# Optimization of Virtual Loudspeakers for Spatial Room Acoustics Reproduction with Headphones
## 1 Introduction  
Spatial audio aims to reproduce a believable illusion for a listener being in a real acoustic space by electronic means [1]. Dozens of different ways exist to record or artificially create the spatial sound signals, which are further reproduced with an array of loudspeakers or with headphones [2]. For good spatial resolution, a high number of reproduction loudspeakers are often used in research facilities or in special venues, but such arrays are impractical in domestic or other daily listening environments. Therefore, the headphone reproduction of spatial sound is gaining interest. Commonly, the headphone-based spatial sound is implemented by virtualizing the reproduction loudspeaker array with the Head-Related Transfer Functions (HRTFs) [3]. In essence, HRTFs are applied to virtually position the sound sources around the listener. The resulting binaural rendering can sound very convincing at the best, but, for some users, the sound is localized inside the head. To achieve better externalization, head-tracking devices are often used to compensate the movement of the listener's head and to keep the reproduced auditory environment stable [4; 5]. This work studies the virtual loudspeaker positioning in headphone-based spatial sound reproduction. The sound rendering is based on the Spatial Decomposition Method (SDM) [6], which in essence analyzes directional information in spatial room impulse responses (RIRs). In brief, the SDM uses a compact array of microphones in a RIR measurement. Based on the time difference of arrivals between microphone pairs in short time windows, it estimates the direction of arrival (DOA) for each audio sample in the captured RIR. Therefore, the SDM allows a wide range of perceptual room acoustics studies, and it has been recently applied to study concert halls [7; 8], studio control rooms [9], car cabins [10; 11], as well as stage acoustics for musicians [12]. To reproduce the spatial sound analyzed with the SDM, audio samples in RIR are assigned to the loudspeakers of a given reproduction array. The assignment yields a number of sparse impulseresponses equal to the number of loudspeakers used. The spatial sound is finally rendered by convolving sound signals with each of these sparse impulse responses. That is, the measured RIR is distributed spatially as convolution reverberators for a defined reproduction loudspeaker array. Although the direction of arriving sound is accurately estimated in the analysis, the practical limits in the real or virtual loudspeaker array introduce varying amounts of angular error to the spatial sound reproduction. In the worst case, the mismatch between original and synthesized DOA may change the spatial image of the acoustic space drastically. In theory, all angular errors could be avoided and the analyzed sound field could be reproduced perfectly by assigning each sample to its own loudspeaker. However, this kind of approach is physically infeasible to implement, especially for real room-acoustic conditions. One popular method to reduce the angular error in spatial sound synthesis is the Vector Base Amplitude Panning (VBAP) [13]. VBAP weights each sound sample between the three closest reproduction loudspeakers so that the resulting sound appears to arrive from the original direction. When applied to all samples in the RIR, the spatial image of the space is reproduced correctly. However, it is known that amplitude panning of samples corresponds to time-averaging of multiple HRTFs, leading to low-pass filter effect on the perceived sound [14]. Earlier studies employing SDM [7; 8; 9; 10; 11; 12] have utilized a more straightforward synthesis method, namely Nearest Loudspeaker Synthesis (NLS), partially to circumvent the potential spectral issues with VBAP. NLS distributes each audio sample of a single monaural RIR to the nearest reproduction loudspeaker based on the estimated DOA information.