# Diffuse-Field Equalisation of Binaural Ambisonic Rendering
## 1 Introduction  
Ambisonics is a three-dimensional spatial audio approach based on the spatial sampling and reconstruction of a sound field using spherical harmonics, first introduced by [PERSON] in the 1970s [1, 2]. Ambisonic sound fields can be recorded using multiple capsule microphone arrays or encoded using point sources, and reproduced over multiple loudspeaker arrays or over headphones. In Ambisonic playback, assuming free-field reproduction conditions and plane wave propagation, the reproduced sound field is accurate only for the region of a head in the centre of the loudspeaker array, for a frequency range up to the spatial aliasing frequency \(f_{alias}\). This is due to the inherent limited spatial accuracy of recording and reproduction of a physical sound field with a finite number of transducers. Above \(f_{alias}\), timbral inconsistencies exist due to comb filtering from the summation of coherent loudspeaker signals with multiple delay paths to the ears, which are not situated in the exact centre of the loudspeaker array. In practice this reduces the accuracy of reproduced timbre and height cues. By increasing the order of Ambisonics, which requires more microphones and encoded channels in production and storage and more loudspeakers in reproduction, \(f_{alias}\) rises improving both localisation and timbre, though for all practical Ambisonic rendering systems at present \(f_{alias}\) is still very much within the human hearing range. Horizontal localisation accuracy can be improved above \(f_{alias}\) by employing different decoding methods for high frequencies [3], though this does not improve timbre. As accurate timbral reproduction has been shown to be more important than localisation when measuring the authenticity of a spatial audio experience [4, 5], it is essential to address timbre above \(f_{alias}\) in Ambisonic reproduction. The traditional method for rendering Ambisonic sound fields over headphones is to decode the sound field to a specified loudspeaker configuration (as in loudspeaker rendering), and then takethe sum of a convolution of the loudspeaker signals with head-related impulse responses (HRIRs) corresponding to each loudspeaker's position [6; 7] (often referred to as the virtual loudspeaker approach). Recent methods for binaural rendering of Ambisonics have moved away from the virtual loudspeaker approach and instead focused on order truncation of a spatially continuous spherical harmonic (SH) represented head-related transfer functions (HRTF) data set [8; 9; 10], which through pre-processing techniques such as equalisation and time-alignment can produce improvements in spectral response at high frequencies [11; 12; 13]. However, this requires a highly dense dataset of HRTFs capable of artefact free Ambisonic rendering (\(>\)2700 to render 35 th-order Ambisonics [10]) and is therefore considered impractical and infeasible for individualisation at present, despite techniques such as reciprocity [14] and multiple swept sine [15] offering faster measurement times. Therefore, this paper focuses on virtual loudspeaker rendering of Ambisonic signals, though with pre-encoding of the virtual loudspeaker HRTFs into the SH domain, which allows for dual-band decoding and loudspeaker configurations with more loudspeakers than SH channels both at no added computational cost. This paper presents a novel method for improving high-frequency reproduction in binaural Ambisonic rendering, without increasing the order of Ambisonics or real-time computational cost. Diffuse-Field Equalisation (DFE) is the removal of the direction-independent aspect of a set of frequency responses measured at all directions on a sphere. By applying the technique of DFE to binaural Ambisonic rendering, high-frequency reproduction can be improved and therefore a more realistic spatial audio experience can be conveyed to the listener. In a preliminary study, DFE was shown to improve timbre between binaural Ambisonic rendering and standard binaural rendering (HRIR convolution) for 1 st-order Ambisonics [16].