Kautz filter: Difference between revisions
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In signal processing, the '''Kautz filter''', named after William H. Kautz, is a fixed-[[Pole (complex analysis)|pole]] traversal [[filter (signal processing)|filter]], published in 1954.<ref> |
In signal processing, the '''Kautz filter''', named after [[William H. Kautz]], is a fixed-[[Pole (complex analysis)|pole]] traversal [[filter (signal processing)|filter]], published in 1954.<ref name="Kautz_1954"/><ref name="Brinker_1998"/> |
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{{cite journal |
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| title = Transient Synthesis in the Time Domain |
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| journal = I.R.E. Transactions on Circuit Theory |
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| volume = 1 |
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| issue = 3 |
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| date = 1954 |
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| author = William H. Kautz |
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| pages = 29–39 |
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}}</ref><ref> |
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{{cite book |
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| chapter = Using Kautz Models in Model Reduction |
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| title = Signal Analysis and Prediction |
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|author1=A. C. den Brinker |author2=H. J. W. Belt |
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|lastauthoramp=yes |editor1=A. Prochazka |editor2=J. Uhlir |editor3=N. G. Kingsbury |editor4=P.J.W. Rayner | publisher = Birkhäuser |
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| year = 1998 |
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| isbn = 978-0-8176-4042-2 |
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| page = 187 |
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| url = https://books.google.com/books?id=qk2LBkKg5zcC&pg=PA187&dq=%22kautz+filter%22 |
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}}</ref> |
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Like [[Laguerre filter]]s, Kautz filters can be implemented using a cascade of [[all-pass filter]]s, with a one-pole [[lowpass filter]] at each tap between the all-pass sections.{{Citation needed|date=April 2009}} |
Like [[Laguerre filter]]s, Kautz filters can be implemented using a cascade of [[all-pass filter]]s, with a one-pole [[lowpass filter]] at each tap between the all-pass sections.{{Citation needed|date=April 2009}} |
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:<math>\Phi_n(s) = \sum_{i=1}^{n} \frac{a_{ni}}{s+\alpha_i}</math> |
:<math>\Phi_n(s) = \sum_{i=1}^{n} \frac{a_{ni}}{s+\alpha_i}</math> |
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For [[discrete-time]] Kautz filters, the same formulas are used, with ''z'' in place of ''s''.<ref> |
For [[discrete-time]] Kautz filters, the same formulas are used, with ''z'' in place of ''s''.<ref name="Karjalainen_2007"/> |
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{{cite journal |
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| journal = EURASIP Journal on Advances in Signal Processing |
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| title = Equalization of Loudspeaker and Room Responses Using Kautz Filters: Direct Least Squares Design |
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|author1=Matti Karjalainen |author2=Tuomas Paatero |
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|lastauthoramp=yes | volume = 2007 |
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| publisher = Hindawi Publishing Corporation |
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| doi = 10.1155/2007/60949 |
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| date = 2007 |
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| url = |
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| pages = 1 }}</ref> |
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== Relation to Laguerre polynomials == |
== Relation to Laguerre polynomials == |
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where ''L<sub>k</sub>'' denotes [[Laguerre polynomial]]s. |
where ''L<sub>k</sub>'' denotes [[Laguerre polynomial]]s. |
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== |
== See also == |
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* [[Kautz code]] |
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== References == |
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<references/> |
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{{reflist|refs= |
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<ref name="Kautz_1954">{{cite journal |title=Transient Synthesis in the Time Domain |journal=I.R.E. Transactions on Circuit Theory |volume=1 |issue=3 |date=1954 |author-first=William H. |author-last=Kautz |author-link=William H. Kautz |pages=29–39}}</ref> |
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<ref name="Brinker_1998">{{cite book |chapter=Using Kautz Models in Model Reduction |title=Signal Analysis and Prediction |author-first1=A. C. |author-last1=den Brinker |author-first2=H. J. W. |author-last2=Belt |editor-first1=A. |editor-last1=Prochazka |editor-first2=J. |editor-last2=Uhlir |editor-first3=N. G. |editor-last3=Kingsbury |editor-first4=P. J. W. |editor-last4=Rayner |publisher=[[Birkhäuser]] |date=1998 |isbn=978-0-8176-4042-2 |page=187 |url=https://books.google.com/books?id=qk2LBkKg5zcC&pg=PA187}}</ref> |
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<ref name="Karjalainen_2007">{{cite journal |journal=EURASIP Journal on Advances in Signal Processing |title=Equalization of Loudspeaker and Room Responses Using Kautz Filters: Direct Least Squares Design |author-first1=Matti |author-last1=Karjalainen |author-first2=Tuomas |author-last2=Paatero |volume=2007 |publisher=Hindawi Publishing Corporation |doi=10.1155/2007/60949 |date=2007 |pages=1}}</ref> |
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}} |
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[[Category:Linear filters]] |
[[Category:Linear filters]] |
Revision as of 01:46, 15 January 2018
In signal processing, the Kautz filter, named after William H. Kautz, is a fixed-pole traversal filter, published in 1954.[1][2]
Like Laguerre filters, Kautz filters can be implemented using a cascade of all-pass filters, with a one-pole lowpass filter at each tap between the all-pass sections.[citation needed]
Orthogonal set
Given a set of real poles , the Laplace transform of the Kautz orthonormal basis is defined as the product of a one-pole lowpass factor with an increasing-order allpass factor:
- .
In the time domain, this is equivalent to
- ,
where ani are the coefficients of the partial fraction expansion as,
For discrete-time Kautz filters, the same formulas are used, with z in place of s.[3]
Relation to Laguerre polynomials
If all poles coincide at s = -a, then Kautz series can be written as,
,
where Lk denotes Laguerre polynomials.
See also
References
- ^ Kautz, William H. (1954). "Transient Synthesis in the Time Domain". I.R.E. Transactions on Circuit Theory. 1 (3): 29–39.
- ^ den Brinker, A. C.; Belt, H. J. W. (1998). "Using Kautz Models in Model Reduction". In Prochazka, A.; Uhlir, J.; Kingsbury, N. G.; Rayner, P. J. W. (eds.). Signal Analysis and Prediction. Birkhäuser. p. 187. ISBN 978-0-8176-4042-2.
- ^ Karjalainen, Matti; Paatero, Tuomas (2007). "Equalization of Loudspeaker and Room Responses Using Kautz Filters: Direct Least Squares Design". EURASIP Journal on Advances in Signal Processing. 2007. Hindawi Publishing Corporation: 1. doi:10.1155/2007/60949.
{{cite journal}}
: CS1 maint: unflagged free DOI (link)