# Active Noise Control over Space: A Subspace Method for Performance Analysis
## 3 Wave-Domain Least Squares Method  
One method for deriving the loudspeaker driving signal \(\mathbf{d}(k)\) is to match the secondary noise field coefficients to the primary noise field coefficients, in the region of interest. Therefore,  
\[\mathbf{\eta}(k)\mathbf{d}(k)=-\mathbf{\beta}(k). \tag{11}\]  
Here, we denote the set of all linear combinations of the columns in \(\mathbf{\eta}(k)\) as column space \(\mathbf{C}\). The rank of matrix \(\mathbf{\eta}(k)\) defines the dimension of the column space \(\mathbf{C}\).  
In Equation (11), (i) the number of loudspeakers, (ii) the rank of \(\mathbf{\eta}(k)\) and (iii) the rank of matrix \(\mathbf{\eta}(k)\) augmented by \(\mathbf{\beta}(k)\) specify the number of solutions for the linear system (11).  
Case 1: \(L=(N+1)^{2}\)  
If the number of loudspeakers is same as the number of modes in the region of interest, (11) has one unique solution, which is given by  
\[d(k)=-(\eta(k))^{-1}\beta(k), \tag{12}\]  
where \((\cdot)^{-1}\) denotes the inverse of a matrix.  
Case 2: \(L>(N+1)^{2}\)  
If the number of loudspeakers is greater than the mode requirement, (11) is an under-determined system. There is either no solution or an infinite number of solutions. In practice, however, this case rarely happens, as extra loudspeakers do not result in better ANC results but increase the device cost and computational cost.  
Case 3: \(L<(N+1)^{2}\)  
If the loudspeaker number \(L\) is less than the number of modes \((N+1)^{2}\) in the region of interest, (11) is an over-determined system. There are more equations than unknowns, resulting in either a single unique solution or no solution.  
When the measurements are in a very special case, which requires  
\[\text{rank}(\eta|\beta)=\text{rank}(\eta), \tag{13}\]  
Equation (11) has an exact solution. Here, \(\text{rank}(\eta)\) denotes the rank of \(\eta\), and \(\text{rank}(\eta|\beta)\) denotes the rank of the matrix \(\eta(k)\) augmented by \(\beta(k)\).  
When  
\[\text{rank}(\eta|\beta)\
eq\text{rank}(\eta), \tag{14}\]  
Equation (11) has no exact solution. The solution can be approximated using the least squares method [30]. The least squares method tries to find the best approximation which results in minimum mean square errors [31], by solving the following problem  
\[\text{min}\|\eta(k)d(k)-(-\beta(k))\|^{2}. \tag{15}\]  
The optimal solution of this minimization problem can be written as  
\[\mathbf{d}(k)=-(\eta(k))^{\dagger}\beta(k), \tag{16}\]  
where \((\cdot)^{\dagger}\) denotes the pseudoinverse of a matrix.  
For some circumstances, \(\beta(k)\) could be totally inside the column space \(C\). While most of time, \(\beta(k)\) have components outside the space \(C\). In general, the result of the driving signal in (16) is achieved by solving the equation as follows:  
\[\mathbf{\eta}(k)\mathbf{d}(k)=-\text{Proj}_{C}\beta(k), \tag{17}\]  
where \(\text{Proj}_{C}\beta(k)\) denotes the projected part of the primary noise field coefficients \(\beta(k)\) in the column space \(C\). The projection matrix can be also written by  
\[\text{Proj}_{C}\beta(k)=\mathbf{\eta}(k)(\eta^{H}(k)\mathbf{\eta}(k))^{-1}\mathbf{\eta}^{ H}(k)\mathbf{\beta}(k). \tag{18}\]  
In most applications, the number of loudspeakers is less than the requirement. Then the driving signals can be designed by least squares solutions in (16). Therefore, this method is here called the 'wave-domain least squares method' (WDLS).