# Adaptive Binaural Cue-Based Amplitude Panning in Irregular Loudspeaker Configurations
## 3 Simulations  
The simulations were conducted using measured RIRs and BRIRs obtained in the listening room of the Acoustics Institute at Nanjing University, with a reverberation time of approximately 0.3 s. The spatial arrangement of the loudspeakers and listening area is shown in Figure 2a. The distance between the left and right loudspeakers is 1.5 m, and the distance from the center of the listening position to the loudspeaker axis is 1.3 m. The center of the listening area is laterally offset by 0.4 m from the loudspeaker centerline, with its height aligned with the loudspeakers. To acquire the BRIRs, the MegaSig AH 262 desktop dummy head is placed at the listening location, oriented directly forward as shown in Figure 2b. This dummy head is interfaced with a Fireface UC sound card connected to a laptop, which serves to control the loudspeakers and record signals from the left and right ear microphones of the dummy head. The sampling rate is set to 44.1 kHz. The RIRs of both loudspeakers, measured with the microphone placed at the listening position, are shown in Figure 3. Irregular loudspeaker placement introduces variations in the distances between each loudspeaker and the measurement position, causing differences in the direct sound components of the RIRs, both in time and amplitude, compared to standard configurations. Additionally, room modes, loudspeaker directivity, and their differences lead to magnitude variations across frequencies. Therefore, the inverse filtering algorithm is applied to mitigate the magnitude frequency response differences between the left and right loudspeakers, with the target response set to a constant. The frequency range of the inverse filter is limited to 150 Hz to 9000 Hz by adjusting the frequency-dependent regularization factor based on the loudspeakers' working frequency bands. This prevents excessive compensation outside the loudspeakers' working range. The regularization factor \(\beta\) is shown in Figure 3c. For the frequency range from 150 Hz to 9000 Hz, \(\beta\) is set to \(1\times 10^{-4}\), and for frequencies below 75 Hz and above 18,000 Hz, it is set to \(1\times 10^{-3}\). The factor for the remaining frequencies is determined through logarithmic interpolation. To improve the robustness of inverse filtering, one-third octave smoothing is applied to the RIRs before calculating the inverse filters. The resulting filtered magnitude frequency responses are shown in Figure 3d. Compared to the raw RIRs shown in Figure 3b, the magnitude differences between the loudspeakers are substantially reduced, with their magnitude frequency curves exhibiting a near-flat characteristic within the designated frequency range. During the gain and delay optimization phase, the gains and inter-loudspeaker delay are determined to accurately reproduce the virtual sound source at various azimuth angles. In this stage, a range of gains is systematically evaluated, with the corresponding ITD of the virtual sound source calculated. The gain matching the target ITD is selected to reproduce the virtual sound source at the desired azimuth angle. In the simulation, the azimuth angle search range extends from \(-15^{\circ}\) to \(40^{\circ}\), with \(5^{\circ}\) intervals in the front horizontal plane. The target ITD values are derived from measurements of a real source acquired in the anechoic chamber at Nanjing University, corresponding to the relevant azimuth angles. During the optimization stage, the left loudspeaker's gain is incrementally increased from 0 to 1 with discrete steps of 0.001, and the right loudspeaker's gain is calculated using Equation (10). The ITD for each gain pair is calculated using Equations (5)-(7), with the input signal set as the unit impulse signal. Prior to the gain optimization process, a preliminary time  
Figure 2: (**a**) Relative positions of the loudspeakers and listening area and (**b**) illustration of the BRIR measurement setup.