# Interaural Level Difference Optimization of Binaural Ambisonic Rendering
## 1 Introduction  
The human auditory system can determine the direction and distance of incoming sounds using three primary binaural localization cues: interaural time difference (ITD), interaural level difference (ILD), and spectral cues. ITDs and ILDs are based on the difference in signals arriving at the left and right ears and help determine the horizontal direction of the sound. Spectral cues are caused by acoustic perturbations such as diffraction and reflections off and around the torso, head and pinnae, and help determine the vertical direction of the sound. Other factors that contribute to a realistic spatial audio experience are externalization and spaciousness. Lower interaural correlation has been shown as necessary to elicit the feeling of spaciousness [1]. Therefore, higher values of ITD and ILD will improve spaciousness. When rendering binaural audio, recreating the spatial cues as realistically as possible will improve the plausibility and authenticity of the auditory experience [2; 3]. Individualized HRIR measurements therefore produce more accurate localization cues and timbre than non-individualed HRIRs [4; 5], such as those acquired from a dummy head. However, to render binaural audio at any position in space requires a highly dense set of HRIRs, which can be difficult to obtain through physical measurements. Additionally, rendering multiple sounds simultaneously or dynamically updating the sound scene to react to changes in head orientation requires computationally costly interpolation [6]. Therefore,other binaural rendering methods are necessary, such as spherical harmonic interpolation of HRIRs using Ambisonics. Ambisonics is a spatial audio technique for recording, storing and reproducing two- or three-dimensional sound fields based on spherical harmonics, initially introduced by [PERSON] in the 1970s [7; 8] and first digitized by [PERSON] and [PERSON] in the 1990s [9]. Binaural Ambisonic reproduction allows spatial audio rendering at any direction with as few as 4 convolutions per ear, and has gained popularity in recent years with virtual reality applications due to the rotational capabilities of spherical harmonics. Theoretically Ambisonics can reproduce the sound field perfectly in the center of a (virtual) loudspeaker array at frequencies up to what is commonly referred to as the'spatial aliasing frequency' \(f_{alias}\)[10], which can be approximated [11; 12] as  
\[f_{alias}=\frac{Mc}{4r(M+1)\sin\frac{\pi}{2M+2}} \tag{1}\]  
where \(M\) is the order of Ambisonic reproduction, \(c\) is the speed of sound, approximated as 343 m/s at 20 \({}^{\circ}\)C in air, and \(r\) is the radius of the listening environment (such as the radius of the human head, in the case of one listener situated in the center of the loudspeaker array). However, at frequencies above \(f_{alias}\), reproduction can be inaccurate due to the limited spatial accuracy of recording and reproducing a physical sound field with a finite number of transducers, which causes localization blur and comb filtering spectral artefacts [13] as well as reduced values of ILD [14] which lead to poor lateralization and spaciousness. Increasing the Ambisonic order allows for exact sound field reproduction up to a higher \(f_{alias}\)[15; 16], though this comes at the expense of more channels needed for storage, more microphone capsules for recording, and more loudspeakers for reproduction. Therefore, it is highly desirable to explore alternative methods of improving Ambisonic reproduction at low orders. Previous attempts to improve ILD reproduction of binaural Ambisonic rendering used additional loudspeakers at the lateral positions in the loudspeaker configuration [17]; however this caused localization issues and worse spectral reproduction due to increased comb filtering from the higher number of virtual loudspeakers [18]. When the soundfield is reconstructed accurately, an Ambisonic rendered HRIR will be equivalent to the original reference HRIR.