Construction of Conjugate Quadrature Filters with Specified Zeros

Let doubleC denote the complex numbers and L denote the ring of complex-valued Laurent polynomial functions on doubleC \ {0}. Furthermore, we denote LsubR, LsubN the subsets of Laurent polynomials whose restriction to the unit circle is real, nonnegative, respectively. We prove that for any two Laurent polynomials Psub1, Psub2 &memberof. LsubN, which have no common zeros in doubleC \ {0} there exists a pair of Laurent polynomials Qsub1, Qsub2 &memberof. LsubN satisfying the equation Qsub1Psub1 + Qsub2Psub2 = 1. We provide some information about the minimal length Laurent polynomials Qsub1 and Qsub2 with theses 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 A subset doubleC \ {0} with the property that for every lambda, mu &memberof. A, lambda notequalto mu implies lambda notequalto - mu and lambda notequalto - 1/line over mu.