Kautz filter

The Kautz filter, developed by William H. Kautz in 1954, is a fixed-pole traversal filter that can be implemented using a cascade of all-pass filters with a one-pole lowpass filter between each tap. It shares similarities with Laguerre filters.

For an orthogonal set of real poles \(-\alpha_1, -\alpha_2, \ldots, -\alpha_n\), the Laplace transform of the Kautz orthonormal basis is defined as a product of a one-pole lowpass factor and an increasing-order allpass factor. The functions Φ₁(s) to Φₙ(s) are given by specific transfer function formulas involving poles and zeros, with time-domain equivalents expressed as sums of exponential terms weighted by coefficients from partial fraction expansions.

Discrete-time Kautz filters use the same formulas but substitute \(z\) for \(s\). When all poles coincide at \(s = -a\), the Kautz series can be written in terms of Laguerre polynomials, resulting in an expression involving exponential and polynomial functions. This relationship connects Kautz filters to Laguerre polynomials, which are orthogonal functions used in signal processing.

The text also references the Kautz code and provides a section for further reading.