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
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.
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.)
Biquad Software Implementations