Construction of conjugate quadrature filters with specified zeros

AbstractLet ℂ denote the complex numbers and $${\mathcal{L}}$$ denote the ring of complex-valued Laurent polynomial functions on ℂ\{0}. Furthermore, we denote by $${\mathcal{L}}_R ,{\mathcal{L}_N} $$ the subsets of Laurent polynomials whose restriction to the unit circle is real, nonnegative, respectively. We prove that for any two Laurent polynomials $$P_1 ,P_2 \in {\mathcal{L}}_N ,$$ , which have no common zeros in ℂ\{0} there exists a pair of Laurent polynomials $$Q_1 ,Q_2 \in {\mathcal{L}}_N $$ satisfying the equation Q1P1 + Q2P2 = 1. We provide some information about the minimal length Laurent polynomials Q1 and Q2 with these properties and describe an algorithm to compute them. We apply this result to design a conjugate quadrature filter whose zeros contain an arbitrary finite subset Λ⊂ℂ\{0} with the property that for every $$\lambda ,\mu \in \Lambda \lambda \ne \mu $$ implies $$\lambda \ne - \mu $$ and $$\lambda \ne - {1 \mathord{\left/ {\vphantom {1 {\bar \mu }}} \right. \kern-\nulldelimiterspace} {\bar \mu }}$$ .