# Blockwise Frequency Domain Active Noise Controller Over Distributed Networks

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Collaborative Distributed Algorithm for an N-Nodes ASN Based on an Incremental Strategy

**Algorithm 1**illustrates the summary of the algorithm instructions, which are executed per block iteration at each node.

Algorithm 1 Distributed FPBP x LMS Algorithm for N-nodes ASN |

1: $\mathbf{for}\phantom{\rule{4pt}{0ex}}\mathbf{all}\phantom{\rule{4pt}{0ex}}node\phantom{\rule{4pt}{0ex}}1\le \phantom{\rule{4pt}{0ex}}k\le N\phantom{\rule{4pt}{0ex}}\mathbf{do}$ |

2: ${\mathbf{W}}_{k}[n-1]={\widehat{\mathbf{W}}}_{k}{[n-1]}_{(:,1+F(k-1):Fk)}$ |

3: ${\mathbf{Y}}_{k}\left[n\right]=\sum _{f=1}^{F}{\mathbf{W}}_{k}[n-1]\circ \mathbf{X}[n-f+1]$ |

4: $\mathbf{for}\phantom{\rule{4pt}{0ex}}\mathbf{all}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}1\le \phantom{\rule{4pt}{0ex}}j\le N\phantom{\rule{4pt}{0ex}}\mathbf{do}$ |

5: ${\mathbf{V}}_{jk}\left[n\right]=\sum _{p=1}^{P}{\mathbf{S}}_{jk}^{p}\circ \mathbf{X}[n-p+1]$ |

6: $\mathbf{end}\phantom{\rule{4pt}{0ex}}\mathbf{for}$ |

7: ${\mathbf{V}}_{k}\left[n\right]=[\phantom{\rule{4pt}{0ex}}{\mathbf{V}}_{1k}\left[n\right],{\mathbf{V}}_{2k}\left[n\right],\cdots \phantom{\rule{0.166667em}{0ex}}{\mathbf{V}}_{Nk}\left[n\right]\phantom{\rule{4pt}{0ex}}]$ |

9: ${\mathbf{E}}_{k}\left[n\right]=\mathrm{FFT}\left\{\left[\phantom{\rule{4pt}{0ex}}{\mathbf{0}}_{B}\phantom{\rule{4pt}{0ex}}\phantom{\rule{8.53581pt}{0ex}}{\mathbf{e}}_{B}\left[n\right]\phantom{\rule{4pt}{0ex}}\right]\right\}$ |

10: ${\underline{\mathbf{E}}}_{k}\left[n\right]={\mathbf{E}}_{k}\left[n\right]\xb7{\mathbf{1}}_{[1\times FN]}$ |

11: ${\widehat{\mathbf{W}}}_{k}[n]={\widehat{\mathbf{W}}}_{k-1}[n]\phantom{\rule{4pt}{0ex}}-\mu \phantom{\rule{4pt}{0ex}}\mathrm{FFT}\{[\phantom{\rule{4pt}{0ex}}{[\phantom{\rule{4pt}{0ex}}\mathrm{IFFT}\{{\underline{\mathbf{E}}}_{k}[n]\circ {{\mathbf{V}}_{k}[n]}^{*}\}\phantom{\rule{4pt}{0ex}}]}_{[1:B,:]}\phantom{\rule{4pt}{0ex}}\phantom{\rule{8.53581pt}{0ex}}{\mathbf{0}}_{[B\times FN]}\phantom{\rule{4pt}{0ex}}]\}$ |

12: $\mathbf{end}\phantom{\rule{4pt}{0ex}}\mathbf{for}$ |

13: $\mathbf{for}\phantom{\rule{4pt}{0ex}}\mathbf{all}\phantom{\rule{4pt}{0ex}}node\phantom{\rule{4pt}{0ex}}0\le \phantom{\rule{4pt}{0ex}}k\le N\phantom{\rule{4pt}{0ex}}\mathbf{do}$ |

14: ${\widehat{\mathbf{W}}}_{k}\left[n\right]={\widehat{\mathbf{W}}}_{N}\left[n\right]$ |

15: $\mathbf{end}\phantom{\rule{4pt}{0ex}}\mathbf{for}$ |

## 3. Prototype Description

^{®}provides multichannel real-time audio recording, processing and reproduction at low latency. Audio objects based on Object Oriented Programming (OOP) have been optimized for iterative computations that process large streams of audio data. Moreover, Audio Stream Input/Output (ASIO) drivers [28] have been incorporated to this software providing a low-latency and high fidelity interface between MATLAB

^{®}and audio card. The hardware implementation is composed by a CPU (Intel Core i7 3.07 GHz) and an audio card (MOTU 24 I/O). The communication between both components is performed by ASIO drivers and it is controlled by using the MATLAB

^{®}System objects provided by the DSP System Toolbox of MATLAB

^{®}software. The audio card stores the input data from the sensor of each node in buffers of size B and send them to the CPU through the ASIO drivers. The CPU, with MATLAB

^{®}support, executes the audio processing, saves the output data in buffers and sends them back to the audio card through the ASIO drivers, to be reproduced by the loudspeaker of each node.

^{®}software.

## 4. Experimental Results

- ANC system over a four-node ideal network controlled by the FPBFxLMS distributed algorithm and its centralized version.
- ANC system over a four-node non-ideal network controlled by the FPBFxLMS distributed algorithm, comparing the results with the same algorithm but introducing constant delays in the data exchanges through the network. Now we assume that the nodes interchange information every Np block iterations, being p a constant positive integer and N the number of single-channel nodes. During the remainder Np-1 block iterations, we assume two cases. In case 1, the nodes just wait for the arrival of new network information trying to simulate a network with limited power, which saves as much energy as possible. In case 2, the nodes will use its local information to update their filter coefficients. Moreover, we assume that the diffusion of the global state of the network ${\widehat{\mathbf{W}}}_{N}\left[n\right]$ from the last node N to the rest of the nodes (see Figure 4) is not considered in this scenario.

**b**) for the case 1 and (

**d**) for the case 2. Therefore, for networks with communication constraints, the behavior of the system can be improved if the nodes are allowed to update their filter coefficients while waiting for the network information. It is important to take into account that the final behavior of both cases depends on the degree of acoustic interaction between nodes, and, therefore, it should be studied in future works.

## 5. Conclusions

^{®}. Results show that the distributed implementation of the algorithm exhibits the same performance as its centralized version when there are no communication constraints in the network.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Scheme of (

**a**) centralized Active Noise Control (ANC) system with a multichannel controller and (

**b**) distributed ANC system with single-channel controllers.

**Figure 3.**A centralized ASN (

**a**) and distributed ASN (

**b**). CU is the central unit in the centralized case.

**Figure 5.**Scheme of the ANC prototype. The incremental communication strategy is represented by dashed lines.

**Figure 6.**Timing diagram of the processes carried out by each node of the network at each block iteration.

**Figure 7.**Photograph of the ANC prototype in the listening room at the Audio Processing Laboratory of the Polytechnic University of Valencia.

**Figure 8.**Noise reduction obtained by both the four-node distributed system and the centralized system with a 1:4:4 configuration.

**Figure 9.**Noise reduction obtained for the distributed FPBPxLMS algorithm using a four-node ASN for different latency values at the node with the best performance (

**a**) (case 1) (

**c**) (case 2) and with the worst performance (

**b**) (case 1) (

**d**) (case 2).

B | Block size |

L | Length of the adaptive filters |

M | Length of the Finite Impulse Response (FIR) filters that model the estimated secondary paths |

F | $L/B$, number of partitions of the adaptive filters |

P | $M/B$, Number of partitions of the estimated secondary paths |

n | index that denotes block iteration |

$f,p$ | super-indexes that denote partition number. |

${\mathbf{s}}_{jk}$ | M-length estimation of the acoustic path that links the actuator at the jth node with the sensor at the kth node. |

${\mathbf{S}}_{jk}^{p}$ | Fast Fourier Transform (FFT) of size $2B$ of the pth partition of the acoustic path ${s}_{jk}$. |

**Table 2.**Total number of multiplications (MUX), additions (ADD) and Fast Fourier Transforms (FFTs) per blockwise iteration of the FPBFxLMS algorithm in both: (1) centralized and (2) distributed ANC systems. L: length of the adaptive filters; N: number of nodes.

Operations | Generic | N = 1 | N = 4 | N = 8 | |
---|---|---|---|---|---|

MUX | 4 $LN$ + 4 $L{N}^{2}$ | 8 L | 80 L | 288 L | |

(1) | ADD | $LN$ + 3 $L{N}^{2}$ | 4 L | 52 L | 200 L |

FFTs | 2 + 6 N | 8 | 26 | 50 | |

MUX | 2 L + 6 $LN$ | 8 L | 26 L | 50 L | |

(2) | ADD | L + 3 $LN$ | 4 L | 13 L | 25 L |

FFTs | 4 + 4 N | 8 | 20 | 36 |

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**MDPI and ACS Style**

Antoñanzas, C.; Ferrer, M.; De Diego, M.; Gonzalez, A.
Blockwise Frequency Domain Active Noise Controller Over Distributed Networks. *Appl. Sci.* **2016**, *6*, 124.
https://doi.org/10.3390/app6050124

**AMA Style**

Antoñanzas C, Ferrer M, De Diego M, Gonzalez A.
Blockwise Frequency Domain Active Noise Controller Over Distributed Networks. *Applied Sciences*. 2016; 6(5):124.
https://doi.org/10.3390/app6050124

**Chicago/Turabian Style**

Antoñanzas, Christian, Miguel Ferrer, Maria De Diego, and Alberto Gonzalez.
2016. "Blockwise Frequency Domain Active Noise Controller Over Distributed Networks" *Applied Sciences* 6, no. 5: 124.
https://doi.org/10.3390/app6050124