A New Representation of Hurwitz's Determinants in the Expansion of Certain Ladder Filters

Abstract

Given a real strictly Hurwitz polynomial H_{n}(s) = a_{0}\prod_{\upsilon = 1}^{n} (s - s_{\upsilon}), n = 3, 4, \cdots , the standard method of calculating the continued fraction expansion of \{[odd H_{n}(s)]/ [even H_{n}(s)]\}^{\pm 1} about its pole at infinity uses Routh's scheme or Hurwitz's determinants \Delta_{r}, r = 1, 2, \cdots , n , in the coefficients of H_{n}(s) (on the equivalence of the two, see [2]). In filter theory, cases are often encountered where knowledge of the zeros of H_{n}(s) precedes that of its coefficients, and one would then prefer to have formulas for the coefficients in the above continued fraction expansion directly in terms of the former rather than the latter. This is achieved by expressing \Delta_{r} as bialternants in the zeros of H_{n}(s) and reads \Delta_{r} = (-)^{r(r+1)/2} a_{0}^{r}A(0, 1, \cdots , n - r - 1, n - r + 1, \cdots , n + r - 1)/A(0, 1, \cdots , n - 1) , where the alternant in the denominator is the Vandermonde in s_{1}, s_{2}, \cdots , s_{n} , whereas the alternant in the numerator is obtained from it on replacing the exponents 0, 1, \cdots , n - 1 by 0, 1, \cdots, n - r - 1, n - r + 1, \cdots , n + r - 1 . Examples include H_{n}(s) = \prod_{\upsilon = 1}^{n} [s - j \exp (2 \upsilon - 1)j \pi /2n] and H_{n}(s) = (s + 1)^{n} .