Kautz filter explained

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

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

In the time domain, this is equivalent to

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

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

\Phi _{n(s)} = \sum ^{n} _{i =1} (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,

\phi _{k(t)} = \sqrt{ 2 a}( -1 ) ^{k-1} e ^{-at} L _{k-1} ( 2 at) ,
where L_k denotes Laguerre polynomials.

See also

• Kautz code