# Upmix B-Format Ambisonic Room Impulse Responses Using a Generative Model
## 2 Review of Ambisonics
Considering the influence of scattering on the surface of a rigid sphere, the transformation matrix needs to incorporate the modal functions, that is  
\[\mathbf{T}^{\prime}=\mathbf{Y}\text{diag}[b_{n}(kr)]. \tag{8}\]  
Here, the modal function for a rigid sphere is represented as  
\[b_{n}=4\pi i^{n}(j_{n}(kr)-\frac{j_{n}^{\prime}(kr))}{(h_{n}^{(2)})^{\prime}(kr )}h_{n}^{(2)}(kr)), \tag{9}\]  
where \(j_{n}(kr)\) is the spherical Bessel function of the first kind, \(h_{n}^{(2)}(\cdot)\) denotes the spherical Hankel function of the second kind in the \(n\)-th order, and \((\cdot)^{\prime}\) denotes the corresponding 1 st derivative. Then, the HOA signals are calculated as follows:  
\[\mathbf{A}_{LS}=\text{diag}[b_{n}^{-1}(kr)]\mathbf{Y}^{\intercal}\mathbf{p}, \tag{10}\]  
where \(\text{diag}[b_{n}^{-1}(kr)]\) is the radial function [28]. Given the decomposed SH coefficients, we can reproduce sound pressure to a specific direction \(\Omega\). The reconstruction process involves the \(N\)th-order transform matrix \(\mathbf{T}_{N}\) and the SH coefficients \(\mathbf{A}_{N}\). Mathematically, the reconstructed signal \(\mathbf{\hat{p}}\) can be expressed as  
\[\mathbf{\hat{p}}=\mathbf{T}_{N}\cdot\mathbf{A}_{N}. \tag{11}\]  
Here, \(\mathbf{T}_{N}\) represents the matrix of \(N\)th-order spherical harmonic basis functions as Equation (6). Due to the limited number of microphones and the finite aperture of the microphone array, accurate capture of the room acoustics above the aliasing frequency is challenging. Spatial aliasing occurs when the frequency of the sound field exceeds the Nyquist frequency of the microphone array, which leads to a roll-off of high-frequency components and subsequently distorts the sound field. The aliasing frequency is determined by the condition \(k\leq N/r\), where \(N\) represents the SH decomposition order and \(r\) represents the array radius. For instance, the aliasing frequency of the 4 th-order spherical array with a radius of \(r=4.2\) cm, i.e., Eigenmike, is approximately 5.2 kHz. When the signals are recorded by the first-order spherical microphone array of the same radius, the aliasing frequency is approximately 1.2 kHz. We conduct a reconstruction of the FOA and HOA RIRs measured by [PERSON] et al. using an Eigenmike [29]. The reconstruction process involves mapping the ARIRs to the multi-channel signals at the positions of the microphones located on the surface of the sphere. The evaluation of the reconstruction quality is performed based on the Mean Absolute Error (MAE) and Normalized Mean Squared Error (NMSE) of the amplitude spectrum as shown in Figure 2. The MAE defined in Equation (12) is calculated as the average magnitude error between the measured sound pressure \(p_{m}(k)\) from \(M\) microphone  
Figure 1: The Ambisonics microphone arrays: (**a**) The FOA microphone array with four cardioid microphones. (**b**) The Eigenmike spherical microphone array with a radius of 4.2 cm and 32 omnidirectional microphones. channels on the surface of the spherical microphone array, and the corresponding estimated sound pressure \(\hat{p}_{m}(k)\) at various frequency points \(k\). The NMSE is defined in Equation (13). These metrics are averaged over reconstruction directions and one-third of octave bands:  
\[Error_{MAE}(k)=\frac{1}{M}\sum_{m=1}^{M}|p_{m}(k)-\hat{p}_{m}(k)|, \tag{12}\]  
\[Error_{NMSE}(k)=\frac{\sum_{m=1}^{M}|p_{m}(k)-\hat{p}_{m}(k)|^{2}}{\sum_{m=1}^ {M}|p_{m}(k)|^{2}}. \tag{13}\]  
It is observed that both the MAE and NMSE increase rapidly after the aliasing frequency is surpassed. The results show that the reconstruction of FOA RIRs has much lower aliasing frequencies and more significant overall errors.