- Source: Kautz filter
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
{
−
α
1
,
−
α
2
,
…
,
−
α
n
}
{\displaystyle \{-\alpha _{1},-\alpha _{2},\ldots ,-\alpha _{n}\}}
, 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:
Φ
1
(
s
)
=
2
α
1
(
s
+
α
1
)
{\displaystyle \Phi _{1}(s)={\frac {\sqrt {2\alpha _{1}}}{(s+\alpha _{1})}}}
Φ
2
(
s
)
=
2
α
2
(
s
+
α
2
)
⋅
(
s
−
α
1
)
(
s
+
α
1
)
{\displaystyle \Phi _{2}(s)={\frac {\sqrt {2\alpha _{2}}}{(s+\alpha _{2})}}\cdot {\frac {(s-\alpha _{1})}{(s+\alpha _{1})}}}
Φ
n
(
s
)
=
2
α
n
(
s
+
α
n
)
⋅
(
s
−
α
1
)
(
s
−
α
2
)
⋯
(
s
−
α
n
−
1
)
(
s
+
α
1
)
(
s
+
α
2
)
⋯
(
s
+
α
n
−
1
)
{\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})}}}
.
In the time domain, this is equivalent to
ϕ
n
(
t
)
=
a
n
1
e
−
α
1
t
+
a
n
2
e
−
α
2
t
+
⋯
+
a
n
n
e
−
α
n
t
{\displaystyle \phi _{n}(t)=a_{n1}e^{-\alpha _{1}t}+a_{n2}e^{-\alpha _{2}t}+\cdots +a_{nn}e^{-\alpha _{n}t}}
,
where ani are the coefficients of the partial fraction expansion as,
Φ
n
(
s
)
=
∑
i
=
1
n
a
n
i
s
+
α
i
{\displaystyle \Phi _{n}(s)=\sum _{i=1}^{n}{\frac {a_{ni}}{s+\alpha _{i}}}}
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,
ϕ
k
(
t
)
=
2
a
(
−
1
)
k
−
1
e
−
a
t
L
k
−
1
(
2
a
t
)
{\displaystyle \phi _{k}(t)={\sqrt {2a}}(-1)^{k-1}e^{-at}L_{k-1}(2at)}
,
where Lk denotes Laguerre polynomials.
See also
Kautz code
References
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