The filter-x least mean square (FxLMS) algorithm is widely used in active noise control (ANC) systems. However, because the algorithm is a feedback control algorithm based on the minimization of the error signal variance to update the filter coefficients, it has a certain delay, usually has a slow convergence speed, and the system response time is long and easily affected by the learning rate leading to the lack of system stability, which often fails to achieve the desired control effect in practice. In this paper, we propose an active control algorithm with nearest-neighbor trap structure and neural network feedback mechanism to reduce the coefficient update time of the FxLMS algorithm and use the neural network feedback mechanism to realize the parameter update, which is called NNR-BPFxLMS algorithm. In the paper, the schematic diagram of the feedback control is given, and the performance of the algorithm is analyzed. Under various noise conditions, it is shown by simulation and experiment that the NNR-BPFxLMS algorithm has the following three advantages: in terms of performance, it has higher noise reduction under the same number of sampling points, i.e., it has faster convergence speed, and by computer simulation and sound pipe experiment, for simple ideal line spectrum noise, compared with the convergence speed of NNR-BPFxLMS is improved by more than 95% compared with FxLMS algorithm, and the convergence speed of real noise is also improved by more than 70%. In terms of stability, NNR-BPFxLMS is insensitive to step size changes. In terms of tracking performance, its algorithm responds quickly to sudden changes in the noise spectrum and can cope with the complex control requirements of sudden changes in the noise spectrum.

There are two different noise control strategies for noise pollution: passive control and active control. Passive noise control, also known as passive noise control, refers to the use of insulation, muffling, sound insulation and other passive methods to attenuate noise [

Generally speaking, ANC is used to adjust the control weights of the adaptive filter by minimizing the error signal. FxLMS is the most well-known adaptive filtering algorithm in the field of ANC. The algorithm works by first estimating the secondary path from the secondary source to the error microphone, and then using the estimated secondary path to filter the reference signal as the input to the controller, effectively reducing the effect of the secondary channel [

The research of adaptive filtering algorithm and the application of neural network are the hot research topics in the field of active noise control in recent years [

In this paper, considering the tracking performance and noise reduction performance when the noise signal is non-stationary, a new active control algorithm NNR-BPFxLMS based on nearest neighbor filter and neural network feedback mechanism is proposed. Simulations and experiments demonstrate that its stability, noise reduction, convergence speed and tracking performance are better than the traditional active control algorithm.

The structure of the feedback ANC system is shown in

The ANC controller consists of an adaptive notch filter and a residual control loop. In adaptive notch filters the line spectra _{1}, _{k}_{1},_{1} by the nearest neighbors regression. An adaptive gain controller (AGC) is used to tune the notch filters output _{1},_{k}

In this paper, we implement the control network parameter update with the help of BP neural network for a given loss function. The basic BP algorithm consists of two processes: forward propagation of the signal and backward propagation of the error. That is, the error output is calculated in the direction from the input to the output, while the adjustment of the weights and thresholds is performed from the output to the input. In forward propagation, the input signal acts on the output node through the implied layer, and after the nonlinear transformation, the output signal is generated, and if the actual output does not match the desired output, it is transferred to the backward propagation process of the error. The error back-propagation is to back-propagate the output error through the hidden layer to the input layer by layer, and to apportion the error to all units in each layer, using the error signal obtained from each layer as the basis for adjusting the weights of each unit. By adjusting the connection strength of the input nodes to the hidden layer nodes and the connection strength of the hidden layer nodes to the output nodes as well as the threshold value, the error decreases along the gradient direction, and the training is stopped after repeated learning and training to determine the network parameters (weights and thresholds) corresponding to the minimum error.

In this section, the noise signal sequence, which has been divided into several overlapping frames, is processed through FFT to obtain the instantaneous spectrum. The spectrum shown in

Firstly, the forward moving average

where

The segmentation threshold can be expressed by the sum of

As shown in

Most ANC systems based on FIR filters are applications of the Wiener filtering theory proposed in 1949. It is believed that the quadratic cost function of a stationary noise signal has a non-concave error surface, which means the control error can be minimized through an optimal filter.

Considering that the acoustic energy of discrete line spectral noise is mainly concentrated in the limited narrow bands, in this section a filter bank is established on the basis of the mapping relationship between the frequency of a pure tone signal and its optimal filter. The NNR is trained by the filter bank to calculate an approximate optimal filter a priori before controlling.

In this section, a set of optimal filters _{1}, _{2}, ···, _{F} of pure tones are calculated at 1 Hz intervals

The coefficients distribution of NNR filter banks is shown in

It should be noted that the filters bank shown in

The Nearest Neighbors Regression is a lazy learning method that uses interpolation to predict unknown filter coefficients with the ‘nearest’ samples in filters bank, whose training cost is zero.

For any line spectral frequency

Since the filter coefficients varies continuously with frequency, as shown in

where

So that the optimal notch filter

In this section, a nonlinear gain controller based on tanh function

where

Additionally, the derivative of the tanh function can be represented by itself, that is

Therefore, the output of AGC can be formulated as

The NNR-based notch filters have obvious control capability for line components within noise spectrum. In this section, an adaptive FIR filter

For a noise signal consisting of

In this section, an internal model

Instead of the input vector

Similarly, the residual control loop output can be expressed as

Therefore, the filtered system output

which should be equal to the actual secondary speaker output

In this section, the mean square error (MSE) is used as the cost function

The steepest descent algorithm is used to update the system parameters, thus the gradients

Expanded by the chain rule,

where:

It can be known from

Similarly,

Moving the system parameters to the negative gradient direction by step

where:

In this section, an acoustic tube model shown in

As shown in

The feedback FxLMS algorithm was selected as the baseline to compare with the NNR-based notch filters algorithm, and average noise reduction (ANR) was used as the evaluation criterion of the control algorithms.

where

In this section, a single-frequency noise signal was used to validate the NNR-based notch filters, whose spectrum was shown in

The ANC systems had the same sampling rate

As demonstrated in

The convergence rate was accelerated linearly with the increase of step size

It can be concluded that under the same conditions, the performance of the NNR-BPFxLMS algorithm in this paper is not easily affected by the change of learning rate and possesses better stability and convergence speed, as shown in the following

ANR(dB) | 5 | 10 | 15 | 20 | 25 | 30 |
---|---|---|---|---|---|---|

Learning rate | ||||||

3 × 10^{−5} |
100 | 200 | 340 | 420 | 550 | 700 |

3 | 5 | 6 | 8 | 13 | 20 | |

4 × 10^{−5} |
80 | 170 | 280 | 362 | 480 | 600 |

3 | 4 | 8 | 10 | 12 | 25 | |

5 × 10^{−5} |
65 | 164 | 252 | 350 | 445 | 500 |

3 | 5 | 7 | 12 | 15 | 19 | |

6 × 10^{−5} |
50 | 100 | 182 | 218 | 280 | 400 |

3 | 6 | 7 | 10 | 13 | 21 |

Note: At each position in the table, the top represents the number of samples required by the traditional LMS algorithm, and the bottom represents the number of samples required by NNR-BPFxLMS.

To sum up, the NNR-BPFxLMS requires only about 3% of the sampling points of the FxLMS to achieve the same amount of noise reduction with the same step size.

The NNR-BPFxLMS algorithm uses the signal frequency as a feature to generate filter banks by interpolating adjacent training samples with 1 Hz interval. The approximate optimal filter coefficients are calculated beforehand, and the coefficients are further fine-tuned by a feedback control mechanism, and the computational effort for the fine-tuning part is small. Therefore, the main work in the system operation is to perform FFT on the signal, because there is a requirement for frequency resolution, so the number of FFT points will be relatively high, but this part of the system can be implemented asynchronously and will not affect the operation of the system.

Furthermore, another experiment was designed to verify the effectiveness of NNR notch filters where the noise signal spectrum had an abrupt change.

As demonstrated in

As demonstrated in

In this section, real noise data downloaded from Signal Processing Information Base (SPIB) was used to further research the feasibility and effectiveness of the NNR-based algorithm for actual control. The noise signal was acquired from a destroyer engine room, whose spectrum was shown in

As demonstrated in

To achieve a noise reduction of 5 dB, the FxLMS algorithm takes 2.13∼4.20 s, while NNR-BPFxLMS takes only 0.2 s or less to complete; to achieve a noise reduction of 5 dB, the FxLMS algorithm takes 9.37∼11.33 s, while NNR-BPFxLMS takes only about 3 s to complete. As demonstrated in

In contrast, the NNR-based notch filter was hardly affected by the change of step size and converged after about 2.64 s, which was optimized by 71.82% compared with the feedback FxLMS algorithm.

In this paper, for the line spectrum noise feedback control problem, the nearest neighbor regulator method is used for adaptive training of the trap filter, so that the ANC system learns an approximate optimal filter from the pre-trained NNR filter bank, and achieves system parameter changes with the help of the neural network feedback mechanism, thus shortening the convergence time.

It is demonstrated experimentally that the maximum noise reduction of the NNR-based trap filter is almost independent of the variation of system parameters such as step size and is almost the same as that of the conventional algorithm. Experimental and simulation results show that the convergence speed is accelerated by more than 95% when the signal spectrum is ideal, and increases by at least 71.82% for real noise.

The authors would like to thank the members of the School of Civil Engineering of Tianjin University.

This work was supported by the National Key R&D Program of China (Grant No. 2020YFA040070).

Study conception and design: Shuiping Zhang, Jun Tang; data collection: Xi Liang,Lei Yan; analysis and interpretation of results: Shuiping Zhang, Lin Shi; draft manuscript preparation: Shuiping Zhang. All authors reviewed the results and approved the final version of the manuscript.

The raw/processed data and materials required to reproduce the above findings cannot be shared at this time due to legal reasons.

The authors declare that they have no conflicts of interest to report regarding the present study.