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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.

Contents

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 } , 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 ) Φ 2 ( s ) = 2 α 2 ( s + α 2 ) ⋅ ( s − α 1 ) ( s + α 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 ) .

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 ,

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

Φ n ( s ) = ∑ i = 1 n a n i s + α 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 ) ,
where Lk denotes Laguerre polynomials.

References

Kautz filter Wikipedia
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