# All About Audio Equalization: Solutions and Frontiers
## 3 Parametric Equalizers
### First Order Shelving Filters
Substituting this result to Equation (5) completes the prototype first order low shelf design:  
\[H_{\text{P}}(z)=\sqrt{G}\frac{G+\sqrt{G}+(G-\sqrt{G})z^{-1}}{G+\sqrt{G}-(G- \sqrt{G})z^{-1}}. \tag{7}\]  
To shift the crossover frequency to \(\omega_{\text{c}}\), which we restrict to be \(0\leq\omega_{\text{c}}\leq\pi\), we apply the lowpass-to-lowpass transformation [4; 29]  
\[z\rightarrow\frac{z-\beta}{1-\beta z}, \tag{8}\]where  
\[\beta=\frac{1-\tan(\omega_{\rm c}/2)}{1+\tan(\omega_{\rm c}/2)}. \tag{9}\]  
Rearranging the terms leads to the transfer function of the first order low-frequency shelving filter  
\[H_{\rm LS}(z)=\frac{G\tan\left(\omega_{\rm c}/2\right)+\sqrt{G}+[G\tan\left( \omega_{\rm c}/2\right)-\sqrt{G}]z^{-1}}{\tan\left(\omega_{\rm c}/2\right)+ \sqrt{G}+[\tan\left(\omega_{\rm c}/2\right)-\sqrt{G}]z^{-1}}. \tag{10}\]  
Due to the choice of gain at crossover frequency, (3), this produces symmetric magnitude frequency responses, above and below the unit gain level, for gains \(G\) and \(1/G\) (_i.e._, equal boost or cut in dB), as wanted. Figure 1 illustrates this for three pairs of low shelf filters. In this and all other examples, the sample rate is \(f_{\rm s}\) = 44,100 Hz and the responses are shown up to the Nyquist limit, \(f_{\rm N}\) = 22,050 Hz. When \(G=1\), the pole and the zero of the transfer function (10) coincide, the magnitude response becomes unity, and the filter's impulse response becomes a non-delayed unit impulse. This way it was possible to derive transfer function coefficients, which are the same for the boost (\(G>1\)) and cut cases (\(0<G<1\)), whereas in the past several shelving filter designs have required separate formulas for the two cases, e.g., [3; 30; 31; 32; 33]. The transfer function (10) is the same as that given in [4] (Equation (4.1)). [PERSON] recently proposed a modified Regalia-Mitra shelving filter design, which also leads to exactly the same transfer function [34]. The complementary high-frequency shelving filter can be obtained by replacing the gain \(G\) in Equation (10) by \(1/G\), which yields the inverse filter, and by multiplying the transfer function (numerator) by \(G\), which shifts the magnitude response vertically so that the gain at zero frequency becomes unity (0 dB), and gain \(G\) is achieved at high frequencies as desired. Additionally, we multiply both the numerator and the denominator by \(\sqrt{G}\) to cancel divisions by \(G\):  
\[H(-1)=1. \tag{2}\]0  
\[H(-1)=1. \tag{2}\]1 Magnitude responses of low-frequency shelving filters with complementary gains: \(-12\) dB and \(12\) dB (green), \(-8\) dB and \(8\) dB (blue), \(-4\) dB and \(4\) dB (red). The vertical dashed line indicates the crossover frequency (\(1\) kHz) where midpoint gain is reached. Comparison with Equation (10) shows that these transfer functions are related as \(H_{\text{HS}}(z)=G/H_{\text{LS}}(z)\), which means that the pole and the zero of the filter can be interchanged to convert a low shelf to a high shelf or vice versa. Furthermore, a scaling by \(G\) must be applied. This implies that a cascade of a low and high shelf with the same crossover frequency \(\omega_{\text{c}}\) and gain \(G\) will produce a constant gain \(G\) across all frequencies. Figure 2 shows this for a pair of low- and high-frequency shelving filters. An alternative way to convert a low shelf filter to a high one is to replace \(z\) with \(-z\) in the transfer function (_i.e.,_ change the sign of every second coefficient), which turns the response over in frequency so that the zero and Nyquist frequencies are interchanged [28]. Additionally, this low-to-high mapping requires replacing \(\omega_{\text{c}}\) with \(\pi-\omega_{\text{c}}\) to restore the crossover frequency.