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 the orthonormal 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 "a_ki" 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 "L_k" denotes Laguerre polynomials.

References

* [1] William H. Kautz, Transient Synthesis in the Time Domain, I-R-E Transactions on Circuit Theory, 1954, 1(3) 29-39