Kautz filter

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 ani 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 Lk denotes Laguerre polynomials.

References

References

1. (1954). "Transient Synthesis in the Time Domain". I.R.E. Transactions on Circuit Theory.
2. (1998). "Signal Analysis and Prediction". [[Birkhäuser]].
3. (2007). "Equalization of Loudspeaker and Room Responses Using Kautz Filters: Direct Least Squares Design". Hindawi Publishing Corporation.