# All About Audio Equalization: Solutions and Frontiers
## 4 Graphic Equalizers
### Parallel Graphic Equalizers
This method requires a target frequency response \(H_{\text{t}}(\omega_{k})\) to be constructed by interpolating between the points defined by the command gains. Hermite interpolation has been shown to be suitable for this, as it does not introduce overshoot [56]. A computationally efficient method combining linear interpolation and constant segments reduces the computational complexity of this design without much reduction in the accuracy [70]. For good accuracy, the target response needs to be evaluated at 10 times as many points as there are pole frequencies [56]. In addition to the target magnitude response, the target phase response must be generated. A natural choice for a graphic equalizer is a minimum phase response. This requires resampling the target response to the linear frequency scale and computing the Hilbert transform of the log magnitude response using FFT and IFFT operations. In the end, the data is sampled again at logarithmic frequency points. A modeling matrix \(\mathbf{M}\) is constructed by sampling the complex frequency responses of the denominators and their delayed versions. A non-negative weighting function \(W(\omega_{k})=1/|H_{\text{t}}(\omega_{k})|^{2}\) is needed to ensure that attenuation is implemented as accurately as amplification [56]. Finally, the optimal parameters \(\mathbf{p_{opt}}\) are obtained as  
\[\mathbf{p_{opt}}=(\mathbf{M}^{H}\mathbf{W}\mathbf{M})^{-1}\mathbf{M}^{H} \mathbf{W}\mathbf{h_{t}}, \tag{45}\]  
where \(\mathbf{M}\) is the modeling matrix, \(\mathbf{M}^{H}\) is its conjugate transpose, \(\mathbf{W}\) is the non-negative diagonal weighting matrix, and \(\mathbf{h_{t}}\) is the column vector containing the target response. As a result, every time a command gain is changed, both the target response and the weighting matrix also change and matrix inversion needs to be executed [56]. Figure 11 shows a design example in which the command gains are set to \(\pm 10\) dB. For the octave equalizer, the target response was interpolated at 100 points on the logarithmic frequency axis from 20 Hz to 22,050 Hz. Now the match to the command points is excellent. The largest deviation at a command point is 0.64 dB, which occurs at the lowest point 31.5 Hz. Figure 11: Magnitude response of an optimized parallel graphic equalizer, when command gains (red circles) at octave bands are set to either 10 dB or \(-10\) dB. Figure 12: Parallel subfilter responses forming the magnitude response of Figure 11: filters at command bands (solid blue lines), slave filters between bands (red dashed lines), extra filter at 20 Hz (green dashed line), and the direct path gain (green dotted line). Figure 12 illustrates the subfilters used to obtain this result. It is interesting to see that even the gains of the main filters having a pole at command frequencies (blue solid lines) do not coincide with the command gains, but are smaller. The intermediate (dashed red lines) filters usually have a gain midway between the neighboring main filters. Additionally, the responses related to the extra pole at 20 Hz and the direct path gain (3.9 dB) are shown. Since only magnitude responses of the subfilters are shown, Figure 12 fails to illustrate the effects of phase. The computational load of the above parallel graphic equalizer is modest: its operation count per sample is only 23% larger than that of a basic graphic equalizer consisting of cascaded biquad filter sections having second order transfer function numerator and denominator [56]. However, the design phase of the parallel structure is quite complicated, requiring many more operations than designing a cascade graphic equalizer. Though this may be an important factor in interactive real-time use, the computational cost is not very critical when designing equalizer preset settings off-line. Alternatively, it is possible to design a parallel graphic equalizer based on higher order IIR filters, although this will increase the computational cost in comparison to designs based on second order band filters.