**NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW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 BiQuad SectionThe term “biquad” is short for “bi-quadratic”, and is a common name for a two-pole, two-zero digital filter. The

transfer functionof a biquad can be defined as

where is called thegainof 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 twozerosof 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 twopolesof the biquad. Using the quadratic formula, the poles are found to be

The biquad forms aresonatorwhen the poles arecomplex, 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

magnitudeof a complex pole determines thedampingorbandwidthof 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 equationfor 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.