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## Peaking Equalizers

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

where is a two-pole resonator:

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

where is approximately the desired gain at the boost (or cut), and is the desired bandwidth as a fraction of the sampling rate. When , a boost is obtained at frequency . For , a cut filter 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.

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.

 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 

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