Construction of conjugate quadrature filters with specified zeros

Let ℂ denote the complex numbers and ℒ denote the ring of complex-valued Laurent polynomial functions on ℂ \ {0}. Furthermore, we denote by ℒR, ℒN the subsets of Laurent polynomials whose restriction to the unit circle is real, nonnegative, respectively. We prove that for any two Laurent polynomials P1, P2 ∈ ℒN, which have no common zeros in ℂ \ {0} there exists a pair of Laurent polynomials Q1, Q2 ∈ ℒN 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 λ, μ ∈ Λ, λ ≠ μ implies λ ≠ -μ and λ ≠ -1/μ̄.