Kautz Filter - Orthogonal Set

Orthogonal Set

Given a set of real poles, 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_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

,

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

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

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