## GUIDE: Mathematics of the Discrete Fourier Transform (DFT) - Julius O. Smith III. The BiQuad Section

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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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