### The BiQuad Section

The term ``biquad'' is short for ``bi-quadratic'', and is a common
name for a two-pole, two-zero digital filter. The
*transfer function* of the biquad can be defined as

where can be called the

*overall gain*of the biquad. Since both the numerator and denominator of this transfer function are quadratic polynomials in (or ), the transfer function is said to be ``bi-quadratic'' in (or ).

As derived in §B.1.3, for real second-order polynomials having complex roots, it is often convenient to express the polynomial coefficients in terms of the radius and angle of the positive-frequency pole. For example, denoting the denominator polynomial by , we have

*resonance frequency*(in radians per sample-- , where is the resonance frequency in Hz), and determines the ``Q'' of the resonance (see §B.1.3). The numerator is less often represented in this way, but when it is, we may think of the zero-angle as the

*antiresonance frequency*, and the zero-radius affects the

*depth*and

*width*of the antiresonance (or

*notch*).

As discussed on page , 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
in
Eq.(B.8) above
.
This zero placement normalizes the peak gain of the resonator if it is
swept using the parameter.

Using the shift theorem for *z* 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. It is essentially the *direct-form I* implementation. (To obtain the official
direct-form I structure, the overall gain must be not be pulled
out separately, resulting in feedforward coefficients
instead. See Chapter 9 for more about
filter implementation forms.)

**Next Section:**

Biquad Software Implementations

**Previous Section:**

Complex Resonator