2) or HRIRs in Equation (3) should be constantly updated accordingly to the temporary position of virtual source with respect to the head.