Peaking Equalizers

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

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

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

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

The transfer function can be written in the normalized form [102]

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

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 $f_s/4$.