Editing Digital filter (section)

==Filter realization==

After a filter is designed, it must be ''realized'' by developing a signal flow diagram that describes the filter in terms of operations on sample sequences.

A given transfer function may be realized in many ways.  Consider how a simple expression such as <math>ax + bx + c</math> could be evaluated &ndash; one could also compute the equivalent <math>x(a + b) + c</math>.  In the same way, all realizations may be seen as ''factorizations'' of the same transfer function, but different realizations will have different numerical properties.  Specifically, some realizations are more efficient in terms of the number of operations or storage elements required for their implementation, and others provide advantages such as improved numerical stability and reduced round-off error.  Some structures are better for [[fixed-point arithmetic]] and others may be better for [[floating-point arithmetic]].

===Direct form I===

A straightforward approach for IIR filter realization is [[Digital biquad filter#Direct form 1|direct form I]], where the difference equation is evaluated directly.  This form is practical for small filters, but may be inefficient and impractical (numerically unstable) for complex designs.<ref>J. O. Smith III, [http://ccrma.stanford.edu/~jos/filters/Direct_Form_I.html Direct Form I]</ref>  In general, this form requires 2N delay elements (for both input and output signals) for a filter of order N.

[[File:Biquad filter DF-I.svg|400px]]

===Direct form II===

The alternate [[Digital biquad filter#Direct form 2|direct form II]] only needs ''N'' delay units, where ''N'' is the order of the filter – potentially half as much as direct form I.  This structure is obtained by reversing the order of the numerator and denominator sections of Direct Form I, since they are in fact two linear systems, and the commutativity property applies. Then, one will notice that there are two columns of delays (<math>z^{-1}</math>) that tap off the center net, and these can be combined since they are redundant, yielding the implementation as shown below.

The disadvantage is that direct form II increases the possibility of arithmetic overflow for filters of high ''Q'' or resonance.<ref>J. O. Smith III, [http://ccrma.stanford.edu/~jos/filters/Direct_Form_II.html Direct Form II]</ref>  It has been shown that as ''Q'' increases,  the round-off noise of both direct form topologies increases without bounds.<ref>L. B. Jackson, "On the Interaction of Roundoff Noise and Dynamic Range in Digital Filters," ''Bell Sys. Tech. J.'', vol. 49 (1970 Feb.), reprinted in ''Digital Signal Process'', L. R. Rabiner and C. M. Rader, Eds. (IEEE Press, New York, 1972).</ref>  This is because, conceptually, the signal is first passed through an all-pole filter (which normally boosts gain at the resonant frequencies) before the result of that is saturated, then passed through an all-zero filter (which often attenuates much of what the all-pole half amplifies).

[[File:Biquad filter DF-II.svg|400px]]

===Cascaded second-order sections===

A common strategy is to realize a higher-order (greater than 2) digital filter as a cascaded series of second-order ''biquadratric'' (or ''biquad'') sections<ref>J. O. Smith III, [http://ccrma.stanford.edu/~jos/filters/Series_Second_Order_Sections.html Series Second Order Sections]</ref> (see [[digital biquad filter]]). The advantage of this strategy is that the coefficient range is limited.  Cascading direct form II sections results in ''N'' delay elements for filters of order ''N''.  Cascading direct form I sections results in ''N'' + 2 delay elements, since the delay elements of the input of any section (except the first section) are redundant with the delay elements of the output of the preceding section.

===Other forms===
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Other forms include:
* Direct form I and II transpose
* Series/cascade lower (typical second) order subsections
* Parallel lower (typical second) order subsections
** Continued fraction expansion
* Lattice and ladder
** One, two and three-multiply lattice forms
** Three and four-multiply normalized ladder forms
** ARMA structures
* State-space structures:
** optimal (in the minimum noise sense): <math>(N+1)^2</math> parameters
** block-optimal and section-optimal: <math>4N-1</math> parameters
** input balanced with Givens rotation: <math>4N-1</math> parameters<ref name=LMXH10>{{cite journal|last=Li|first=Gang|author2=Limin Meng |author3=Zhijiang Xu |author4=Jingyu Hua |title=A novel digital filter structure with minimum roundoff noise|journal=Digital Signal Processing|date=July 2010|volume=20|issue=4|pages=1000–1009|doi=10.1016/j.dsp.2009.10.018|bibcode=2010DSP....20.1000L }}</ref>
* Coupled forms: Gold Rader (normal), State Variable (Chamberlin), Kingsbury, Modified State Variable, Zölzer, Modified Zölzer
* Wave Digital Filters (WDF)<ref name=Fet86>{{cite journal|last=Fettweis|first=Alfred|title=Wave digital filters: Theory and practice|journal=Proceedings of the IEEE|date=Feb 1986|volume=74|issue=2|pages=270–327|doi=10.1109/proc.1986.13458|s2cid=46094699}}</ref>
* Agarwal–Burrus (1AB and 2AB)
* Harris–Brooking
* ND-TDL
* Multifeedback
* Analog-inspired forms such as Sallen-key and state variable filters
* [[Systolic array]]s