# Designing Audio Equalization Filters by Deep Neural Networks
## 6 Results  
### Alfa Romeo Giulia  
Binaural equalization results are shown in Table 2 for the Alfa Romeo Giulia. The two proposed methods based on deep neural networks outperform significantly any other method in the test, while the MLP does not reach the same performance as the FD and the SD. The CNN achieves slightly better results compared to the convolutional AE despite being simpler in terms of implementation and computational cost. Best overall results have been achieved using the CNN with FIR filters of order 1024. Their magnitude frequency response is shown in Figure 6. Shorter filters designed by the convolutional methods are subject to a slight performance degradation, however, their MSE remains very low.  
In Figure 7, we compare the non-equalized (green) and equalized (blue) magnitude frequency response at the dummy head left and right microphone obtained from the filters designed with the CNN and the baseline approaches. The CNN filters correct the frequency responses obtaining an exceptionally flat magnitude. No relevant peaks or notches are present in the equalized frequency response. The FD method achieves a rather flat spectrum, but peaks and notches are still visible. The SD presents the higher \(\overline{MSE}\), while its \(\overline{\sigma}\) is lower than the FD. Indeed, the frequency responses have less peaks, but the magnitude response is biased and sits below 0 dB. The same happens for other FIR filter orders.  
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.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Filter Order**} & \multicolumn{3}{c}{**MLP**} & \multicolumn{3}{c}{**AE**} & \multicolumn{3}{c}{**CNN**} & \multicolumn{3}{c}{**FD (\(\beta\) = 0.1)**} & \multicolumn{3}{c}{**SD**} \\ \cline{2-13}  & **Conf.  
The performance of the FD method is known to be dependent on the \(\beta\) parameter, which can be adjusted as a fixed constant or a frequency-dependent parameter, usually having dominance in the denominator for very low and high frequencies, to avoid excessive gain in the inverse filter in those ranges or to avoid equalization at all. We have tested different configurations of \(\beta\) to search for the best performance of the FD method for a given filter order. Table 3 reports the MSE and sigma for several values of \(\beta\) and for two frequency-dependent \(\beta\) with filter order 1024. Although, theoretically, with lower \(\beta\) the inversion should get closer to ideal, thus reaching a lower MSE, the filter order constraints the performance by truncating the very long ideal impulse response. A sweet spot is obtained for \(\beta\) in the range \(10^{-2}<\beta<10^{-1}\). With larger \(\beta\) the performance decreases, as expected. Some frequency-dependent configurations for \(\beta\) have been selected that obtain good results. The V-shaped one is able to reduce the MSE by a tiny amount, however, no significant improvement can be found by using a frequency-dependent \(\beta\). Overall, the MSE values do not change much from those of Table 2, thus confirming that the choice of \(\beta\) in the experiments above is not adversely affecting the performance.  
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.  
As seen above, even though the elimination of the regularization term \(\beta\) should lead to an almost perfect inversion, the ideal inverse response is limited by the filter order, thus increasing the MSE for very low \(\beta\). On the contrary, the proposed approach seems to achieve a very low error even with short filters.