# All About Audio Equalization: Solutions and Frontiers
## 4 Graphic Equalizers
### Parallel Graphic Equalizers  
Instead of using a cascade structure, a graphic equalizer often uses a collection of bandpass filters arranged in parallel, as shown in Figure 10[47; 54; 55; 56]. The audio input is split and sent to the input of every bandpass filter. It is common to include a direct path, which passes the scaled input signal to the output, as seen in Figure 10. Each filter allows only a narrow frequency band to pass through. Ideally, the center frequencies and bandwidths are configured so that if all the outputs were added together, the original signal would be reconstructed. The controls on the graphic equalizer are then implemented by changing the gain of each bandpass filter output before the signals are summed together. In theory, the parallel and cascade implementations are mathematically identical but their design must be treated differently. Parallel connection of bandpass filters avoids the accumulating phase errors and, potentially, quantization noise, found in the cascade. The parallel filter structure is also well suited to parallel computing [67]. In the parallel structure, the direct path with gain \(G_{0}\) allows easy implementation of flat command gain configurations, just like in the cascade structure. For example, the constant 10-dB gain at all command bands, which was shown in Figures 8 and 9, can be easily implemented using the parallel graphic equalizer by setting \(G_{0}=3.2\) and by zeroing all other filter gains. However, the parallel structure is equally prone to interaction between band filters as the cascade structure, when command  
Figure 10: Parallel implementation of a graphic equalizer in which all bandpass filters receive the same input signal. The output is obtained as a sum of individual filter outputs weighted by command gains. gains are set to a non-flat configuration. The transfer function of the parallel graphic equalizer is obtained as the sum of all band filters and the direct path:  
\[H_{\text{PGEQ}}(z)=G_{0}+\sum_{m=1}^{M}G_{m}H_{m}(z). \tag{42}\]  
This implies that also the phase, not only the magnitude response, of each bandpass filter affects the total frequency response of the parallel graphic equalizer. Therefore, for good accuracy it is necessary to optimize the gain and phase of all filters for each command gain setting. This is inevitably more complicated than design of a cascade structure. [PERSON] _et al._ recently developed a parallel graphic equalizer, which can accurately follow the command gain settings [56]. This method is based on Bank's fixed-pole parallel filters [68; 69]. In this structure, each IIR filter transfer function \(H_{m}(z)\) has a second order denominator but a first order numerator:  
\[H_{m}(z)=\frac{b_{m,0}+b_{m,1}z^{-1}}{1-2|p_{m}|\cos(\theta_{m})z^{-1}+|p_{m}|^ {2}z^{-2}},\text{ for }m=1,2,...,M, \tag{43}\]  
where \(p_{m}\) and \(\theta_{m}\) are the pole radius and pole angle of the \(m\)th bandpass filter. The poles are set in advance at pre-designed frequencies determined by the frequency resolution of the graphic equalizer. When an octave graphic equalizer is designed, the poles are set at the 10 standard frequencies from 31.5 Hz to 16 kHz, which are listed in Table 1. To obtain high accuracy, additional poles are assigned between each two standard center frequencies (at their geometric mean frequency) and below the lowest standard frequency (for example at 20 Hz), so that there will be altogether 20 predesigned poles. The pole radii are chosen so that the magnitude responses associated with neighboring poles meet at their \(-3\)-dB points:  
\[p_{m}=e^{\frac{\Delta\theta_{m}}{2}}e^{\pm\text{j}\theta_{m}}, \tag{44}\]  
where \(\Delta\theta_{m}\) are the differences between the neighboring pole frequencies [56]. This graphic equalizer design uses least squares optimization to adjust the direct path gain \(G_{0}\) and the numerator coefficients \(b_{m,0}\) and \(b_{m,1}\) for each filter.