## 1 Introduction  
Active noise control (ANC) mitigates disturbances by generating sound waves that interfere destructively with the primary noise field. The destructively interfering sound field is generated by secondary sources driven to minimize the sum-of-squared pressures at the error microphone locations [1]. Although ANC is widely implemented in headphones, it is only in recent years that ANC has been applied to control noise in larger 3D spaces, such as in infant incubators [2], the interior of automobiles [3; 4], and through apertures [5; 6].  
In light of the increasing awareness of health risks associated with long-term exposure to environmental noise [7; 8; 9], sustainable noise mitigation methods need to be implemented to increase the aural comfort in affected urban accommodation. Because windows are the main entry points for environmental noise in a building, active mitigation methods are most effective when targeted at such openings. Since windows have to provide natural ventilation--especially in tropical climates--passive methods of control (e.g., double-glazed panels, barriers, etc.) are less favorable as they obstruct both sunlight and ventilation. Hence, ANC techniques that can mitigate noise through open windows are potentially sustainable solutions to environmental noise in tropical urban high-rise environments [10].  
ANC systems have been developed and studied for fully-open [11; 12; 13; 14; 15; 16; 17], and partially-open windows [6; 18; 19; 20]. For practical implementations, however, the requirement of error microphones in the system poses placement challenges, computation issues, and privacy concerns. Although global noise attenuation of around 10 dB has been demonstrated experimentally with a fixed set of control filter coefficients in some cases [11; 12], the mechanism behind the formulation of a single filter, which is optimal for different noise scenarios (e.g., different incidence angles), has not been defined.  
Whilst non-adaptive ANC systems that do not employ error microphones have been realized in a multi-dimensional reverberant room, the optimal filters used in such cases must be recalculated when the primary or secondary path transfer functions change (e.g., human movements in the room) [1]. However, if the window problem is formulated such that the total acoustic power transmitted through the window is the cost function to be minimized, the ANC performance will be robust to variations in the secondary path for optimal filters that do not employ error microphones [21]. This formulation also assumes that the window opening is the noise source, which is collocated with the reference microphones and secondary sources, yielding a favorable control scenario. Thus, optimal filters can be designed using the inversion method with free-field secondary path measurements for each window design (i.e., without room reverberations). The problem is how feasible it is to design such a fixed optimal filter that can work well in various scenarios of noise in open windows.  
Hence, the feasibility of the optimal filter design is investigated here through a separation of the multichannel ANC problem. To this end, the global attenuation performance of the theoretical formulations in controlling noise through an open aperture is examined with numerical experiments.