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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]

$\displaystyle H(s) = 1 + H_R(s)

where $ H_R(s)$ is a two-pole resonator:

$\displaystyle H_R(s) \isdef R\cdot \frac{Bs}{s^2+Bs + 1}

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

$\displaystyle H(s) = \frac{s^2 + gBs + 1}{s^2+Bs + 1},

where $ g$ is approximately the desired gain at the boost (or cut), and $ B$ is the desired bandwidth. When $ g>1$, a boost is obtained at frequency $ \omega=1$. For $ g<1$, a cut filter is obtained at that frequency. In particular, when $ g=0$, there are infinitely deep notches at $ \omega=\pm 1$, and when $ g=1$, the transfer function reduces to $ H(s)=1$ (no boost or cut). The parameter $ B$ 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 $ s$ plane, and when $ g<1$ (a ``cut''), the zeros are closer to the $ j\omega$ 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:
The frequency-response utility myfreqz, listed in Fig.7.1, can be substituted for freqz.
Figure B.15: Matlab function for designing (and optionally testing) a peaking equalizer section.

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;


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
  title('Boost Frequency Response')

Figure B.16: Frequency response of a second-order peaking equalizer section tuned to give a 6 dB peak of width $ \approx f_s/10$ at center frequency $ f_s/4$.
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
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Time-Varying Two-Pole Filters
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Low and High Shelving Filters