## 6 Conclusions  
This paper introduced an improved cascade design for auditory graphic equalizers having either a third-octave or a Bark band division. Each band was assigned a single second-order IIR section having a symmetric magnitude response and a controllable Nyquist-limit gain. When a logarithmic band division was used, such as in the third-octave case, the band filters with adjustable gain at the bandwidth boundaries allowed strict control of their response at the neighboring band centers. This appeared to be beneficial for accurate GEQ design. Since the Bark frequency division was non-logarithmic, we suggested here that the gain was controlled at the upper boundary of each band filter's response. The WLS design was applied to solve for the optimal filter gains to account for the interaction between the band filters. The LS optimization was based on an interaction matrix, which was determined with an optimized prototype gain for each band filter. The interaction matrix contained the band filter gains at all center frequencies and their logarithmic mid-points. Since the interaction matrix values varied slightly with filter gain, i.e., the basis functions were nonconstant, an iteration step was needed for the best accuracy. The interaction matrix was updated using the filter gains obtained at the previous iteration. The resulting maximum magnitude-response error of the proposed third-octave GEQ design called SGE3 was 0.81 dB, which was sufficient even for demanding audio applications. It is often stated that \(\pm\)1 dB is as much gain error as can be accepted in Hi-Fi audio [2, 5, 6]. The maximum error of the proposed Bark GEQ design called SGEB was 1.3 dB, indicating that the non-logarithmic frequency division used in the auditory Bark scale was more challenging in terms of filter optimization than the third-octave scale. Furthermore, we demonstrated how the WLS designs could be replaced with a neural network, as suggested in our previous works for earlier designs [19, 20]. A multilayer perceptron with two hidden layers was trained for both GEQs, using 1500 pairs of command gain vectors and corresponding optimized filter gains vectors. In the prediction mode, the perceptron could precisely compute the filter gains for any command gain setting in the range \(\pm\)12 dB with a mean accuracy of about 0.5 dB \begin{table}
\begin{tabular}{l c c c} \hline \hline
**Method** & **Gain** & **Coefficient** & \\  & **Computation** & **Update** & **Total** \\ \hline SGE3 (iterative WLS design) & 408 μs & 126 μs & 534 μs \\ NSGE3 (neural network) & **5.1 μs** & 126 μs & **131 μs** \\ \hline SGEB (iterative WLS design) & 526 μs & 101 μs & 627 μs \\ NSGEB (neural network) & **4.4 μs** & 101 μs & **105 μs** \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Comparison of computation times required for parameter update in MATLAB. The average of 100,000 test runs is shown in each case. The fastest gain computation times and total times are emphasized. for the third-octave and of about 0.6 dB for the Bark case, whilst considerably speeding up the gain computation. In fact, the trained neural network computed the filter gains 80 times faster than the WLS design method in the third-octave case and about 120 times faster in the Bark case. In addition to speeding up the gain computation, contrary to the WLS based GEQ algorithms, the neural-network approach did not require the GEQ hardware to perform matrix inversions and Fourier transforms in order to solve the LS problem and create the interaction matrix during the iterative step, respectively. The auditory equalizers described in this work are widely applicable for audio signal processing, such as equalization of loudspeakers, headphones, or sound reproduction systems in rooms. Since the filter design was fully automated and highly accurate, it is perfectly suited to novel intelligent audio applications, such as automatic mixing schemes [32; 33], where the graphic equalizer is controlled with artificial intelligence.