Kautz filter

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.

Orthogonal set

Given a set of real poles $\{-\alpha _{1},-\alpha _{2},\ldots ,-\alpha _{n}\}$ , 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:

\Phi _{1}(s)={\frac {{\sqrt {2\alpha _{1}}}}{(s+\alpha _{1})}}

\Phi _{2}(s)={\frac {{\sqrt {2\alpha _{2}}}}{(s+\alpha _{2})}}\cdot {\frac {(s-\alpha _{1})}{(s+\alpha _{1})}}

\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

\phi _{n}(t)=a_{{n1}}e^{{-\alpha _{1}t}}+a_{{n2}}e^{{-\alpha _{2}t}}+\cdots +a_{{nn}}e^{{-\alpha _{n}t}}

,

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

\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.^[3]

Relation to Laguerre polynomials

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

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. 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. Hindawi Publishing Corporation. 2007: 1. doi:10.1155/2007/60949.

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[Kautz_1954-1] Kautz, William H. (1954). "Transient Synthesis in the Time Domain". I.R.E. Transactions on Circuit Theory. 1 (3): 29–39.

[Brinker_1998-2] 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. Signal Analysis and Prediction. Birkhäuser. p. 187. ISBN 978-0-8176-4042-2.

[Karjalainen_2007-3] 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. Hindawi Publishing Corporation. 2007: 1. doi:10.1155/2007/60949.

Kautz filter

Orthogonal set

Relation to Laguerre polynomials

See also

References