# A Review on Head-Related Transfer Function Generation for Spatial Audio
## 1 Introduction  
The mathematical model describing the acoustic path between a sound source and a listener's ear is represented by the head-related transfer function (HRTF) in the frequency domain and by the head-related impulse response (HRIR) in the time domain. Therefore, for each position of the sound source, two HRTFs are defined, one for the left ear and another for the right ear. HRTFs are unique to each individual, as they are heavily influenced by the reflections of sound waves on the upper body; as such, they are also influenced by the shape of the head, pinnae geometry, and torso dimensions. HRTFs encapsulate all the auditory cues that the human sound perception system (ears and brain) uses to discern the direction and distance of a sound source, such as the perceived sound differences between the ears (i.e., interaural time difference (ITD) and interaural level difference (ILD)) and the spectral cues [1]. The properties of these functions enable their application in the development of spatial audio systems. These systems can be derived, for example, by filtering an audio signal with specific HRTFs [2]. When these signals are reproduced through headphones, the listener perceives the sound source as originating from the original position encoded in the HRTFs. This technique is known as binaural synthesis. HRTFs are also exploited in the reproduction of binaural signals through loudspeakers. In this scenario, the soundemitted from each loudspeaker is perceived by both ears, and a network of filters, known as a crosstalk canceller, is typically employed to mitigate this issue. The crosstalk canceller is obtained by the inversion of the 2 \(\times\) 2 HRTFs matrix related to the position of the two loudspeakers with respect to the two ears [3]. In recent years, considerable research efforts have been focused on developing advanced crosstalk cancellation techniques that utilise HRTFs to achieve optimal sound immersion as described, for instance, in [4; 5; 6; 7; 8; 9; 10; 11; 12]. These techniques aim to eliminate the undesired effects caused by sound reaching both ears from multiple loudspeakers, thereby enhancing spatial audio accuracy. By leveraging HRTFs, crosstalk cancellation methods can deliver a more precise and immersive listening experience, making them crucial for applications in virtual and augmented reality. Numerous studies have investigated the effectiveness of individual HRTFs in spatial audio systems. Tailor-made HRTFs are considered to provide the most accurate 3D audio experience. Furthermore, research, such as that conducted by [PERSON] et al. [13], has demonstrated that using generic HRTFs often leads to suboptimal perception of sound source localisation. The use of non-personalised HRTFs can introduce unnatural sound colouration and other artefacts as highlighted, for example, by [PERSON] [14]. In [15; 16], it has been demonstrated that source localisation is enhanced when individual HRTFs are employed. However, in studies such as [17], it has been observed that personalised HRTFs are not always the preferred subjective choice; this may be attributed to issues arising during the measurement process. In [18], the impact of head movements during HRTF measurements has been assessed, revealing that pitch movements (i.e., the upward or downward motions of the head) have a more significant influence than roll, and yaw. These movements can introduce spectral variations in the HRTFs of up to 6 dB. To mitigate this issue, a neck support can be used for the listener during the measurement process [19], alternatively, databases of individual HRTFs can be developed [20; 21], incorporating information about the azimuth and elevation of the sound source, as well as the listener's head orientation as recorded by a tracking system. In this context, it is evident that HRTFs are essential for creating immersive 3D audio experiences, with several factors requiring careful consideration. This extended review covers a wide range of approaches and technologies aimed at generating HRTFs.