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A NEW CLASS OF SIGNATURE SYMMETRIC REALIZATIONS

P.A. Fuhrmann  U. Helmke
Department of Mathematics  Department of Mathematics
Ben Gurion University of the Negev  University of Regensburg
Beer Sheva, Israel  Regensburg, Germany

November 1993 1. Introduction Segal has described a remarkable connection between real rational transfer functions of McMillan degree 2n and complex transfer functions of McMillan degree n. The connection enables one to parametrize real rational transfer functions by complex ones of half of the degree, thus providing a particularly efficient coding of the real parameters in terms of complex coefficients. In this paper we will extend and refine Segal’s construction by describing it on the level of realizations. Let 𝔽 = ℝ, ℂ denote the field of real or complex numbers respectively. Let Rat(n, 𝔽) denote the set of rational transfer functions

g(s) = \frac{p(s)}{q(s)}, p(s), q(s) ∈ 𝔽[s],

of McMillan degree n which are strictly proper. Thus g(∞) = 0 and p(s), q(s) are coprime polynomials, q(s) monic, with deg p < deg q = n. Using the coefficients of p(s) and q(s) as parameters, the space Rat(n, 𝔽) is a Zariski-open subset of 𝔽^{2n}, being the complement of the algebraic subvariety consisting of pairs of polynomials (p, q) with vanishing resultant. The set Rat(n, ℂ) is connected while Rat(n, ℝ) has several connected components. Brockett has shown that Rat(n, ℝ) splits into n+1 connected components Rat(k, ℓ), k + ℓ = n, k ≥ 0, ℓ ≥ 0, which are dinstinguished by the Cauchy-Index CI(g) = k − ℓ. Here CI(g) can be interpreted as the winding number of the curve

α_{pq}: ℝ ⟶ ℂt ⟼ q(t) + ip(t)