title: Kautz filter
text: # 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 $\{-\alpha _{1},-\alpha _{2},\ldots ,-\alpha _{n}\}$, 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:
  
      $\Phi _{1}(s)={\frac {\sqrt {2\alpha _{1}}}{(s+\alpha _{1})}}$
      $\Phi _{2}(s)={\frac {\sqrt {2\alpha _{2}}}{(s+\alpha _{2})}}\cdot {\frac {(s-\alpha _{1})}{(s+\alpha _{1})}}$
      $\Phi _{n}(s)={\frac {\sqrt {2\alpha _{n}}}{(s+\alpha _{n})}}\cdot {\frac {(s-\alpha _{1})(s-\alpha _{2})\cdots (s-\alpha _{n-1})}{(s+\alpha _{1})(s+\alpha _{2})\cdots (s+\alpha _{n-1})}}$.
  
  In the time domain, this is equivalent to
  
      $\phi _{n}(t)=a_{n1}e^{-\alpha _{1}t}+a_{n2}e^{-\alpha _{2}t}+\cdots +a_{nn}e^{-\alpha _{n}t}$,
  
  where $a_{ni}$ are the coefficients of the partial fraction expansion as,
  
      $\Phi _{n}(s)=\sum _{i=1}^{n}{\frac {a_{ni}}{s+\alpha _{i}}}$
  
  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,
  
  $\phi _{k}(t)={\sqrt {2a}}(-1)^{k-1}e^{-at}L_{k-1}(2at)$, where $L_k$ denotes Laguerre polynomials.
  
  