## 2 Separation of the Multichannel Active Noise Control Problem into Two Subproblems
### The Underdetermined Subproblem 2 and Possible Solutions
\right.. \tag{10}\]  
Thus,  
where the convergence condition is \(\mu<\frac{1}{\lambda_{\text{max}}^{\prime}}\), where \(\lambda_{\text{max}}^{\prime}\) is the maximum eigenvalue of \(\widetilde{\mathbf{X}}\mathbf{S}^{\text{H}}\mathbf{S}\widetilde{\mathbf{X}}^{ \text{H}}\). Compared to the convergence condition in the fully-coupled multichannel FxLMS, we find that, generally, the convergence condition in the colocated method will have a lower bound because the maximum eigenvalue \(\lambda_{\text{max}}^{\prime}\) will likely be greater than the maximum eigenvalue \(\lambda_{\text{max}}\). Comparing \(\widetilde{\mathbf{W}}_{\infty}\) with the optimal solution given in (7), we can verify that the colocation-based FxLMS converges to its optimal solution. Following the above derivation, we can derive the secondary source output as  
\[\widetilde{\mathbf{Y}}_{\infty}=\widetilde{\mathbf{X}}^{\text{H}}\widetilde{ \mathbf{W}}_{\infty}=-\Big{(}\mathbf{S}^{\text{H}}\mathbf{S}\Big{)}^{-1} \mathbf{S}^{\text{H}}\mathbf{D}=\mathbf{Y}_{\text{opt}}. \tag{13}\]  
These results suggest that with the colocated method, we could achieve identical secondary source output and hence equivalent noise reduction performance as the fully-coupled multichannel FxLMS method. More importantly, the colocated method simplifies the computational complexity of the ANC system by a factor of \(K\) times. A simulation is conducted to illustrate the findings above. Noise reduction (NR) performance for different tonal noises from 20 Hz to 2000 Hz is computed using  
\[\text{NR}=10\log_{10}\frac{\mathbf{D}^{\text{H}}\mathbf{D}}{\mathbf{E}^{\text {H}}\mathbf{E}}. \tag{14}\]  
Time-domain free-field simulation of a 2-2-4 overdetermined system (for problem 1) with 2 reference microphones, 2 secondary sources, and 4 error microphones is considered using a setup illustrated in Figure 3a, which is similar to that in [14]. The horizontal distance from the primary source to the reference microphones, secondary sources, and error microphones are 0.2 m, 0.1 m, and 0.2 m, respectively. The spacing between the two reference microphones and secondary sources is 0.2 m, whereas the four error microphones are spaced at 0.1 m, 0.2 m, and 0.1 m. The primary noise signal is captured by the reference microphones as \(x_{1}(n),x_{2}(n)\), and by the error microphones as \(d_{1}(n),d_{2}(n),d_{3}(n),d_{4}(n)\). The secondary sources \(y_{1}(n),y_{2}(n)\) are generated by filtering the reference microphone signals using the adaptive filters based on the above two FxLMS algorithms, which then go through the secondary paths and are summed with the primary noise at the error microphones, resulting in the error microphone signals \(e_{1}(n),e_{2}(n),e_{3}(n),e_{4}(n)\). The noise reduction performance (after convergence) of the above two adaptive FxLMS algorithms is shown in Figure 3b. The results reveal that the noise reduction performance of both the fully-coupled based on the adaptation equation (8) and the colocated method based on the adaptation Equation (11) are identical with the theoretical performance, verifying the analyses above, in particular, Equations (3) and (13).