# A Review on Head-Related Transfer Function Generation for Spatial Audio
## 6 Interpolation
### Nearest Neighbour Interpolation
Figure 9: Comparison of HRTF interpolation methods with minimum-phase HRTF, considering (**a**) the left ear and the (**b**) right ear for the position \((\theta=45^{\circ},\phi=10^{\circ})\) using the MIT-KEMAR database of [41]. requires pre-processing to align the impulse responses. In this context, in 2018, [PERSON] and [PERSON] [33] proposed a new interpolation method applied to BRIRs that divides the impulse responses into early reflections and reverberant tail. The idea of splitting the impulse responses was already proposed by [PERSON] et al. in [208] for room impulse responses. In that case, the dynamic time warping (DTW) algorithm is applied for the temporal alignment, and the reverberant part is synthesised following the approach of [209]. In a different way, [PERSON] and [PERSON] [33] apply the time-splitting to binaural room impulse responses, where the early reflections part is divided into two frequency bands. The low-frequency portions are linearly interpolated, while peak detection and alignment algorithms are reserved for high-frequency parts, which contain most of the information. This method presents a lower computational cost than the DTW algorithm of [208]. In 2020, [PERSON] et al. [172] improved the approach of [PERSON] by changing the peak detection algorithm and evaluating the performance on semi-anechoic measurements. Recently, [PERSON] et al. proposed a spherical array interpolation by time alignment (SARTTA) for upsampling the BRIR dataset [173]. Similarly, signal time alignment is performed by separating early reflections and the direct sound into overlapping blocks, and a linear interpolation is then applied [173]. When nearest neighbour interpolation is applied in the frequency domain, Equation (19) can be directly applied to the complex HRTF; however, this method does not always guarantee good performance, depending on the employed HRTFs. For this reason, some approaches suggest applying the nearest neighbour interpolation to the frequency magnitude response before reconstructing the phase response, as shown in the scheme of Figure 7b. In [28], [PERSON] et al. performed a comparison between IDW and spherical spline interpolation [210], both implemented in time and frequency domains. The results have shown the better performance of the frequency domain approaches, particularly with the spherical spline technique, which involves not only the nearby positions but the entire dataset, thus requiring a higher computational cost. Another frequency domain technique was proposed by [PERSON] in 2013 [24]. In this case, the interpolation is performed in azimuth, elevation, and also distance, and an interpolated HRTF is obtained by applying proper gains to the nearby HRTFs of the database, forming a tetrahedron. Interpolation is only executed only on the magnitude frequency response, and a spherical head model is applied to derive the phase response. In 2017, [PERSON] et al. compared tetrahedral interpolation with triangular and rectangular bilinear interpolation [23]. The results show better performance with minimum-phase responses and with tetrahedral interpolation. However, the latter requires a 3D database that varies with distance and has a higher computational complexity.