# Adaptive Binaural Cue-Based Amplitude Panning in Irregular Loudspeaker Configurations
## 3 Simulations
Within the time delay range where the mean absolute ITD error approaches zero, the mean IACC initially increases and subsequently decreases, reaching a maximum value at 1.04 ms. Outside this range, the highest mean IACC occurs at 0.77 ms. At this particular time delay, the ITDs for each azimuth angle are restricted to either 0.16 ms or \(-\)0.43 ms, corresponding to the ITDs when the left or right loudspeakers work individually. This indicates that during the search process, the sound image abruptly shifts from the left loudspeaker to the right, failing to synthesize the virtual sound source between loudspeakers. Consequently,  
Figure 4: (**a**) ITD corresponding to different gains and (**b**) optimal gains obtained at various desired azimuth angles for three different time delays. Figure 5: (**a**) ITD, (**b**) absolute ITD error, and (**c**) IACC of virtual source under different time delays. compared to the virtual sound source reproduced by two loudspeakers, the IACC value for a single loudspeaker is higher. Therefore, a time delay of 1.04 ms is selected as the optimal setting, and the corresponding gains obtained during the search are applied to reproduce the virtual sound source at the desired azimuth angles, as shown in Figure 6b. As the desired azimuth angle increases, the gain of the left loudspeaker decreases, and that of the right loudspeaker increases. The proposed algorithm uses a uniform time delay for all azimuth angles, thereby simplifying the creation of continuously moving virtual sound sources compared to methods with variable time delays. For instance, when the virtual sound source is intended to move continuously from \(-15^{\circ}\) to \(-10^{\circ}\), interpolation techniques can be used to achieve the desired spatial trajectory. When the delays for \(-15^{\circ}\) and \(-10^{\circ}\) are consistent, the loudspeaker gains can be easily interpolated. However, if the time delays for \(-15^{\circ}\) and \(-10^{\circ}\) differ, both the time delay and gain would need simultaneous adjustments when moving the virtual sound source. Since time delay adjustments occur in discrete steps (at least a single time sample), maintaining continuous changes becomes challenging and complicates the interpolation process. The performance of the proposed algorithm was evaluated in comparison with the VBAP [13] and DBAP [19] algorithms. As the VBAP algorithm is inherently designed for loudspeaker arrangements with equidistant placement, amplitude and time delay compensation are required when applying it to setups with unequal loudspeaker distances [28]. In the simulation, measured RIRs were used to implement delay compensation, aligning the direct sound peaks of the two loudspeakers. Additionally, the amplitude was adjusted to equalize the peak values of the direct sound. Regarding the DBAP algorithm, which simultaneously controls both the azimuth angle and distance of the virtual sound source, the virtual source was restricted to the loudspeaker axis. Figure 7a presents the ITD simulation results for the three evaluated methods. The VBAP algorithm accurately reproduces virtual sound sources with ITD errors within 0.02 ms (equivalent to one sample delay) and azimuth angle deviations within \(5^{\circ}\) for desired angles between \(-15^{\circ}\) and \(10^{\circ}\), as well as at \(40^{\circ}\). However, in the range of \(15^{\circ}\) to \(35^{\circ}\), the reproduced ITDs are consistently lower than the target values. In particular, at \(20^{\circ}\), the ITD error reaches 0.14 ms (equivalent to six sample delays), with corresponding azimuth errors approaching \(15^{\circ}\). For the DBAP algorithm, the ITDs remain nearly constant at approximately 0.14 ms for azimuth angles between \(-15^{\circ}\) and \(15^{\circ}\). However, at \(20^{\circ}\) and \(25^{\circ}\), the ITDs exhibit a sudden increase to 0.91 ms, while for angles above \(30^{\circ}\), they drop sharply to \(-0.41\) ms.