# Interpolating the Directional Room Impulse Response for Dynamic Spatial Audio Reproduction
## 2 DRIR Interpolation Framework
### Overview of the Interpolation Approach  
The FOA at location \(L_{m}(x,y,z)\) (assuming a 3D acoustics model) at discrete time \(n\) and in a noise-free environment is assumed to be:  
\[B_{j}(n,L_{m})=\sum_{j=1}^{J}s_{j}\big{(}n,L_{j}\big{)}*h_{j}(n,L_{m}) \tag{1}\]  
Figure 1: FOA interpolation based on a grid of known FOAs for a single source where \(j=1\): (**a**) locations of known FOAs; (**b**) example FOA interpolation using every second known FOAs (FOA 1 and FOA 2) to find the interpolated FOA (FOA 1). Red rectangles indicate chosen grid points.  
where \(s_{j}(n,L_{j})\) is the \(j\)th (\(j=1\), \(2\), \(\ldots\), \(J\)) source originating at location \(L_{j}=\left(x_{j},y_{j},z_{j}\right)\), \(h_{j}(n,L_{m})\) is the corresponding RIR between the source at location \(L_{j}\) and recording location \(L_{m}\), and \((*)\) is the convolution operator. For recording the RIR signal \(h_{j}(n,L_{m})\) as A-format using four microphones in the standard tetrahedral arrangement [22] surrounding the central location \(L_{m}\), we assume the RIR for source \(j\) at each of the four channels are given by \(h_{j,1}(n,L_{m})\), \(h_{j,2}(n,L_{m})\), \(h_{j,3}(n,L_{m})\), and \(h_{j,A}(n,L_{m})\) (labeled as \(h_{j,1}\), \(h_{j,2}\), \(h_{j,3}\), and \(h_{j,A}\) for short in the following descriptions), where the relative locations of each microphone can be found by standard equations for B-format. The resulting B-format channel signals converted from the A-format recordings are represented as [22]:  
\[\left\{\begin{array}{c}W_{j}(n,L_{m})=\frac{1}{\sqrt{2}}s_{j}\big{(}n,L_{j} \big{)}*\big{(}h_{j,1}+h_{j,2}+h_{j,3}+h_{j,A}\big{)}\\ X_{j}(n,L_{m})=s_{j}\big{(}n,L_{j}\big{)}*\big{(}h_{j,1}+h_{j,2}-h_{j,3}-h_{j, A}\big{)}\\ Y_{j}(n,L_{m})=s_{j}\big{(}n,L_{j}\big{)}*\big{(}h_{j,1}-h_{j,2}+h_{j,3}-h_{j, A}\big{)}\\ Z_{j}(n,L_{m})=s_{j}\big{(}n,L_{j}\big{)}*\big{(}h_{j,1}-h_{j,2}-h_{j,3}+h_{j, A}\big{)}\\ \end{array}\right. \tag{2}\]  
In a compact form, the FOA signals can be represented as:  
\[B_{j}(n,L_{m})=\big{(}X_{j}(n,L_{m}),Y_{j}(n,L_{m}),\ Z_{j}(n,L_{m}),W_{j}(n,L _{m})\big{)} \tag{3}\]  
Figure 1 illustrates the DRIR interpolation approach for a single source (\(j=1\)) when using two FOA recordings at locations \(L_{K1}\) and \(L_{K2}\) to find the interpolated FOA signal at location \(L_{I}\). The proposed method can be summarized as follows:  
1. Separate the FOA signals of DRIR into the direct sound, the specular reflection signals and the chaotic diffuse components;
2. Analyze the \(W\) channel of the separated direct sound and specular reflection FOA signals to locate peaks corresponding to estimated image sources;
3. Form a new FOA signal for this image source using a segment of the signal surrounding this peak for the \(W\) channel and corresponding \(X\), \(Y\), and \(Z\) channels;
4. Repeat until a desired number of image sources (peaks) have been extracted;
5. Interpolate such FOA signals and their extracted DOA information at two closest recording positions to the listening position using a weighted average approach;
6. Synthesize the FOA signal of DRIR at the listening position utilizing the interpolated parameters and a known FOA signal recorded closely.