en>FrescoBot: Bot: links syntax

2010-10-24T02:26:29Z

Bot: links syntax

New page

In signal processing, the '''Kautz filter''', named after William H. Kautz, is a fixed-[[Pole (complex analysis)|pole]] traversal [[filter (signal processing)|filter]], published in 1954.<ref>
{{cite journal
| title = Transient Synthesis in the Time Domain
| journal = I.R.E. Transactions on Circuit Theory
| volume = 1
| issue = 3
| date = 1954
| author = William H. Kautz
| pages = 29–39
}}</ref><ref>
{{cite book
| chapter = Using Kautz Models in Model Reduction
| title = Signal Analysis and Prediction
| author = A. C. den Brinker and H. J. W. Belt
| editor = A. Prochazka, J. Uhlir, N. G. Kingsbury, and P.J.W. Rayner
| publisher = Birkhäuser
| year = 1998
| isbn = 978-0-8176-4042-2
| page = 187
| url = http://books.google.com/?id=qk2LBkKg5zcC&pg=PA187&dq=%22kautz+filter%22
}}</ref>

Like [[Laguerre filter]]s, Kautz filters can be implemented using a cascade of [[all-pass filter]]s, with a one-pole [[lowpass filter]] at each tap between the all-pass sections.{{Citation needed|date=April 2009}}

== Orthogonal set ==

Given a set of real poles <math>\{-\alpha_1, -\alpha_2, \ldots, -\alpha_n\}</math>, 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:

:<math>\Phi_1(s) = \frac{\sqrt{2 \alpha_1}} {(s+\alpha_1)}</math>

:<math>\Phi_2(s) = \frac{\sqrt{2 \alpha_2}} {(s+\alpha_2)} \cdot \frac{(s-\alpha_1)}{(s+\alpha_1)}</math>

:<math>\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})}</math>.

In the time domain, this is equivalent to

:<math>\phi_n(t) = a_{n1}e^{-\alpha_1 t} + a_{n2}e^{-\alpha_2 t} + \cdots + a_{nn}e^{-\alpha_n t}</math>,

where ''ani'' are the coefficients of the [[Partial fraction|partial fraction expansion]] as, 

:<math>\Phi_n(s) = \sum_{i=1}^{n} \frac{a_{ni}}{s+\alpha_i}</math>

For [[discrete-time]] Kautz filters, the same formulas are used, with ''z'' in place of ''s''.<ref>
{{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 = Matti Karjalainen and Tuomas Paatero
| volume = 2007
| publisher = Hindawi Publishing Corporation
| doi = 10.1155/2007/60949
| date = 2007
| url =
| pages = 1 }}</ref>

== Relation to Laguerre polynomials ==

If all poles coincide at ''s = -a'', then Kautz series can be written as, 
<math>\phi_k(t) = \sqrt{2a}(-1)^{k-1}e^{-at}L_{k-1}(2at)</math>, 
where ''Lk'' denotes [[Laguerre polynomial]]s.

== References ==

<references/>

[[Category:Linear filters]]

Zonal polynomial - Revision history

en>FrescoBot: Bot: links syntax