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Active Noise Control Over Space: a Subspace Method for Performance Analysis applied sciences Article Active Noise Control over Space: A Subspace Method for Performance Analysis Jihui Zhang 1,* , Thushara D. Abhayapala 1 , Wen Zhang 1,2 and Prasanga N. Samarasinghe 1 1 Audio & Acoustic Signal Processing Group, College of Engineering and Computer Science, Australian National University, Canberra 2601, Australia; [email protected] (T.D.A.); [email protected] (W.Z.); [email protected] (P.N.S.) 2 Center of Intelligent Acoustics and Immersive Communications, School of Marine Science and Technology, Northwestern Polytechnical University, Xi0an 710072, China * Correspondence: [email protected] Received: 28 February 2019; Accepted: 20 March 2019; Published: 25 March 2019 Abstract: In this paper, we investigate the maximum active noise control performance over a three-dimensional (3-D) spatial space, for a given set of secondary sources in a particular environment. We first formulate the spatial active noise control (ANC) problem in a 3-D room. Then we discuss a wave-domain least squares method by matching the secondary noise field to the primary noise field in the wave domain. Furthermore, we extract the subspace from wave-domain coefficients of the secondary paths and propose a subspace method by matching the secondary noise field to the projection of primary noise field in the subspace. Simulation results demonstrate the effectiveness of the proposed algorithms by comparison between the wave-domain least squares method and the subspace method, more specifically the energy of the loudspeaker driving signals, noise reduction inside the region, and residual noise field outside the region. We also investigate the ANC performance under different loudspeaker configurations and noise source positions. Keywords: active noise control (ANC); performance analysis; wave domain; spatial noise; reverberant room 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 [2–5], 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 [11–13] and (cylindrical/spherical) harmonic-based wave-domain algorithms [14–19], 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 the Appl. Sci. 2019, 9, 1250; doi:10.3390/app9061250 www.mdpi.com/journal/applsci Appl. Sci. 2019, 9, 1250 2 of 20 loudspeakers 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, Laugesen 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, Chen et al. investigated ANC performance by noise pattern analysis of the primary noise field [22,23]. Buerger 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 Section2, we formulate the ANC problem in a 3-D room. We investigate the maximum ANC performance using the wave-domain least squares method in Section3, and investigate the maximum ANC performance using the subspace method in Section4. The simulation validation is demonstrated in Section5. We draw some conclusions in Section6. 2. Problem Formulation In this section, we formulate the ANC problem in the wave domain to cancel the noise over a 3-D spatial region inside rooms. As shown in Figure1, let the quiet zone of interest (blue area) be a spherical region (S) with a radius R1. Assume that the noise sources (black stars) and the secondary sources (black loudspeakers) are located outside the region of interest. In the ANC process, we measure the noise field by placing a spherical microphone array (dark blue stars) on the boundary of the control region (i.e., the desired quiet zone). Any arbitrary observation point within the control region is denoted as x ≡ fr, fx, yxg. Here, y and f are the elevation angle and the azimuthal angle, respectively. In the ANC system, the residual signal at this point e(x, k) is given by e(x, k) = n(x, k) + s(x, k), (1) where k = 2p f /c is the wave number, f is the frequency, and c is the speed of sound propagation. n(x, k) and s(x, k) denote the primary noise field and the secondary noise field observed at point x, respectively. Appl. Sci. 2019, 9, 1250 3 of 20 The secondary noise field generated by a discrete loudspeaker array with L loudspeakers (A simple model of the loudspeaker is applied here. More models characterizing the acoustic, mechanic and electric system of the loudspeakers are investigated in [25].) can be represented by L s(x, k) = ∑ dl(k)G(xjyl, k), (2) l=1 th th where dl(k) is the driving signal for the l loudspeaker, yl denotes the location of the l loudspeaker, th and G(xjyl, k) denotes the acoustic transfer function (ATF) between the l loudspeaker and the observation point x. Please note that in the reverberant environment, G(xjyl, k) includes the room reflections. Figure 1. ANC system in a 3-D room. Black stars represent primary sources, loudspeakers represent secondary sources, the blue sphere represents the control region, and dark blue stars represent error microphones over the control region. Instead of using measurements of the microphone signals directly, the wave-domain approach employs the wave equation solutions as basis functions to express any wave field over the spatial region, and designs the secondary noise field based on the wave-domain decomposition coefficients. We then represent the primary noise field and secondary noise field in the wave domain. The spherical harmonics-based wave equation solution decomposes any homogeneous incident wave field n(x, k) observed at x into ¥ u n(x, k) = ∑ ∑ bum(k)ju(kr)Yum(fx, yx), (3) u=0 m=−u where ju(·) is the spherical Bessel function of order u and Yum(·) denotes the spherical harmonics. Therefore, the decomposition coefficients bum(k) represent the primary noise field in the wave domain. Within the region of interest r ≤ R1, a finite number of modes can be used to approximate the noise field [26]. Thus, the primary noise field in (3) can be truncated by N u n(x, k) ≈ ∑ ∑ bum(k)ju(kr)Yum(fx, yx), (4) u=0 m=−u Appl. Sci. 2019, 9, 1250 4 of 20 where the truncation order of N = dekR1/2e [26–28]. In the vector version, b(k) = [b0,0(k), b1,−1(k), T ..., bN,N(k)] . Using the spherical harmonic expansion, the secondary noise field within the quiet zone can also be represented by ¥ u s(x, k) = ∑ ∑ gum(k)ju(kr)Yum(fx, yx), (5) u=0 m=−u where the coefficients gum(k) represent the secondary noise field in the wave domain. Similar to the primary noise field, inside the region of interest with the radius of R1, the secondary noise field can be truncated by N u s(x, k) ≈ ∑ ∑ gum(k)ju(kr)Yum(fx, yx). (6) u=0 m=−u The ATF in (2) can be parameterized in the wave domain [29] as N u (l) G(xjyl, k) ≈ ∑ ∑ hum(k)ju(kr)Yum(fx, yx), (7) u=0 m=−u (l) where hum(k) is the ATF in wave domain for each loudspeaker.
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