# Designing Audio Equalization Filters by Deep Neural Networks

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## Abstract

**:**

## 1. Introduction

## 2. Problem Statement

## 3. Proposed Method

#### 3.1. Multilayer Perceptron

#### 3.2. Convolutional Neural Networks

#### 3.3. Autoencoder

## 4. Baseline Methods

#### 4.1. Frequency Deconvolution Method

#### 4.2. Steepest Descent Method

## 5. Experiments

## 6. Results

#### 6.1. Alfa Romeo Giulia

#### 6.2. Jeep Renegade

#### 6.3. Sensitivity to Head Movements

#### 6.4. Sensitivity to the Input

#### 6.5. Over-Determined Case

#### 6.6. Remarks

#### 6.7. Results Summary

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Multi-point equalization problem: $\mathcal{S}$ loudspeakers are displaced in an environment together with $\mathcal{M}$ microhones. The equalizing filters ${g}_{s}$ are designed to invert the environment impulse responses ${h}_{m,s}$.

**Figure 2.**Scheme of the proposed method using an Multilayer Perceptron (MLP). The impulse responses are all concatenated into a vector and fed to the first layer, which must have size $\mathcal{S}\times \mathcal{M}\times L$.

**Figure 5.**Top view of the Alfa Romeo Giulia (

**a**) and the Jeep Renegade (

**b**) showing the placement of the $\mathcal{S}$ loudspeakers and the $\mathcal{M}$ microphones. D indicates the dummy head. The three yellow labels around M2 are the proximity test microphone PM1, PM2, PM3.

**Figure 6.**Magnitude frequency response of the 1024-th order FIR filters designed by the CNN for each one of the Alfa Romeo Giulia loudspeakers S1-S7 shown in Figure 5a.

**Figure 7.**Magnitude frequency responses at the left and right microphones of the dummy head in the Alfa Romeo Giulia after applying filters obtained from the CNN (

**a**,

**b**), Frequency Deconvolution (

**c**,

**d**) and Steepest Descent (

**e**,

**f**) methods. The original magnitude frequency response is shown in green while the equalized frequency response is shown in blue. The target magnitude response is shown in black.

**Figure 8.**Frequency response at microphone M2 (

**a**); microphones PM1 and PM2 (

**b**,

**c**), corresponding to small forward and backward head movements; microphones PM3 (

**d**), corresponding to a large lateral head movement.

**Figure 9.**Phase response of one of the filters achieved with the CNN method (FIR order 1024) and a linear fitting. Frequency is normalized according to Nyquist.

**Table 1.**The CNN and MLP configurations used in the experiments. The number of parameters are referred to filters of 1024-th order.

CNN | MLP | |||||
---|---|---|---|---|---|---|

Configuration | Number of Kernels | Number of Units | Trainable Parameters | Configuration | Number of Units | Trainable Parameters |

Conv #1 | [48, 24] | [10] | 7,481,943 | MLP #1 | [10] | 6,798,935 |

Conv #2 | [10, 5] | [100, 10] | 3,826,153 | MLP #2 | [100, 10] | 67,280,035 |

Conv #3 | [100, 25] | [100, 100] | 12,483,433 | MLP #3 | [100, 100] | 67,934,875 |

Conv #4 | [10] | [1000] | 3,825,863 | MLP #4 | [1000] | 679,183,175 |

MLP #5 | [100] | 67,924,775 | ||||

MLP #6 | [100, 100, 100] | 67,944,975 | ||||

MLP #7 | [5] | 36,003,713 | ||||

MLP #8 | [10, 1000, 1000] | 14,914,185 |

**Table 2.**Audio equalization results for the Alfa Romeo Giulia with binaural microphones. Please note that the $\overline{MSE}$ in the absence of equalization is 2.19, with $\overline{\sigma}$ 3.52. Best results for each column are highlighted in bold.

Filter Order | MLP | AE | CNN | FD ($\mathit{\beta}=0.1$) | SD | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Conf. | $\overline{MSE}$ | $\overline{\mathit{\sigma}}$ | Conf. | $\overline{MSE}$ | $\overline{\mathit{\sigma}}$ | Conf. | $\overline{MSE}$ | $\overline{\mathit{\sigma}}$ | $\overline{MSE}$ | $\overline{\mathit{\sigma}}$ | $\overline{MSE}$ | $\overline{\mathit{\sigma}}$ | |

512 | MLP #5 | 0.32 | 2.877 | Conv #1 | 9.72$\xb7{10}^{-4}$ | 0.136 | Conv #2 | 7.90$\xb7{10}^{-4}$ | 0.122 | 0.18 | 2.52 | 0.40 | 1.95 |

640 | MLP #8 | 0.36 | 2.730 | Conv #1 | 3.80$\xb7{10}^{-4}$ | 0.085 | Conv #2 | 3.74$\xb7{10}^{-4}$ | 0.084 | 0.15 | 2.34 | 0.35 | 1.72 |

768 | MLP #5 | 0.46 | 2.796 | Conv #1 | 1.66$\xb7{10}^{-4}$ | 0.056 | Conv #2 | 1.79$\xb7{10}^{-4}$ | 0.058 | 0.14 | 2.23 | 0.33 | 1.60 |

896 | MLP #2 | 0.45 | 2.799 | Conv #1 | 1.07$\xb7{10}^{-4}$ | 0.045 | Conv #1 | 1.02$\xb7{10}^{-4}$ | 0.044 | 0.12 | 2.07 | 0.31 | 1.54 |

1024 | MLP #7 | 0.32 | 2.746 | Conv #1 | 6.85$\xb7{\mathbf{10}}^{-\mathbf{5}}$ | 0.036 | Conv #1 | 6.31$\xb7{\mathbf{10}}^{-\mathbf{5}}$ | 0.034 | 0.10 | 1.93 | 0.30 | 1.50 |

**Table 3.**Effect of the parameter $\beta $ on the performance. The V-shaped configuration refers to a frequency-dependent $\beta $ with a minimum of ${10}^{-4}$ at 1 kHz and maxima of ${10}^{-1}$ at DC and Nyquist, varying linearly on a dB scale. The U-shaped configuration takes a value of ${10}^{-4}$ in the range 100 Hz–10 kHz and one elsewhere. Best results for each column are highlighted in bold.

$\mathit{\beta}$ | $\overline{MSE}$ | $\overline{\mathit{\sigma}}$ |
---|---|---|

${10}^{-4}$ | 0.123 | 1.83 |

${10}^{-3}$ | 0.118 | 1.82 |

${10}^{-2}$ | 0.108 | 1.81 |

${10}^{-1}$ | 0.108 | 1.93 |

1 | 0.281 | 2.71 |

10 | 0.686 | 4.2 |

100 | 0.937 | 5.09 |

V-shaped | 0.101 | 1.829 |

U-shaped | 0.124 | 1.86 |

**Table 4.**Audio equalization results for the Jeep Renegade with binaural microphones and four microphones (one per seat). The FIR order is 1024.

Setup | CNN | FD $\mathit{\beta}=0.1$ | |||
---|---|---|---|---|---|

Conf | $\overline{MSE}$ | $\overline{\mathit{\sigma}}$ | $\overline{MSE}$ | $\overline{\mathit{\sigma}}$ | |

Binaural | #1 | $6.19\xb7{10}^{-5}$ | 0.035 | 0.05 | 1.21 |

4 seats | #1 | $5.7\xb7{10}^{-4}$ | 0.106 | 0.15 | 1.95 |

**Table 5.**Audio equalization results for microphone M2 and microphones PM1, PM2 and PM3. The evaluation is achieved by the experiments performed using the Jeep Renegade with four microphones (see Table 4).

Mic. | CNN | FD | ||
---|---|---|---|---|

$\overline{MSE}$ | $\overline{\mathbf{\sigma}}$ | $\overline{MSE}$ | $\overline{\mathbf{\sigma}}$ | |

M2 | $5.07\xb7{10}^{-4}$ | 0.10 | 0.14 | 1.82 |

PM1 | 0.61 | 2.88 | 1.2 | 2.9 |

PM2 | 0.50 | 3.31 | 0.57 | 3.07 |

PM3 | 0.80 | 3.09 | 0.84 | 3.12 |

**Table 6.**Effect of the input type on the results of the CNN (filter order 1024). For each case, the best result and the related configuration is reported.

Input | $\overline{MSE}$ | $\overline{\mathit{\sigma}}$ | Conf. |
---|---|---|---|

Impulse Responses | $6.31\xb7{10}^{-5}$ | 0.034 | Conv #1 |

Random Iteration | $0.14$ | 2.152 | Conv #1 |

Random Fixed | $1.35\xb7{10}^{-4}$ | 0.052 | Conv #1 |

All 1s | $1.17\xb7{10}^{-4}$ | 0.049 | Conv #1 |

All 0s | ill-conditioned |

**Table 7.**Audio equalization in the single-channel and over-determined cases. Setup is $\mathcal{M}\times \mathcal{S}$.

Car | Setup | CNN | FD | ||
---|---|---|---|---|---|

$\overline{MSE}$ | $\overline{\mathbf{\sigma}}$ | $\overline{MSE}$ | $\overline{\mathbf{\sigma}}$ | ||

Giulia | $1\times 1$ | 0.52 | 8.57 | 0.62 | 9.84 |

$2\times 1$ | 0.57 | 7.81 | 0.64 | 9.19 | |

Renegade | $1\times 1$ | 0.03 | 1.34 | 0.12 | 2.01 |

$4\times 1$ | 0.22 | 2.76 | 0.44 | 3.62 |

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## Share and Cite

**MDPI and ACS Style**

Pepe, G.; Gabrielli, L.; Squartini, S.; Cattani, L. Designing Audio Equalization Filters by Deep Neural Networks. *Appl. Sci.* **2020**, *10*, 2483.
https://doi.org/10.3390/app10072483

**AMA Style**

Pepe G, Gabrielli L, Squartini S, Cattani L. Designing Audio Equalization Filters by Deep Neural Networks. *Applied Sciences*. 2020; 10(7):2483.
https://doi.org/10.3390/app10072483

**Chicago/Turabian Style**

Pepe, Giovanni, Leonardo Gabrielli, Stefano Squartini, and Luca Cattani. 2020. "Designing Audio Equalization Filters by Deep Neural Networks" *Applied Sciences* 10, no. 7: 2483.
https://doi.org/10.3390/app10072483