## 3 Proposed Graphic Equalizer Designs  
An improved design method for the third-octave GEQ using symmetric band filters, called SGE3, is proposed in this section. Furthermore, similar design methods are used to provide a design for a novel Bark band GEO (SGEB). In the proposed GEO designs, the band filters have a symmetric magnitude response on the logarithmic frequency scale, as suggested in [22]. This symmetry affects primarily the shape of the band filter magnitude response at high frequencies, which allows the band filters to have a specific gain at the Nyquist limit. In our previous GEO designs, the Nyquist gain of the band filters was 1.0, i.e., 0 dB, which forced some of the band filters to have a very asymmetric magnitude-response peak [3,6].  
The design formulas for the symmetric band filters were obtained by modifying those suggested by [PERSON] [31] so that the filter gain at DC (i.e., 0 Hz) was unity and the gain \(G_{\text{Nq},m}\) at the Nyquist limit was adjustable instead of prescribed based on the analog filter counterpart [22,31]. The transfer function of the resulting second-order band filter, implemented with the structure in Figure 2, is:  
\[H_{m}(z)=k_{0,m}\frac{1+b_{1,m}z^{-1}+b_{2,m}z^{-2}}{1+a_{1,m}z^{-1}+a_{2,m}z^{ -2}}, \tag{9}\]  
where \(m=1,2,\dots,M\). For the third-octave design \(M=31\), and for the Bark design \(M=24\). The filter coefficients are calculated as:  
\[k_{0,m} =V(G_{\text{Nq},m}+W^{2}+A_{2}), \tag{10}\] \[b_{1,m} =-2\frac{G_{\text{Nq},m}-W^{2}}{G_{\text{Nq},m}+W^{2}+A_{2}},\] (11) \[b_{2,m} =\frac{G_{\text{Nq},m}+W^{2}-A_{2}}{G_{\text{Nq},m}+W^{2}+A_{2}},\] (12) \[a_{1,m} =-2V(1-W^{2}),\] (13) \[a_{2,m} =V(1+W^{2}-A_{1}), \tag{14}\]  
using the following variables:  
\[A_{1} =\sqrt{\frac{C+D}{|G_{m}^{2}-G_{\text{B},m}^{2}|}}, A_{2}= \sqrt{\frac{G_{m}^{2}C+G_{\text{B},m}^{2}D}{|G_{m}^{2}-G_{\text{B},m}^{2}|}},\] \[W^{2} =\sqrt{\frac{|G_{m}^{2}-G_{\text{Nq},m}^{2}|}{|G_{m}^{2}-1|}} \tan\Big{(}\frac{\omega_{\text{c},m}}{2}\Big{)}^{2}, V= \frac{1}{1+W^{2}+A_{1}}.\]  
The following variables are also used to simplify the above formulas:  
\[C =|G_{\text{B},m}^{2}-G_{\text{Nq},m}^{2}|\Delta\Omega^{2}-2W^{2} \Big{(}|G_{\text{B},m}^{2}-G_{\text{Nq},m}|-\sqrt{|G_{\text{B},m}^{2}-1||G_{ m}^{2}-G_{\text{Nq},m}^{2}|}\Big{)},\] \[D =2W^{2}\Big{(}|G_{m}^{2}-G_{\text{Nq},m}|-\sqrt{|G_{m}^{2}-1||G_{ m}^{2}-G_{\text{Nq},m}^{2}|}\Big{)},\] \[\Delta\Omega =\Big{(}1+\sqrt{\frac{|G_{\text{B},m}^{2}-1|}{|G_{\text{B},m}^{2} -G_{\text{Nq},m}^{2}|}}W^{2}\Big{)}\tan\Big{(}\frac{B_{m}}{2}\Big{)},\]  
where \(G_{m}\) is the linear peak gain of the filter, \(G_{\text{B},m}\) is the gain of the filter at the bandwidth boundaries, \(B_{m}=2\pi f_{\text{B},m}/f_{\text{s}}\) is the bandwidth, \(\omega_{\text{c},m}=2\pi f_{\text{c},m}/f_{\text{s}}\) is the center frequency, and \(f_{\text{s}}=44.1\,\text{kHz}\) is the sampling rate. The linear filter gains \(G_{m}\) are obtained by converting the optimized dB gains \(g_{m}=20\log(G_{m})\) to linear gain factors:  
\[G_{m}=10^{g_{m}/20}. \tag{15}\]The Nyquist gain for each band filter was calculated with the help of a sufficiently highly oversampled parametric EQ filter emulating the shape of an analog filter, as proposed in [22]. The sample rate was set to 10 MHz, and the filter gain was evaluated at 22.05 kHz, which was the Nyquist limit \(f_{\mathrm{g}}/2\) at the sample rate of 44.1 kHz. The gain estimated this way was used as the Nyquist gain \(G_{\mathrm{Nq},m}\) of the band filter. A polynomial approximation for the dependency of the Nyquist gain could be used to simplify the design of the symmetric band filters [22]. Thus, the oversampling method for the Nyquist gain was used only to determine the required polynomial coefficients.