Construction of conjugate quadrature filters with specified zeros
Let \({\mathcal L}\) denote the ring of complex-valued Laurent polynomial functions on \(\mathbb{C} \backslash \{0\}\). Let \({\mathcal L}_R \subset {\mathcal L}\) denote the set of Laurent polynomials whose restriction to the unit circle is real valued and let \({\mathcal L}_N \subset {\mathcal L}\) be the set of Laurent polynomials whose restriction to the unit circle is non-negative. The authors prove that for any two Laurent polynomials \(P_1\), \(P_2\) \(\in {\mathcal L}_N\), which have no common zeros in \(\mathbb{C} \backslash \{0\}\) there exists a pair of Laurent polynomials \(Q_1\), \(Q_2 \in {\mathcal L}_N\) satisfying the equation \(Q_1 P_1 + Q_2 P_2 = 1\). The authors provide some information about the minimal length Laurent polynomials \(Q_1\) and \(Q_2\) with this property and describe an algorithm to compute them. The authors apply this result to design a conjugate quadrature filter whose zeros contain an arbitrary finite subset \(\Lambda \subset C \backslash \{0\}\) with the property that \((\Lambda \cup \bar {\Lambda}^{-1}) \cap (\Lambda \cup \bar {\Lambda}^{-1}) = \emptyset\).
scientific article; zbMATH DE number 1078346