# Active Noise Control over Space: A Subspace Method for Performance Analysis
## 2 Problem Formulation  
In this section, we formulate the ANC problem in the wave domain to cancel the noise over a 3-D spatial region inside rooms. As shown in Figure 1, let the quiet zone of interest (blue area) be a spherical region (S) with a radius \(R_{1}\). Assume that the noise sources (black stars) and the secondary sources (black loudspeakers) are located outside the region of interest. In the ANC process, we measure the noise field by placing a spherical microphone array (dark blue stars) on the boundary of the control region (i.e., the desired quiet zone). Any arbitrary observation point within the control region is denoted as \(\mathbf{x}\equiv\{r,\phi_{x},\psi_{x}\}\). Here, \(\psi\) and \(\phi\) are the elevation angle and the azimuthal angle, respectively. In the ANC system, the residual signal at this point \(e(\mathbf{x},k)\) is given by  
\[e(\mathbf{x},k)=\
u(\mathbf{x},k)+s(\mathbf{x},k), \tag{1}\]  
where \(k=2\pi f/c\) is the wave number, \(f\) is the frequency, and \(c\) is the speed of sound propagation. \(\
u(\mathbf{x},k)\) and \(s(\mathbf{x},k)\) denote the primary noise field and the secondary noise field observed at point \(\mathbf{x}\), respectively. The secondary noise field generated by a discrete loudspeaker array with \(L\) loudspeakers (A simple model of the loudspeaker is applied here. More models characterizing the acoustic, mechanic and electric system of the loudspeakers are investigated in [25].) can be represented by  
\[s(\mathbf{x},k)=\sum_{l=1}^{L}d_{l}(k)G(\mathbf{x}|\mathbf{y}_{l},k), \tag{2}\]  
where \(d_{l}(k)\) is the driving signal for the \(l^{\text{th}}\) loudspeaker, \(\mathbf{y}_{l}\) denotes the location of the \(l^{\text{th}}\) loudspeaker, and \(G(\mathbf{x}|\mathbf{y}_{l},k)\) denotes the acoustic transfer function (ATF) between the \(l^{\text{th}}\) loudspeaker and the observation point \(\mathbf{x}\). Please note that in the reverberant environment, \(G(\mathbf{x}|\mathbf{y}_{l},k)\) includes the room reflections. Instead of using measurements of the microphone signals directly, the wave-domain approach employs the wave equation solutions as basis functions to express any wave field over the spatial region, and designs the secondary noise field based on the wave-domain decomposition coefficients. We then represent the primary noise field and secondary noise field in the wave domain. The spherical harmonics-based wave equation solution decomposes any homogeneous incident wave field \(\
u(\mathbf{x},k)\) observed at \(\mathbf{x}\) into  
\[\
u(\mathbf{x},k)=\sum_{u=0}^{\infty}\sum_{m=-u}^{u}\beta_{um}(k)j_{u}(kr)Y_{um}( \phi_{\mathbf{x}},\psi_{\mathbf{x}}), \tag{3}\]  
where \(j_{u}(\cdot)\) is the spherical Bessel function of order \(u\) and \(Y_{um}(\cdot)\) denotes the spherical harmonics. Therefore, the decomposition coefficients \(\beta_{um}(k)\) represent the primary noise field in the wave domain. Within the region of interest \(r\leq R_{1}\), a finite number of modes can be used to approximate the noise field [26]. Thus, the primary noise field in (3) can be truncated by  
\[\
u(\mathbf{x},k)\approx\sum_{u=0}^{N}\sum_{m=-u}^{u}\beta_{um}(k)j_{u}(kr)Y_{um}( \phi_{\mathbf{x}},\psi_{\mathbf{x}}), \tag{4}\]  
Figure 1: ANC system in a 3-D room. Black stars represent primary sources, loudspeakers represent secondary sources, the blue sphere represents the control region, and dark blue stars represent error microphones over the control region. where the truncation order of \(N=\lceil ekR_{1}/2\rceil\)[26; 27; 28]. In the vector version, \(\mathbf{\beta}(k)=[\beta_{0,0}(k),\beta_{1,-1}(k),\)\(\ldots,\beta_{N,N}(k)]^{T}\). Using the spherical harmonic expansion, the secondary noise field within the quiet zone can also be represented by  
\[s(\mathbf{x},k)=\sum_{u=0}^{\infty}\sum_{m=-u}^{u}\gamma_{um}(k)j_{u}(kr)Y_{um}( \phi_{\mathbf{x}},\psi_{\mathbf{x}}), \tag{5}\]  
where the coefficients \(\gamma_{um}(k)\) represent the secondary noise field in the wave domain.