## Peaking Equalizers

A peaking equalizer filter section provides a boost or cut in the vicinity of some center frequency. It may also be called a parametric equalizer section. The gain far away from the boost or cut is unity, so it is convenient to combine a number of such sections in series. Additionally, a high and/or low shelf (§B.4 above) are nice to include in series with one's peaking eq sections.

The analog transfer function for a peak filter is given by [103,5,6]

where is a two-pole resonator:

The transfer function can be written in the normalized form [103]

where is approximately the desired gain at the boost (or cut), and is the desired bandwidth. When , a boost is obtained at frequency . For , a cut filter is obtained at that frequency. In particular, when , there are infinitely deep notches at , and when , the transfer function reduces to (no boost or cut). The parameter controls the width of the boost or cut.

It is easy to show that both zeros and both poles are on the unit circle in the left-half plane, and when (a cut''), the zeros are closer to the axis than the poles.

Again, the bilinear transform can be used to convert the analog peaking equalizer section to digital form.

Figure B.15 gives a matlab listing for a peaking EQ section. Figure B.16 shows the resulting plot for an example call:

boost(2,0.25,0.1);

The frequency-response utility myfreqz, listed in Fig.7.1, can be substituted for freqz.

 function [B,A] = boost(gain,fc,bw,fs); %BOOST - Design a boost filter at given gain, center % frequency fc, bandwidth bw, and sampling rate fs % (default = 1). % % J.O. Smith 11/28/02 % Reference: Zolzer: Digital Audio Signal Processing, p. 124 if nargin<4, fs = 1; end if nargin<3, bw = fs/10; end Q = fs/bw; wcT = 2*pi*fc/fs; K=tan(wcT/2); V=gain; b0 = 1 + V*K/Q + K^2; b1 = 2*(K^2 - 1); b2 = 1 - V*K/Q + K^2; a0 = 1 + K/Q + K^2; a1 = 2*(K^2 - 1); a2 = 1 - K/Q + K^2; A = [a0 a1 a2] / a0; B = [b0 b1 b2] / a0; if nargout==0 figure(1); freqz(B,A); title('Boost Frequency Response') end 

A Faust implementation of the peaking equalizer is available as the function peak_eq in filter.lib distributed with Faust (Appendix K) starting with version 0.9.9.4k-par.

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Time-Varying Two-Pole Filters
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Low and High Shelving Filters