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Kautz filter

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In signal processing, the Kautz filter, named after William H. Kautz, is a fixed-pole traversal filter, published in 1954.

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.

Orthogonal set

Given a set of real poles { − α 1 , − α 2 , … , − α n } {\displaystyle \{-\alpha _{1},-\alpha _{2},\ldots ,-\alpha _{n}\}} , 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:

Φ 1 ( s ) = 2 α 1 ( s + α 1 ) {\displaystyle \Phi _{1}(s)={\frac {\sqrt {2\alpha _{1}}}{(s+\alpha _{1})}}}
Φ 2 ( s ) = 2 α 2 ( s + α 2 ) ⋅ ( s − α 1 ) ( s + α 1 ) {\displaystyle \Phi _{2}(s)={\frac {\sqrt {2\alpha _{2}}}{(s+\alpha _{2})}}\cdot {\frac {(s-\alpha _{1})}{(s+\alpha _{1})}}}
Φ n ( s ) = 2 α n ( s + α n ) ⋅ ( s − α 1 ) ( s − α 2 ) ⋯ ( s − α n − 1 ) ( s + α 1 ) ( s + α 2 ) ⋯ ( s + α n − 1 ) {\displaystyle \Phi _{n}(s)={\frac {\sqrt {2\alpha _{n}}}{(s+\alpha _{n})}}\cdot {\frac {(s-\alpha _{1})(s-\alpha _{2})\cdots (s-\alpha _{n-1})}{(s+\alpha _{1})(s+\alpha _{2})\cdots (s+\alpha _{n-1})}}} .

In the time domain, this is equivalent to

ϕ n ( t ) = a n 1 e − α 1 t + a n 2 e − α 2 t + ⋯ + a n n e − α n t {\displaystyle \phi _{n}(t)=a_{n1}e^{-\alpha _{1}t}+a_{n2}e^{-\alpha _{2}t}+\cdots +a_{nn}e^{-\alpha _{n}t}} ,

where ani are the coefficients of the partial fraction expansion as,

Φ n ( s ) = ∑ i = 1 n a n i s + α i {\displaystyle \Phi _{n}(s)=\sum _{i=1}^{n}{\frac {a_{ni}}{s+\alpha _{i}}}}

For discrete-time Kautz filters, the same formulas are used, with z in place of s.

Relation to Laguerre polynomials

If all poles coincide at s = -a, then Kautz series can be written as,
ϕ k ( t ) = 2 a ( − 1 ) k − 1 e − a t L k − 1 ( 2 a t ) {\displaystyle \phi _{k}(t)={\sqrt {2a}}(-1)^{k-1}e^{-at}L_{k-1}(2at)} ,
where Lk denotes Laguerre polynomials.

See also

References

  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.
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