In signal processing, the Kautz filter, named after
William H. Kautz 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.
Like Laguerre filters, Kautz filters can be implemented using a cascade ...
, is a fixed-
pole traversal
filter, published in 1954.
Like
Laguerre filter
Edmond Nicolas Laguerre (9 April 1834, Bar-le-Duc – 14 August 1886, Bar-le-Duc) was a French mathematician and a member of the Académie des sciences (1885). His main works were in the areas of geometry and complex analysis. He also investigate ...
s, Kautz filters can be implemented using a cascade of
all-pass filter
An all-pass filter is a signal processing filter that passes all frequencies equally in gain, but changes the phase relationship among various frequencies. Most types of filter reduce the amplitude (i.e. the magnitude) of the signal applied to it ...
s, 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
In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform
In mathematics, an integral transform maps a function from its original function space into another function space via integra ...
of the Kautz
orthonormal
In linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal (or perpendicular along a line) unit vectors. A set of vectors form an orthonormal set if all vectors in the set are mutually orthogonal and all of un ...
basis is defined as the product of a one-pole lowpass factor with an increasing-order allpass factor:
:
:
:
.
In the time domain, this is equivalent to
:
,
where ''a
ni'' are the coefficients of the
partial fraction expansion
In algebra, the partial fraction decomposition or partial fraction expansion of a rational fraction (that is, a fraction such that the numerator and the denominator are both polynomials) is an operation that consists of expressing the fraction as ...
as,
:
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,
,
where ''L
k'' denotes
Laguerre polynomial In mathematics, the Laguerre polynomials, named after Edmond Laguerre (1834–1886), are solutions of Laguerre's equation:
xy'' + (1 - x)y' + ny = 0
which is a second-order linear differential equation. This equation has nonsingular solutions only ...
s.
See also
*
Kautz code
The reflected binary code (RBC), also known as reflected binary (RB) or Gray code after Frank Gray, is an ordering of the binary numeral system such that two successive values differ in only one bit (binary digit).
For example, the representati ...
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