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Peaking Equalizers

The analog transfer function for a peak filter is given by [102,5]

$\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 [102]

$\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 as a fraction of the sampling rate. 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 10.16 gives a matlab listing for a peaking EQ section. Figure 10.17 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.

Figure 10.16: 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;

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

Figure 10.17: 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$.
\includegraphics[width=\textwidth]{eps/tboost}


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[How to cite this work] [Order a printed hardcopy]

``Introduction to Digital Filters with Audio Applications'', by Julius O. Smith III, (August 2006 Edition).
Copyright © 2007-02-02 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA  [Automatic-links disclaimer]