# Active Noise Control over Space: A Subspace Method for Performance Analysis

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## Abstract

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## 1. Introduction

## 2. Problem Formulation

## 3. Wave-Domain Least Squares Method

- Case 1: $L={(N+1)}^{2}$

- Case 2: $L>{(N+1)}^{2}$

- Case 3: $L<{(N+1)}^{2}$

## 4. Subspace Method

#### 4.1. Principal Component Analysis of the Secondary Path

#### 4.2. Projection of the Primary Noise Field into the Subspace

#### 4.3. Noise Control in the Subspace

- Case 1: $B=L$When we reserve all the information in the PCA, (31) has only one unique solution. In that case, the driving signals can be represented by$$\mathit{d}=-{\left(\mathbf{\kappa}\right)}^{-1}\mathit{y}.$$
- Case 2: $B<L$When we only use the largest components to generate the subspaces, instead of solving the over-determined system in (11), Equation (31) solves an under-determined system. In that case, the driving signals $\mathit{d}$ can be derived by$$\mathit{d}=-{\left(\mathbf{\kappa}\right)}^{\u2020}\mathit{y},$$

## 5. Simulation Results

#### 5.1. Simulation Setup

#### 5.2. Cancellation Performance Using Different Methods

#### 5.3. Comparison of the Effect of Different Noise Source Positions

#### 5.4. Comparison of the Effect of Different Loudspeaker Placements

#### 5.5. Summary and Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

3-D | Three-dimensional |

ANC | Active noise control |

ATF | Acoustic transfer function |

PCA | Principal component analysis |

WFS | Wave field synthesis |

WD | Wave domain |

WDLS | Wave-domain least squares method |

SNR | Signal to noise ratio |

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**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.

**Figure 2.**ANC system setup, where the pink point is the noise source position, blue points are loudspeaker positions, and red points are microphone positions: (

**a**) case 1; (

**b**) case 2.

**Figure 3.**Energy of the primary noise field, where pink point is the projection of the primary source on the x-y plane, blue points are the loudspeaker points located on the x-y plane, and the red dashed circle is the boundary of the region of interest: (

**a**) case 1; (

**b**) case 2.

**Figure 4.**Energy of the residual noise field, when the noise field is generated by one primary source using different methods. Here, pink point is the projection of the primary source on the x-y plane and blue points are the loudspeaker points located on the x-y plane: (

**a**) the WDLS method in case 1; (

**b**) the WDLS method in case 2; (

**c**) the subspace method in case 1; (

**d**) the subspace method in case 2.

**Figure 5.**Two different array setups, when the noise source moves around a sphere, where in both setups, pink points are the primary source positions, blue points are loudspeaker positions, and red points are microphone positions: (

**a**) Case 3; (

**b**) Case 4.

**Figure 6.**Noise reduction performance in case 3, when the noise field is generated by one primary source moving around the sphere using different methods: (

**a**) with SNR = 60 dB white noise on the microphone recordings; (

**b**) with SNR = 30 dB white noise on the microphone recordings.

**Figure 7.**Energy of the driving signals in case 3, when the noise field is generated by one primary source moving around the sphere using different methods: (

**a**) with SNR = 60 dB white noise on the microphone recordings; (

**b**) with SNR = 30 dB white noise on the microphone recordings.

**Figure 8.**Noise reduction over the region using different loudspeaker setups, when the noise field generated by one primary source moving around the sphere using different methods: (

**a**) case 3; (

**b**) case 4.

**Figure 9.**Energy of the driving signals generated by one primary source moving around the sphere using different methods: (

**a**) case 3; (

**b**) case 4.

**Table 1.**Loudspeaker array setup and noise source location in 4 cases, which are given in Figures 2 and 5, respectively.

Loudspeaker Array | Non-Symmetry | Symmetry | |
---|---|---|---|

Noise Source Position | |||

$(2,{315}^{\circ},{45}^{\circ})$ | case 1 | ||

$(2,{315}^{\circ},{90}^{\circ})$ | case 2 | ||

24 position candidates | case 3 | case 4 |

Loudspeakers in the x-y Plane | Loudspeakers Outside the x-y Plane | |||
---|---|---|---|---|

No. | Non-Symmetry | Symmetry | No. | Non-Symmetry or Symmetry |

1 | (4, 3, 2.5) | (4.5, 3, 2.5) | 9 | (0.5, 0.5, 4.5) |

2 | (1.8, 3, 2.5) | (1.5, 3, 2.5) | 10 | (5.5, 5.5, 4.5) |

3 | (3, 2, 2.5) | (3, 1.5, 2.5) | 11 | (5.5, 0.5, 4.5) |

4 | (3, 4.2, 2.5) | (3, 4.5, 2.5) | 12 | (0.5, 5.5, 4.5) |

5 | (4.3, 3.2, 2.5) | (4.2, 1.8, 2.5) | ||

6 | (1.7, 2.8, 2.5) | (1.8, 1.8, 2.5) | ||

7 | (3.2, 1.7, 2.5) | (4.2, 4.2, 2.5) | ||

8 | (2.8, 4.2, 2.5) | (1.8, 4.2, 2.5) |

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**MDPI and ACS Style**

Zhang, J.; Abhayapala, T.D.; Zhang, W.; Samarasinghe, P.N.
Active Noise Control over Space: A Subspace Method for Performance Analysis. *Appl. Sci.* **2019**, *9*, 1250.
https://doi.org/10.3390/app9061250

**AMA Style**

Zhang J, Abhayapala TD, Zhang W, Samarasinghe PN.
Active Noise Control over Space: A Subspace Method for Performance Analysis. *Applied Sciences*. 2019; 9(6):1250.
https://doi.org/10.3390/app9061250

**Chicago/Turabian Style**

Zhang, Jihui, Thushara D. Abhayapala, Wen Zhang, and Prasanga N. Samarasinghe.
2019. "Active Noise Control over Space: A Subspace Method for Performance Analysis" *Applied Sciences* 9, no. 6: 1250.
https://doi.org/10.3390/app9061250