- Kautz filter
In signal processing, Kautz filter, named after William H. Kautz, is a fixed pole traversal filter in 1954ref|Kautz1954.
Like
Laguerre filter , Kautz filter can be implemented as a cascade of all-pass filters.Orthogonal set
Given a set of real poles alpha_1, -alpha_2, ldots, -alpha_n}, the
Laplace transform of theorthonormal basis is defined as,
Phi_1(s) = sqrt{2 alpha_1} frac{1}{(s+alpha_1)},
Phi_2(s) = sqrt{2 alpha_2} frac{(s-alpha_1)}{(s+alpha_1)(s+alpha_2)},
Phi_n(s) = sqrt{2 alpha_n} frac{(s-alpha_1)(s-alpha_2) cdots (s-alpha_{n-1})}{(s+alpha_1)(s+alpha_2) cdots (s+alpha_n)}.In time domain, this is equivalent to
phi_k(t) = a_{n1}e^{-alpha_1 t} + a_{n2}e^{-alpha_2 t} + cdots + a_{nn}e^{-alpha_n t},
where "aki" is the coefficients of the partial fraction expansion as,
Phi_k(s) = sum_{i=1}^{k} frac{a_{ki{s+alpha_i}Relation to Laguerre Polynomials
If all poles coincide at "s = -a", then Kautz series can be written as,
phi_k(t) = sqrt{2a}(-1)^{k-1}e^{-at}L_{k-1}(2at),
where "Lk" denotesLaguerre polynomial s.References
* [1] William H. Kautz, Transient Synthesis in the Time Domain, I-R-E Transactions on Circuit Theory,
1954 , 1(3) 29-39
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