# Audio EQ Cookbook

The following is the Audio EQ Cookbook, containing the most widely used formulas for biquad filters. Synthizer's internal implementation of most filters either follows these exactly or is composed of cascaded/parallel sections.

There are several versions of this document on the web. This version is from http://music.columbia.edu/pipermail/music-dsp/2001-March/041752.html.

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Cookbook formulae for audio EQ biquad filter coefficients
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by Robert Bristow-Johnson <rbj at gisco.net> a.k.a. <robert at audioheads.com>
All filter transfer functions were derived from analog prototypes (that
are shown below for each EQ filter type) and had been digitized using the
Bilinear Transform. BLT frequency warping has been taken into account
for both significant frequency relocation and for bandwidth readjustment.
First, given a biquad transfer function defined as:
b0 + b1*z^-1 + b2*z^-2
H(z) = ------------------------ (Eq 1)
a0 + a1*z^-1 + a2*z^-2
This shows 6 coefficients instead of 5 so, depending on your architechture,
you will likely normalize a0 to be 1 and perhaps also b0 to 1 (and collect
that into an overall gain coefficient). Then your transfer function would
look like:
(b0/a0) + (b1/a0)*z^-1 + (b2/a0)*z^-2
H(z) = --------------------------------------- (Eq 2)
1 + (a1/a0)*z^-1 + (a2/a0)*z^-2
or
1 + (b1/b0)*z^-1 + (b2/b0)*z^-2
H(z) = (b0/a0) * --------------------------------- (Eq 3)
1 + (a1/a0)*z^-1 + (a2/a0)*z^-2
The most straight forward implementation would be the Direct I form (using Eq 2):
y[n] = (b0/a0)*x[n] + (b1/a0)*x[n-1] + (b2/a0)*x[n-2]
- (a1/a0)*y[n-1] - (a2/a0)*y[n-2] (Eq 4)
This is probably both the best and the easiest method to implement in the 56K.
Now, given:
sampleRate (the sampling frequency)
frequency ("wherever it's happenin', man." "center" frequency
or "corner" (-3 dB) frequency, or shelf midpoint frequency,
depending on which filter type)
dBgain (used only for peaking and shelving filters)
bandwidth in octaves (between -3 dB frequencies for BPF and notch
or between midpoint (dBgain/2) gain frequencies for peaking EQ)
_or_ Q (the EE kind of definition)
_or_ S, a "shelf slope" parameter (for shelving EQ only). when S = 1,
the shelf slope is as steep as it can be and remain monotonically
increasing or decreasing gain with frequency. the shelf slope, in
dB/octave, remains proportional to S for all other values.
First compute a few intermediate variables:
A = sqrt[ 10^(dBgain/20) ]
= 10^(dBgain/40) (for peaking and shelving EQ filters only)
omega = 2*PI*frequency/sampleRate
sin = sin(omega)
cos = cos(omega)
alpha = sin/(2*Q) (if Q is specified)
= sin*sinh[ ln(2)/2 * bandwidth * omega/sin ] (if bandwidth is specified)
beta = sqrt(A)/Q (for shelving EQ filters only)
= sqrt(A)*sqrt[ (A + 1/A)*(1/S - 1) + 2 ] (if shelf slope is specified)
= sqrt[ (A^2 + 1)/S - (A-1)^2 ]
Then compute the coefficients for whichever filter type you want:
The analog prototypes are shown for normalized frequency.
The bilinear transform substitutes:
1 1 - z^-1
s <- -------------- * ----------
tan(omega/2) 1 + z^-1
and makes use of these trig identities:
sin(w)
tan(w/2) = ------------
1 + cos(w)
1 - cos(w)
(tan(w/2))^2 = ------------
1 + cos(w)
LPF: H(s) = 1 / (s^2 + s/Q + 1)
b0 = (1 - cos)/2
b1 = 1 - cos
b2 = (1 - cos)/2
a0 = 1 + alpha
a1 = -2*cos
a2 = 1 - alpha
HPF: H(s) = s^2 / (s^2 + s/Q + 1)
b0 = (1 + cos)/2
b1 = -(1 + cos)
b2 = (1 + cos)/2
a0 = 1 + alpha
a1 = -2*cos
a2 = 1 - alpha
BPF (constant skirt gain): H(s) = s / (s^2 + s/Q + 1)
b0 = Q*alpha
b1 = 0
b2 = -Q*alpha
a0 = 1 + alpha
a1 = -2*cos
a2 = 1 - alpha
BPF (constant peak gain): H(s) = (s/Q) / (s^2 + s/Q + 1)
b0 = alpha
b1 = 0
b2 = -alpha
a0 = 1 + alpha
a1 = -2*cos
a2 = 1 - alpha
notch: H(s) = (s^2 + 1) / (s^2 + s/Q + 1)
b0 = 1
b1 = -2*cos
b2 = 1
a0 = 1 + alpha
a1 = -2*cos
a2 = 1 - alpha
APF: H(s) = (s^2 - s/Q + 1) / (s^2 + s/Q + 1)
b0 = 1 - alpha
b1 = -2*cos
b2 = 1 + alpha
a0 = 1 + alpha
a1 = -2*cos
a2 = 1 - alpha
peakingEQ: H(s) = (s^2 + s*(A/Q) + 1) / (s^2 + s/(A*Q) + 1)
b0 = 1 + alpha*A
b1 = -2*cos
b2 = 1 - alpha*A
a0 = 1 + alpha/A
a1 = -2*cos
a2 = 1 - alpha/A
lowShelf: H(s) = A * (s^2 + beta*s + A) / (A*s^2 + beta*s + 1)
b0 = A*[ (A+1) - (A-1)*cos + beta*sin ]
b1 = 2*A*[ (A-1) - (A+1)*cos ]
b2 = A*[ (A+1) - (A-1)*cos - beta*sin ]
a0 = (A+1) + (A-1)*cos + beta*sin
a1 = -2*[ (A-1) + (A+1)*cos ]
a2 = (A+1) + (A-1)*cos - beta*sin
highShelf: H(s) = A * (A*s^2 + beta*s + 1) / (s^2 + beta*s + A)
b0 = A*[ (A+1) + (A-1)*cos + beta*sin ]
b1 = -2*A*[ (A-1) + (A+1)*cos ]
b2 = A*[ (A+1) + (A-1)*cos - beta*sin ]
a0 = (A+1) - (A-1)*cos + beta*sin
a1 = 2*[ (A-1) - (A+1)*cos ]
a2 = (A+1) - (A-1)*cos - beta*sin
```