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Digital Biquad Filter




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 Function s:

:\ H(z)= rac{b_0+b_1z^{-1}+b_2z^{-2}} {1+a_1z^{-1}+a_2z^{-2} }

High-order recursive 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).


IMPLEMENTATION



Direct Form 1

The most straightforward implementation is the Direct Form 1, which has the following Difference Equation :

:\ y(n) = b_0x(n) + b_1x(n-1) + b_2x(n-2) - a_1y(n-1) - a_2y(n-2)

Here the b_0, b_1 and b_2 coefficients determine zeros, and a_1, a_2 determine the position of the poles.

Flow graph of biquad filter in Direct Form 1:


Direct Form 2

The Direct Form 1 implementation requires four delay registers. An equivalent circuit is the Direct Form 2 implementation, which requires only two delay registers:

The Direct Form 2 implementation is called the canonical form, because it uses the minimal amount of delays, adders and multipliers, yielding in the same transfer function as the Direct Form 1 implementation. The Difference Equations for DF2 are:

:\ y(n)=b_0 w(n)+b_1 w(n-1)+b_2 w(n-2),

where

:\ w(n)=x(n)-a_1 w(n-1)-a_2 w(n-2).


REFERENCES



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