Kautz filter
In signal processing, the Kautz filter, named after William H. Kautz, is a fixed-pole traversal filter, published in 1954.[1][2] 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.[citation needed]
Orthogonal set
Given a set of real poles [math]\displaystyle{ \{-\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]\displaystyle{ \Phi_1(s) = \frac{\sqrt{2 \alpha_1}} {(s+\alpha_1)} }[/math]
- [math]\displaystyle{ \Phi_2(s) = \frac{\sqrt{2 \alpha_2}} {(s+\alpha_2)} \cdot \frac{(s-\alpha_1)}{(s+\alpha_1)} }[/math]
- [math]\displaystyle{ \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]\displaystyle{ \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 expansion as,
- [math]\displaystyle{ \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.[3]
Relation to Laguerre polynomials
If all poles coincide at s = -a, then Kautz series can be written as,
[math]\displaystyle{ \phi_k(t) = \sqrt{2a}(-1)^{k-1}e^{-at}L_{k-1}(2at) }[/math],
where Lk denotes Laguerre polynomials.
See also
- Kautz code
References
- ↑ "Transient Synthesis in the Time Domain". I.R.E. Transactions on Circuit Theory 1 (3): 29–39. 1954.
- ↑ "Using Kautz Models in Model Reduction". Signal Analysis and Prediction. Birkhäuser. 1998. p. 187. ISBN 978-0-8176-4042-2. https://books.google.com/books?id=qk2LBkKg5zcC&pg=PA187.
- ↑ "Equalization of Loudspeaker and Room Responses Using Kautz Filters: Direct Least Squares Design". EURASIP Journal on Advances in Signal Processing (Hindawi Publishing Corporation) 2007: 1. 2007. doi:10.1155/2007/60949.
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