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Kautz filter

From Wikipedia, the free encyclopedia

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

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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

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If all poles coincide at s = -a, then Kautz series can be written as,
,
where Lk denotes Laguerre polynomials.

See also

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References

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  1. ^ Kautz, William H. (1954). "Transient Synthesis in the Time Domain". I.R.E. Transactions on Circuit Theory. 1 (3): 29–39.
  2. ^ 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.
  3. ^ 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.