Biquad filter

"For the digital implementation of a biquad filter, check digital biquad filter."

A biquad filter is a type of linear filter that implements a transfer function that is the ratio of two quadratic functions. The name "biquad" is short for "biquadratic".

Biquad filters are typically active and implemented with a single-amplifier biquad (SAB) or two-integrator-loop topology.
* The SAB topology uses feedback to generate complex poles and possibly complex zeros. In particular, the feedback moves the real poles of an RC circuit in order to generate the proper filter characteristics.
* The two-integrator-loop topology is derived from rearranging a biquadratic transfer function. The rearrangement will equate one signal with the sum of another signal, its integral, and the integral's integral. In other words, the rearrangement reveals a state variable filter structure. By using different states as outputs, any kind of second-order filter can be implemented. The SAB topology is sensitive to component choice and can be more difficult to adjust. Hence, usually the term biquad refers to the two-integrator-loop state variable filter topology.

Tow-Thomas Biquad Example

For example, the basic configuration in Figure 1 can be used as either a low-pass or bandpass filter depending on where the output signal is taken from. The second-order low-pass transfer function is given by
$H(s)=frac\{G\_\{lpf\}omega^\{2\}\_\{0\{s^\{2\}+frac\{omega\_\{0\{Q\}s+omega^\{2\}\_\{0$
where low-pass gain $G\_\{lpf\}=-R\_\{2\}/R\_\{1\}$. The second-order bandpass transfer function is given by
$H(s)=frac\{G\_\{bpf\}frac\{omega\_\{0\{Q\}s\}\{s^\{2\}+frac\{omega\_\{0\{Q\}s+omega^\{2\}\_\{0$.
with bandpass gain $G\_\{bpf\}=-R\_\{3\}/R\_\{1\}$. In both cases, the
* Natural frequency is $omega\_\{0\}=1/sqrt\{R\_\{2\}R\_\{4\}C\_\{1\}C\_\{2$.
* Quality factor is $Q=sqrt\{frac\{R\_\{3\}^\{2\}C\_\{1\{R\_\{2\}R\_\{4\}C\_\{2\}$.The bandwidth is approximated by $B=omega\_\{0\}/Q$, and Q is sometimes expressed as a damping constant $zeta=2/Q$. If a noninverting low-pass filter is required, the output can be taken at the output of the second operational amplifier. If a noninverting bandpass filter is required, the order of the second integrator and the inverter can be switched, and the output taken at the output of the inverter's operational amplifier.

ee also

* Linear filter
* State variable filter