# All About Audio Equalization: Solutions and Frontiers
## 4 Graphic Equalizers
### Cascade Graphic Equalizers
(However, [PERSON] and [PERSON] [51] used parametric sections characterized by crossover frequencies at which the filter gain is the square root of its extreme value. Doing so slightly better maintains the self similarity property, as \(Q\) varies a little with filter center frequency.) Thus, the magnitude responses of the peak/notch filters can be used as basis functions to approximate the magnitude response of the graphic equalizer at the center frequencies. This can be written in the form \(\mathbf{\hat{h}}=\mathbf{B}\mathbf{g}\), where \(\mathbf{\hat{h}}\) is an \(M\)-by-1 vector of estimated dB magnitude response values at command frequencies, \(\mathbf{B}\) is the \(M\)-by-\(M\) interaction matrix representing how much the response of each band filter leaks to other center frequencies in dB, and \(\mathbf{g}\) is an \(M\)-by-1 vector of command gains in dB,  
\[\mathbf{g}=\left[ \begin{array}{cccc}20\log(G_{1})&20\log(G_{2})&...&20\log( G_{M})\end{array} \right]^{T}, \tag{39}\]  
where \(T\) denotes transposition. The \(\mathbf{B}\) matrix models the leakage of the filters to other center frequencies when all filter gains are 1 dB [51]. We have estimated the following ten-by-ten \(\mathbf{B}\) matrix by setting all filter gains of the octave graphic equalizer to 10 dB (an arbitrarily chosen fairly large gain), estimating the dB magnitude response of the filters at the 10 center frequencies, and by dividing the gain values by 10:  
Figure 8: Magnitude responses of an octave graphic equalizer implemented using the cascade structure (thick line) and its two shelving and eight band filters with \(Q=1.41\) (colored curves). The red circles indicate the command gains, which are now all at 10 dB. \[\mathbf{B}=\left[ \begin{array}{cccccccccccc}0.80&0.23&0.02&0&0&0&0&0&0&0\\ 0.19&1&0.21&0.04&0.01&0&0&0&0&0\\ 0.04&0.21&1&0.20&0.04&0.01&0&0&0&0\\ 0.01&0.04&0.20&1&0.20&0.04&0.01&0&0&0\\ 0&0.01&0.04&0.20&1&0.20&0.04&0.01&0&0\\ 0&0&0.01&0.04&0.20&1&0.20&0.04&0.01&0\\ 0&0&0&0.01&0.04&0.20&1&0.20&0.03&0\\ 0&0&0&0&0.01&0.04&0.21&1&0.18&0.01\\ 0&0&0&0&0&0.01&0.06&0.25&1&0.10\\ 0&0&0&0&0&0&0&0.01&0.14&0.94\\ \end{array} \right]. \tag{40}\]  
Using the inverse matrix of \(\mathbf{B}\), it is now possible to determine the filter gains \(\mathbf{g}_{\mathrm{opt}}\) which approximate in the least squares sense the magnitude response specified by command gain values \(\mathbf{g}\)[51]:  
\[\mathbf{g}_{\mathrm{opt}}=\mathbf{B}^{-1}\mathbf{g}. \tag{41}\]  
The inverse matrix \(\mathbf{B}^{-1}\) can be computed off-line and then stored. It is applied every time a command gain is changed. This means that all filter gains will be modified even if one command gain is changed, and so all band filters must also be redesigned using Equation (29). Figure 9 shows an example usage of the above method. The gain of all peak filters is about 6 dB although all command gains are at 10 dB, because the modeling technique accounts for the gain buildup caused by interaction. The overall magnitude response of the graphic equalizer is, however, quite close to the command points. The maximum error, which occurs at the 125-Hz point, is only 0.5 dB--a remarkable improvement over the example of Figure 8. Nevertheless, the overall fit is not highly accurate, because the response still oscillates between the command points. The peak-to-peak extent of this ripple (about 2 dB in this case) is proportional to filter gains. [PERSON] and [PERSON] [63] mitigated this issue by setting the global gain \(G_{0}\) to the median of all command gain values and offsetting these values accordingly. Figure 9: Magnitude response of the cascade graphic equalizer when its filter gains have been solved accounting for the interference between band filters. Note that peak gains of band filters are not the same as command gains (red circles). Cf. Figure 8.  
To reduce the ripple in a cascade graphic equalizer's magnitude response, [PERSON] [59] has proposed to use extra filter sections, which he calls 'opposite' filters, between the band filters in a cascade graphic equalizer.