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 
where is approximately the desired gain at the boost (or cut), and is the desired bandwidth at the normalized peak frequency (cf. Eq. (E.8)). When , a boost is obtained at frequency . For , a cut 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.
The bilinear transform (§I.3.1) can be used to convert the analog peaking equalizer section to digital form. As derived in §I.3.2, the mapping constant is best chosen as , where is the desired peak frequency and is the sampling interval.
Figure B.15 gives a matlab listing for a peaking equalizer section. Figure B.16 shows the resulting plot for the example ``boost(2,0.25,0.1).'' The frequency-response display utility myfreqz, listed in Fig.7.1, can be substituted for freqz (better for Octave).
function [B,A] = boost(g,fc,bw,fs); %BOOST - Design a digital boost filter at given gain g, % center frequency fc in Hz, % bandwidth bw in Hz (default = fs/10), and % sampling rate fs in Hz (default = 1). if nargin<4, fs = 1; end if nargin<3, bw = fs/10; end c = cot(pi*fc/fs); % bilinear transform constant cs = c^2; csp1 = cs+1; Bc=(bw/fs)*c; gBc=g*Bc; nrm = 1/(csp1 + Bc); % 1/(a0 before normalization) b0 = (csp1 + gBc)*nrm; b1 = 2*(1 - cs)*nrm; b2 = (csp1 - gBc)*nrm; a0 = 1; a1 = b1; a2 = (csp1 - Bc)*nrm; A = [a0 a1 a2]; B = [b0 b1 b2]; if nargout==0 figure(1); myfreqz(B,A); % /l/mll/myfreqz.m dstr=sprintf('boost(%0.2f,%0.2f,%0.2f,%0.2f)',g,fc,bw,fs); subplot(2,1,1); title(['Boost Frequency Response: ',... dstr],'fontsize',24); end
A Faust implementation of the second-order peaking equalizer is available as the function peak_eq in filter.lib distributed with Faust (Appendix K).