# Interpolating the Directional Room Impulse Response for Dynamic Spatial Audio Reproduction
## 2 DRIR Interpolation Framework
### Deriving the Interpolated FOA Signals at the Virtual Listening Location
To first evaluate if this was a problem in practice, an analysis was conducted for 500 interpolated DRIRs corresponding to 15 different positions in the classroom to be introduced in Section 3. This analysis compared the TOA for interpolated specular reflections with the TOA for the same specular reflection in the known (ground truth) DRIR at the same location. It was found from this analysis that there were only 10 cases (2%) where the TOA differences indicated a possible mismatch in the specular reflections. For these 10 mismatch cases, it is found that the TOA of the interpolated specular reflection is closer to the TOA of the next specular reflection in the known DRIR (e.g., the first interpolated specular reflection corresponds to the second specular reflection in the known DRIR), which is automatically corrected in the proposed algorithm. Hence, for the dataset to be experimented in Section 3, it was determined that the vast majority of early reflections extracted from the known DRIR locations were matched, and the few mismatched specular reflections in this paper have been automatically corrected. Further research can be conducted to evaluate this issue for a wider variety of locations in the room utilizing more information of the recording environment. For example, if knowledge of the wall locations is available, it will be possible to use the estimated DOAs of two specular reflections extracted at two locations to check if they triangulate at the expected virtual image source location. It is noted that an alternative approach to interpolating the TOA for the specular reflection that takes account of differences in path length may also lead to improved accuracy and will be considered in future work. The proposed method is highly efficient because rather than having to transmit a large set of densely located DRIRs, a smaller set of more widely spaced DRIRs can be transmitted to cover the current region where a user is located within a virtual room. Hence, rather than transmitting a new DRIR to the end device to match the new listening position, interpolation is used to find the new DRIR from the smaller set of transmitted DRIRs. To further evidence the high efficiency, it is worth analyzing the time computational complexity (TCC) of the algorithm. 1. Signal decomposition: the convolution operation in (4) and (5) is implemented using an overlap-add approach based on Fast Fourier Transform, which possesses a TCC of \(O(N_{w}logN_{w})\) with \(N_{w}\) being the window length [25]. The weighted average filters in (6) and (7) show a TCC of \(O(N_{w})\), followed by the peak salience calculation in (9) and value assignments in (10) and (11), with all having a TCC of \(O(N)\) where \(N\) is the length of recording. 2. Deriving DOA and TOA information: (12) and (13) include STFT calculations and thus have a TCC of \(O(N_{w}^{\prime}logN_{w}^{\prime})\) with \(N_{w}^{\prime}\) being the length of the window adopted. 3. Interpolating and synthesizing FOA signals: the summation in signal synthesis shows a TCC of \(O(N)\), which is not increased by the efficient interpolation operation. Therefore, the proposed algorithm concatenated by the three steps has an overall TCC of \(O(N_{w}logN_{w})\), which is highly efficient and suitable for end VR devices. The efficiency of the proposed method is compared with the weighted average approach in [4] as a baseline due to their similar interpolating processes. The baseline method does not decompose signals or utilize the DOA and TOA information, which solely utilizes the third step of the proposed algorithm, leading to a TCC of \(O(N)\). The baseline method is highly efficient, but note that the TCC of \(O(N)\) is not necessarily smaller than \(O(N_{w}logN_{w})\), as \(N\) is usually much larger than the window length \(N_{w}\). Comparison results of the two TCCs depend on the specific values of \(N\) and \(N_{w}\). The other comparative approach in [16] shows parametric calculation of the matrix order, singular value decomposition, regular ized pseudoinverse and so on.