A New Representation of Hurwitz's Determinants in the Expansion of Certain Ladder Filters
Israel Navot · IEEE Transactions on Circuit Theory · 1968
Given a real strictly Hurwitz polynomialH_{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 ofH_{n}(s)(on the equivalence of the two, see [2]). In filter theory, cases are often encountered where knowledge of the zeros ofH_{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 ofH_{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 ins_{1}, s_{2}, \cdots , s_{n}, whereas the alternant in the numerator is obtained from it on replacing the exponents0, 1, \cdots , n - 1by0, 1, \cdots, n - r - 1, n - r + 1, \cdots , n + r - 1. Examples includeH_{n}(s) = \prod_{\upsilon = 1}^{n} [s - j \exp (2 \upsilon - 1)j \pi /2n]andH_{n}(s) = (s + 1)^{n}.