# All About Audio Equalization: Solutions and Frontiers
## 4 Graphic Equalizers
### Cascade Graphic Equalizers  
One way to construct a graphic equalizer is by cascading several peaking/notch filters, as shown in Figure 7, where \(M\) is the number of filters. Usually the lowest and highest band filters, \(H_{1}(z)\) and \(H_{M}(z)\), are a low and a high shelf filter, respectively [63]. It is possible to include an optional overall gain term \(G_{0}\), which may be set to the average or median command gain value [63]. This helps in the cases when many command gains are set to the same value. The transfer function of the cascade graphic equalizer can be written as  
\[H_{\text{CGEQ}}(z)=G_{0}\prod_{m=1}^{M}H_{m}(z), \tag{38}\]  
where \(H_{m}(z)\) is the equalizing filter for the \(m\)th band (\(m=1,2,...,M\)). From this form it can be deduced that the total transfer function \(H_{\text{CGEQ}}(z)\) of the cascade structure has all the same poles and zeros as the individual filters \(H_{m}(z)\). This implies that when all band filters are minimum phase, the whole graphic equalizer will also have the minimum-phase property. Ideally, each equalizing filter \(H_{m}(z)\) has magnitude of the desired gain \(G_{m}\) inside the band and gain \(\sqrt{G_{m}}\) at band edges, so that at the crossover frequency, when \(G_{m}=G_{m+1}=G\), we have \(|H_{m}(e^{j\omega_{m}})H_{m+1}(e^{j\omega_{m+1}})|=G\). However, in practice other band filters also affect the gain at that frequency, causing interaction, so the total gain may differ from this value. Figure 8 shows an example magnitude response of a 10-band cascade graphic equalizer. The lowest and highest filters are second order shelving filters designed using Equations (18) and (19), respectively. The rest of the filters are peak/notch filters whose center frequencies have been selected according to Table 1 and bandwidths according to Equation (37) with \(R=2\). The bandwidths of band filters \(H_{2}(z)\) to \(H_{9}(z)\) are 44.5 Hz, 88.4 Hz, 176.8 Hz, 353.6 Hz, 707.1 Hz, 1414 Hz, 2828 Hz, and 5657 Hz. The crossover frequency of the low shelf has been chosen to be 46 Hz (instead of 44 Hz, which is the nominal upper band edge frequency) and that of the high shelf is 11,360 Hz, the nominal lower band edge of the highest band (see Table 1). In Figure 8, the filter gains are set equal to the command gains. Here we neglect the overall gain factor, which is equivalent to fixing it to \(G_{0}=1\). It is seen that using command gains directly as filter gains leads to severe gain buildup, as the overall magnitude response exceeds the command points. The error is about 5 dB at several center frequencies. This is caused by the skirts of the neighboring band filter responses, which is visible in Figure 8. For example, at 63 Hz (the second command point from the left in Figure 8), the low shelf filter and the 125-Hz peak filter contribute a 2.4 dB and 2.1 dB gain increase, respectively. Additionally, a few other peak filters centered at higher frequencies contribute a few tenths of a dB to that frequency. Together they lead to a 5.0-dB overshoot at 63 Hz. To reduce the overshoot, it seems tempting to change the filter bandwidths to be narrower. In fact, this helps in reducing the gain buildup but increases the ripple by making the valleys between the command frequencies deeper. Figure 7: Cascade implementation of a graphic equalizer. Equalizing filters \(H_{m}(z)\) are controlled with command gain parameters \(G_{m}\), as indicated with blue arrows. The accuracy of a graphic equalizer can be improved by allowing filter gains to be different from command gains, and by then optimizing the filter gains for each command gain configuration. First [PERSON] and [PERSON] [51] and recently [PERSON] and [PERSON] [63] have proposed to optimize the filter gains of the cascade graphic equalizer by solving a system of linear equations. This is made possible by the fact that the magnitude responses of the peak/notch filters at different gain settings (but with fixed center frequency and \(Q\)) are self-similar on the dB scale [51].