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Concept# Digital biquad filter

Summary

In signal processing, a digital biquad filter is a second order recursive linear filter, containing two poles and two zeros. "Biquad" is an abbreviation of "biquadratic", which refers to the fact that in the Z domain, its transfer function is the ratio of two quadratic functions:
The coefficients are often normalized such that a0 = 1:
High-order infinite impulse response filters can be highly sensitive to quantization of their coefficients, and can easily become unstable. This is much less of a problem with first and second-order filters; therefore, higher-order filters are typically implemented as serially-cascaded biquad sections (and a first-order filter if necessary). The two poles of the biquad filter must be inside the unit circle for it to be stable. In general, this is true for all discrete filters i.e. all poles must be inside the unit circle in the Z-domain for the filter to be stable.
The most straightforward implementation is the direct form 1, which has the following difference equation:
or, if normalized:
Here the , and coefficients determine zeros, and , determine the position of the poles.
Flow graph of biquad filter in direct form 1:
When these sections are cascaded for filters of order greater than 2, efficiency of implementation can be improved by noticing the delay of a section output is cloned in the next section input. Two storage delay components may be eliminated between sections.
The direct form 2 implements the same normalized transfer function as direct form 1, but in two parts:
and using the difference equation:
Flow graph of biquad filter in direct form 2:
The direct form 2 implementation only needs N delay units, where N is the order of the filter – potentially half as much as direct form 1. This structure is obtained by reversing the order of the numerator and denominator sections of direct Form 1, since they are in fact two linear systems, and the commutativity property applies. Then, one will notice that there are two columns of delays () that tap off the center net, and these can be combined since they are redundant, yielding the implementation as shown.

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