مرکز منطقه ای اطلاع رساني علوم و فناوري - 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 $H_{n}(s) = (s + 1)^{n}$ .

Keywords :

Alternants and bialternants; Continued-fraction expansion of ladder filters; Hurwitz determinants; Ladder filters; Realizations; Argon; Circuit theory; H infinity control; Influenza; Microwave circuits; Microwave filters; Network synthesis; Polynomials; Reflection; Transfer functions;