# Active Noise Control over Space: A Subspace Method for Performance Analysis
## 4 Subspace Method
### Principal Component Analysis of the Secondary Path  
Let \(\mathbf{\eta}^{(l)}(k)=[\eta^{(l)}_{00}(k),\eta^{(l)}_{-11}(k),\ldots,\eta^{(l)}_{ NN}(k)]^{T}\) the wave-domain secondary-path coefficients for the \(l_{\text{th}}\) loudspeaker. Matrix \(\mathbf{\eta}(k)\) in (10) for the entire loudspeaker array represents the secondary path, where  
\[\mathbf{\eta}(k)=[\mathbf{\eta}^{(1)}(k),\ldots,\mathbf{\eta}^{(L)}(k)]. \tag{19}\]  
In an arbitrary loudspeaker array setup, each column of matrix \(\mathbf{\eta}(k)\) is not necessarily orthogonal. We perform the principal component analysis (PCA) of the correlation matrix \(E\{\mathbf{\eta}^{H}(k)\mathbf{\eta}(k)\}\) to obtain an orthonormal eigen-basis for the space of the secondary path in the wave domain.  
We take the correlation matrix \(E\{\mathbf{\eta}^{H}(k)\mathbf{\eta}(k)\}\), and then decompose this matrix into a set of orthonormal eigenvectors and their corresponding eigenvalues, as follows:  
\[E\{\mathbf{\eta}^{H}\mathbf{\eta}\}=\mathbf{u}\mathbf{\lambda}\mathbf{v}, \tag{20}\]  
where \(\mathbf{u}=[\mathbf{u}_{1},\ldots,\mathbf{u}_{i},\ldots,\mathbf{u}_{L}]\) are the eigenvectors of the wave-domain ATF, \(\mathbf{v}=\mathbf{u}^{T}\), and the \(i^{\text{th}}\) column corresponds to the eigenvalue \(\lambda_{i}\). Here onwards, the frequency dependent \(k\) is omitted for notational simplicity. The eigenvalues in the matrix form are  
\[\mathbf{\lambda}=\begin{bmatrix}\lambda_{1}&0&\cdots&0\\ 0&\lambda_{2}&\cdots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\cdots&\lambda_{L}\end{bmatrix}. \tag{21}\]  
Here, the vectors \(\mathbf{u}_{1},\ldots,\mathbf{u}_{i},\ldots,\mathbf{u}_{L}\) are written in order of descending eigenvalues \(\mathbf{\lambda}\)[32]. On this basis \(\mathbf{u}\), the first few largest eigenvalues correspond to the principal components of the secondary path in wave domain (\(\mathbf{\eta}\)), which contain the most useful information.  
Depending on the acoustic environment and the loudspeaker placement, the first \(B\) components are used to represent the loudspeakers, then the corresponding eigenvectors are \(\mathbf{u}^{\diamond}=[\mathbf{u}_{1},\ldots,\mathbf{u}_{B}]\). The subspace \(O\) spanned by the wave-domain secondary-path coefficients \(\mathbf{\eta}\) is defined as  
\[\mathbf{O}=\mathbf{\eta}\mathbf{u}^{\diamond}, \tag{22}\]  
where the dimension of \(\mathbf{u}^{\diamond}\) is \(L\times B\), and \(B\leq L\).  
By normalizing each column of matrix \(\mathbf{O}\), the orthonormal vectors \(\mathbf{o}_{1},\ldots,\mathbf{o}_{B}\) are obtained. These vectors generate a subspace, which represents the loudspeaker array and the acoustic environment. The dimensions of basis \(\mathbf{O}\) are \((N+1)^{2}\times B\).  
For the \(l^{\text{th}}\) loudspeaker, the average ATF coefficients over certain short frames can be represented in this space as  
\[\bar{\mathbf{\eta}}^{(l)}=\sum_{b=1}^{B}\kappa^{(l)}_{b}\mathbf{o}_{b}, \tag{23}\]  
where \(\kappa^{(l)}_{b}\) are the projection coefficients. In vector form, (23) can be written by  
\[\bar{\mathbf{\eta}}^{(l)}=\mathbf{O}\mathbf{\kappa}^{(l)}, \tag{24}\]where \(\kappa^{(l)}=\{\kappa^{(l)}_{1},\ldots,\kappa^{(l)}_{b},\ldots,\kappa^{(l)}_{B}\}^{T}\) and  
\[\kappa^{(l)}_{b}=<\mathbf{\eta}^{(l)},\mathbf{o}_{b}>. \tag{25}\]  
Here, \(<\cdot,\cdot>\) is the inner product of two vectors.  
Therefore, \(\kappa=\{\kappa^{(1)},\ldots,\kappa^{(L)}\}\) is the secondary-path coefficients in the subspace \(\mathbf{O}\), with the dimension of \(B\times L\).