### 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

*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.)

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Biquad Software Implementations

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Complex Resonator