Kautz filter

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 $\{-{\alpha }_1,-{\alpha }_2,\dots ,-{\alpha }_n\}$, 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:

${\mathrm{\Phi }}_1(s)=\frac{\sqrt{2{\alpha }_1}}{(s+{\alpha }_1)}$

${\mathrm{\Phi }}_2(s)=\frac{\sqrt{2{\alpha }_2}}{(s+{\alpha }_2)}\cdot \frac{(s-{\alpha }_1)}{(s+{\alpha }_1)}$

${\mathrm{\Phi }}_n(s)=\frac{\sqrt{2{\alpha }_n}}{(s+{\alpha }_n)}\cdot \frac{(s-{\alpha }_1)(s-{\alpha }_2)\cdots (s-{\alpha }_{n-1})}{(s+{\alpha }_1)(s+{\alpha }_2)\cdots (s+{\alpha }_{n-1})}$.

In the time domain, this is equivalent to

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

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

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

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

See also

• Kautz code