NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW
VERSION: "Mathematics of the Discrete
Fourier Transform (DFT), with Audio Applications --- Second
Edition", by Julius
O. Smith III, W3K
Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright ©
2017-09-28 by Julius O. Smith III -
Center for Computer Research
in Music and Acoustics (CCRMA), Stanford University
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The term ''biquad'' is short for ''bi-quadratic'', and is a
common name for a two-pole, two-zero digital filter.
Thetransfer function of a biquad can be defined as
where 
is
called the gain of the biquad. Since both the numerator and
denominator of this transfer function are quadratic polynomials in
, the
transfer function is said to be ''bi-quadratic''.
The parameters 
and 
are called the numerator coefficients, and they
determine the two zeros of the biquad. Using the quadratic
formula for finding the roots of a second-order polynomial, we
find that the zeros are located at
The parameters 
and 
are called the denominator coefficients, and they
determine the two poles of the biquad. Using the quadratic
formula, the poles are found to be
The biquad forms a resonator when the poles are
complex, i.e., when 
.
If a complex pole is denoted by 
, then the resonancefrequency
(in
Hz) is related to pole angle 
and
sampling rate 
by the relation 
, i.e., 
.
The magnitude 
of a complex pole determines the damping or bandwidth of the
resonator. (The damping may be defined as the reciprocal of the
bandwidth.) A common definition for this relationship is
where is the
pole radius, 
is
the bandwidth in Hertz (cycles per second), and 
is the
sampling interval in seconds.
The denominator coefficients of a resonator may be expressed in
terms of and
as
Thus,
depends only on the damping and is independent of the resonance
frequency, while is a function of both.
A common setting for the zeros when making a resonator is to
place one at 
(dc) and the other at 
(half the sampling rate), i.e.,
and
. This placement of the zeros normalizes
the peak gain of the resonator if it is swept using the
parameter.
Using the shift
theorem for
transforms, the difference
equation for the biquad can be written by inspection of the
transfer function as
where 
denotes the input signal sample at time 
, and
is the
output signal. This is the form that is typically implemented in
software.
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