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Introduction to Digital Filters
    Elementary Audio Digital Filters
       Peaking Equalizers

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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 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.

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;

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

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Previous: Exercise
Next: Time-Varying Two-Pole Filters
written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.


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