Linear Phase Two-Dimensional FIR Digital Filter Functions Generated by applying Christoffel-Darboux Formula for Orthonormal Polynomials

摘要

Filter theory represents one of the strictest disciplines with the possibilities of applications in various frequency ranges and technologies [1-4]. In this theory, successful applications of powerful orthogonal polynomials are well-known [4-7]. A number of problems in various scientific and technical areas have been solved by applying the classical Christoffel-Darboux formula for all classic orthogonal polynomials [8, 9]. New class explicit filter functions for continuous signals generated by the classical Christoffel-Darboux formula for classical Jacobi and Gegenbauer orthonormal polynomials are described in detail [10, 11]. On the other hand, there have been a number of attempts to solve the complex problem of generating linear phase two-dimensional finite impulse response (FIR) digital filters of lower order, e.g. [12]. They are based on either a transformation of one-dimensional FIR filters or direct application of the approximation techniques in two dimensions. Further generalization of the previous research [10, 11] in two dimensions is presented in this paper. The global Christoffel-Darboux formula for four orthonormal polynomials on two equal finite segments for generating filter functions is proposed here in a compact explicit form. A new class of the linear phase two-dimensional FIR digital filters generated by the proposed formula is given.