## 5 Conclusions  
In this paper, we revisit the multichannel active noise control problem. By separating this problem into two subproblems, we find that in a general multiple-input-multiple-output setup, the subproblem 2, i.e., obtaining the optimal control filter from reference signals and optimal secondary outputs, is usually underdetermined with more than one solution. The underdetermined system presents an opportunity to simplify the control filter that achieves equivalent performance or even extends the capability of a fixed optimal filter to handle complex scenarios. For example, we may simplify the control filter by only activating the diagonal elements in the control filter matrix, arriving at the so-called colocated control, which yields the exact optimal secondary output and hence achieves the same noise reduction performance as the fully-coupled multichannel system. Similar observations were found in their adaptive realizations. Furthermore, the underdetermined system is leveraged to develop the full-rank and overdetermined realization by introducing constraints to derive a fixed optimal control filter, which could be employed to handle noise from multiple angles of incidence. The performance of the full-rank and overdetermined optimal control filter is investigated numerically using FEM simulations. The attenuation performance of the full-rank optimal filter is optimal for incidences utilized in its realization but degrades as the noise incidence deviates from those selected directions. Notably, a full-rank method that evenly selects all eight directions from either the entirely positive or negative direction between \(-90^{\circ}\) to \(90^{\circ}\) could achieve optimal performance in their entire respective incidence ranges. This implies that a single incidence used in the formulation of the filter could yield optimal attenuation to the neighboring incidence within \(\pm 5^{\circ}\), and the extreme incidences (near \(\pm 90^{\circ}\)) are not required to achieve good global attenuation. Therefore, we can employ two full-rank optimal filters to cover the entire \(180^{\circ}\) range. In contrast, the overdetermined optimal filter achieves a better average performance across the entire range of incidences with a single filter. The realization of the overdetermined optimal filter does not demand a fine resolution of incidences, where an interval of \(10^{\circ}\) across the entire range of incidence angles was shown to be sufficient. Finally, we show that the overdetermined optimal filter outperforms the full-rank optimal filter in a more complex noise environment such as with multiple noise sources. Moving forward, it is necessary to understand the feasibility of applying the proposed overdetermined optimal control method in a practical setup. Furthermore, it is interesting to understand how the overdetermined optimal control method compares to the stochastic optimal control method under the various practical use cases with varying noise incidence. The analysis and simulation studies presented in this paper could facilitate the development of the optimal fixed filter for ANC in more  
Figure 13: Noise reduction performance of the full-rank optimal filter and the overdetermined optimal filter for plane wave noise from two directions, at frequency (**a**) 400 Hz, (**b**) 1000 Hz. dynamic scenarios, not only for multichannel systems, but also for single channel ANC systems such as ANC headphones. **Author Contributions:** Conceptualization, J.H. and B.L. ; methodology, J.H. and B.L. ; software, B.L. ; validation, J.H., B.L. and D.S. ; formal analysis, J.H. ; investigation, J.H. and D.S. ; resources, B.L. ; data curation, B.L. ; writing--original draft preparation, J.H. and B.L. ; writing--review and editing, J.H., B.L. D.S. and W.S.G; visualization, J.H. and B.L. ; supervision, W.S.G. ; project administration, W.S.G. ; funding acquisition, W.S.G.