# Designing Audio Equalization Filters by Deep Neural Networks
## 7 Conclusions  
In this work, we have shown a binaural and a multipoint audio equalization system based on a deep neural network approach to tune FIR filter coefficients. We proposed the use of the back-propagation algorithm as an optimization method in order to train a neural network to produce FIR coefficients able to satisfy specific criteria provided as loss function.  
Three neural network architectures--MLP, CNN, and AE--are compared with state-of-the-art methods. Results show that deep learning approaches outperform other techniques by several orders of magnitude, yielding extremely flat magnitude frequency responses with a quasi linear phase. Among the networks, the CNN provided best results. Additional experiments highlighted the ability of the CNN to converge to a solution that is slightly superior to the least-squares one even when the system to solve is over-determined, motivating further studies on non-convex optimization methods for audio equalization. Finally, the effect of head movements has been studied using additional microphones. The proposed technique cannot be used in a real-time context, thus other techniques can be envisioned to tune the filters adaptively by tracking the head movements, as suggested in Section 6.3. Another possibility is the extension of the current work to a broader area by using multiple microphones in the vicinity of the head.  
Although the training stage can be heavy in computational cost, the convergence speed is quite fast, allowing a user to set a desired error threshold to stop the iterations as soon as the objective is reached.  
Since the deep neural network approach has shown to be capable in the design of audio filters meeting the expected goals, this research topic may be expanded in the future to different applications and constraints.  
Several topics have been left for future works and need to be addressed, such as a subjective evaluation and the design of IIR filters. Given their lower computational cost, compared to FIR filters, they may be suitable for real-time implementation. The use of psychoacoustically oriented metrics, such as 1/3 octave band smoothed frequency responses, may drive the optimization to a frequency response that better represents the human auditory perception. Finally, a thorough exploration of the hyperparameters, the input features and their size, may lead to smaller neural networks with the same performance, improving the filter design speed.  
Figure 10: The \(\overline{MSE}\) and \(\overline{\sigma}\) of the best among the proposed approaches (CNN) and the best among the comparative methods (FD), from the Alfa Romeo Giulia experiments (**a**), and the Jeep Rengeade experiments (**b**). Refer to Tables 2 and 4 for more details.  
**Author Contributions:** Conceptualization, L.G. and S.S.; Software, G.P. and L.G.; Investigation, G.P. and L.G.; Writing--original draft, L.G. and G.P.; Writing--review and editing, L.C.; supervision, S.S. and L.C. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This project is founded under the Italian Ministry of Economic Development (MiSE) Ministerial Decree 24/05/2017 (Agreements for Innovation).  
**Acknowledgments:** The authors would like to thank ASK industries S.P.A. for financial support and technical assistance.  
**Conflicts of Interest:** The authors declare no conflicts of interest.