# Active Noise Control over Space: A Subspace Method for Performance Analysis
## 1 Introduction  
Active noise control (ANC) over a spatially extended region, which is termed as _'Spatial ANC'_, is a challenging field of research as the aim is to create a large quiet zone for multiple listeners in three-dimensional (3-D) spaces. In spatial ANC applications, such as noise cancellation in aircraft [1] and automobiles [2; 3; 4; 5], multichannel ANC systems equipped with multiple sensors and multiple secondary sources are adopted [6]. In literature, both time-domain [7; 8] and frequency-domain [9; 10] algorithms have been implemented in multichannel ANC systems, which can cancel the noise at error sensor positions and their close surroundings [10]. Recently, ANC over space has been approached via Wave field synthesis (WFS)-based wave-domain algorithms [11; 12; 13] and (cylindrical/spherical) harmonic-based wave-domain algorithms [14; 15; 16; 17; 18; 19], with which the noise over entire region of interest can be cancelled directly. Here onwards, we use the terminology _'wave-domain ANC'_ to refer to harmonics-based wave-domain ANC. In wave-domain ANC, the number of secondary sources/loudspeakers is required to be no less than the number of modes in the spatial region, so that all the modes can be controlled. When the number of loudspeakers cannot control all the modes in the spatial region, using the wave-domain adaptive algorithms and conventional adaptive algorithms, the noise reduction performance in the steady state degrades significantly [16]. In practical applications, numbers and locations of theloudspeakers have more constraints compared to simulation setups in [16]. For instance, in a vehicle, the numbers and positions of the loudspeakers are highly constrained by the vehicle dimensions and passenger convenience. It is valuable for design engineers to estimate whether the available numbers and positions of the loudspeakers are sufficient to the noise reduction requirements, before they implement an ANC system in a real environment. Since the noise reduction performance varies with different ANC algorithms, it is important to investigate the maximum achievable performance for the given system, which is dependent on the coherence between the reference sensors and the error sensors [20], secondary source characteristics and locations, room environments, and the primary noise field characteristics. In the literature, [PERSON] et al. provided the theoretical prediction for multichannel ANC in a small reverberant room based on the primary signal on the error sensors [21]. For spatial ANC performance estimation over an entire region, [PERSON] et al. investigated ANC performance by noise pattern analysis of the primary noise field [22; 23]. [PERSON] et al. investigated the coherence between two observation points in the noise field evoked by given continuous source distributions, which can be applied to predict the upper bound of ANC performance in the region of interest [24]. However, the capability of secondary sources, in particular room environments, has not yet been explored. In this paper, we investigate the maximum noise control performance over a spatial region by investigating capability of secondary sources, in particular room environments. We define the subspace spanned by the wave-domain secondary-path coefficients, and evaluate the ANC performance in the subspace. Using the proposed subspace method, the design engineers can predict the noise cancellation performance before an actual ANC system is implemented in a product. Simulations are conducted in a 3-D room environment under different noise source positions, when the loudspeakers have constraints on numbers and positions. The proposed subspace method is more feasible than the wave-domain least squares method. The rest of this paper is organized as follows. In Section 2, we formulate the ANC problem in a 3-D room. We investigate the maximum ANC performance using the wave-domain least squares method in Section 3, and investigate the maximum ANC performance using the subspace method in Section 4.