Kautz filter

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

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

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

${\displaystyle \Phi _{n}(s)={\frac {\sqrt {2\alpha _{n}}}{(s+\alpha _{n})}}\cdot {\frac {(s-\alpha _{1})(s-\alpha _{2})\cdots (s-\alpha _{n-1})}{(s+\alpha _{1})(s+\alpha _{2})\cdots (s+\alpha _{n-1})}}}$.

In the time domain, this is equivalent to

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

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

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

For discrete-time Kautz filters, the same formulas are used, with z in place of s.^[3]

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

• Kautz code