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| In [[signal processing]], a '''digital biquad filter''' is a second-order [[recursive filter|recursive]] [[linear filter]], containing two [[Pole-zero_diagram|poles]] and two [[Pole-zero_diagram|zeros]]. "Biquad" is an abbreviation of "''biquadratic''", which refers to the fact that in the [[Z-transform|Z domain]], its [[transfer function]] is the ratio of two [[quadratic function]]s:
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| :<math>\ H(z)=\frac{b_0+b_1z^{-1}+b_2z^{-2}} {1+a_1z^{-1}+a_2z^{-2} }</math>
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| High-order recursive filters can be highly sensitive to [[Quantization (signal processing)|quantization]] of their coefficients, and can easily become [[BIBO stability|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 filters i.e. all poles must be inside the unit circle for the filter to be stable.
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| ==Implementation== | |
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| ===Direct Form 1===
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| The most straightforward implementation is the Direct Form 1, which has the following [[difference equation]]:
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| :<math>\ y[n] = b_0x[n] + b_1x[n-1] + b_2x[n-2] - a_1y[n-1] - a_2y[n-2] </math>
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| Here the <math>b_0</math>, <math>b_1</math> and <math>b_2</math> coefficients determine zeros, and <math>a_1</math>, <math>a_2</math> determine the position of the poles.
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| Flow graph of biquad filter in Direct Form 1:
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| [[File:Biquad filter DF-I.svg|400px|]]
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| ===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:
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| [[File:Biquad filter DF-II.svg|400px|]]
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| 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 equation]]s for DF2 are:
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| :<math>\ y[n]=b_0 w[n]+b_1 w[n-1]+b_2 w[n-2],</math>
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| where
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| :<math>\ w[n]=x[n]-a_1 w[n-1]-a_2 w[n-2].</math>
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| ===Transposed direct forms===
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| [[File:Biquad direct form 2 transposed .svg|right|thumb|Transposed direct form biquadratic section]] | |
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| See: https://ccrma.stanford.edu/~jos/fp/Transposed_Direct_Forms.html
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| ==References==
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| *[http://www.wiley.com/WileyCDA/WileyTitle/productCd-047003551X.html Sen M. Kuo, Bob H. Lee, Wenshun Tian: ''Real-Time Digital Signal Processing: Implementations and Applications, 2nd Edition'', Wiley & Sons Ltd, 2006 (E-book)]
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| *[http://www.musicdsp.org/files/Audio-EQ-Cookbook.txt Cookbook formulae for audio EQ biquad filter coefficients]
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| *[http://peabody.sapp.org/class/350.838/lab/biquad/ Biquad filter]
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| *[http://ccrma.stanford.edu/~jos/filters/BiQuad_Section.html JOS BiQuad section]
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| ==See also==
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| *[[Electronic_filter_topology#Biquad_filter|Biquad filter]]
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| *[[Digital filter]]
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| [[Category:Linear filters]]
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| [[Category:Audio engineering]]
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| [[Category:Filter theory]]
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