William H. Kautz

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 $\$, 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:

:$\Phi_1(s) = \frac$

:$\Phi_2(s) = \frac \cdot \frac$

:$\Phi_n(s) = \frac \cdot \frac$.

In the time domain, this is equivalent to

:$\phi_n(t) = a_e^ + a_e^ + \cdots + a_e^$,

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

:$\Phi_n(s) = \sum_^ \frac$

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,

$\phi_k(t) = \sqrt(-1)^e^L_(2at)$,

where ''L_k'' denotes Laguerre polynomials.

See also

* Kautz code

References

{{reflist, refs=
{{cite journal , title=Transient Synthesis in the Time Domain , journal=I.R.E. Transactions on Circuit Theory , volume=1 , issue=3 , date=1954 , author-first=William H. , author-last=Kautz , author-link=William H. Kautz , pages=29–39
{{cite book , chapter=Using Kautz Models in Model Reduction , title=Signal Analysis and Prediction , author-first1=A. C. , author-last1=den Brinker , author-first2=H. J. W. , author-last2=Belt , editor-first1=A. , editor-last1=Prochazka , editor-first2=J. , editor-last2=Uhlir , editor-first3=N. G. , editor-last3=Kingsbury , editor-first4=P. J. W. , editor-last4=Rayner , publisher=Birkhäuser , date=1998 , isbn=978-0-8176-4042-2 , page=187 , chapter-url=https://books.google.com/books?id=qk2LBkKg5zcC&pg=PA187
{{cite journal , journal=EURASIP Journal on Advances in Signal Processing , title=Equalization of Loudspeaker and Room Responses Using Kautz Filters: Direct Least Squares Design , author-first1=Matti , author-last1=Karjalainen , author-first2=Tuomas , author-last2=Paatero , volume=2007 , publisher=Hindawi Publishing Corporation , doi=10.1155/2007/60949 , date=2007 , pages=1, doi-access=free

Linear filters