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The approximation problem of a filter function of even and odd order is solved mathematically in this paper most directly applying the proposed Christoffel-Darboux formula for two continual orthogonal polynomials on the equal finite segment. As a result, a linear phase low-pass digital finite impulse response (FIR) filter function is generated in compact explicit form. In addition, a new difference equation and structure of digital FIR filter are proposed. Two examples of the extremely economic FIR filters (with four adders and without multipliers) designed by the proposed technique are presented. The proposed solutions are efficient in regard to energy consumption and have a high selectivity.

A new original formulation of all pole low-pass filter functions is proposed in this article. The starting point in solving the approximation problem is a direct application of the Christoffel-Darboux formula for the set of orthogonal polynomials, including Gegenbauer orthogonal polynomials in the finite interval [71, þ1] with the application of a weighting function with a single free parameter. A general solution for the filter functions is obtained in a compact explicit form, which is shown to enable generation of the Gegenbauer filter functions in a simple way by choosing the value of the free parameter. Moreover, the proposed solution with the same criterion of approximation could be used to generate Legendre and Chebyshev filter functions of the first and second kind as well. The examples of proposed filter functions of even (10th) and odd (11th) order are illustrated. The approximation is shown to yield a good compromise solution with respect to the filter frequency characteristics (magnitude as well as phase characteristics). The influence of tolerance of the filter critical component (inductor) on the proposed magnitude and group delay characteristics of a resistively terminated LC lossless ladder filter is analysed as well. The proposed filter functions are superior in terms of the excellent magnitude characteristic, which approximates an ideal filter almost perfectly over the entire pass-band range and exhibits the summed sensitivity function better than that of a Butterworth filter. In the article, we present the filter function solution that exhibits optimum amplitude as well as optimum group delay characteristics that are of crucial importance for implementation of digital processing as well as RF analogue parts of communication networks. Derivation of the other band range filter functions, which could be realised either by continuous or digital filters, is also generally possible with the procedure proposed in this article.

The new originally capital general solution of determining the prototype filter function as the response that satisfies the specifications of all pole low-pass continual time filter functions of odd and even order is presented in this article. In this article, two new classes of filter functions are proposed using orthogonal and orthonormal Jacobi polynomials. The approximation problem of filter function was solved mathematically, most directly applying the summed Christoffel-Darboux formula for the orthogonal polynomials. The starting point in solving the approximation problem is a direct application of the Christoffel-Darboux formula for the initial set of continual Jacobi orthogonal polynomials in the finite interval ½À1, þ1 in full respect to the weighting function with two free real parameters. General solution of the filter functions is obtained in a compact explicit form, which is shown to enable generation the Jacobi filter functions in a simple way by choosing the numerical values of the free real parameters. For particular specifications of free parameters, the proposed solution is used with the same criterion of approximation to generate the appropriate particular filter functions as are: the Gegenbauer, Legendre and Chebyshev filter functions of the first and second kind as well. The examples of proposed filter functions of even and odd order are illustrated and compared with classical solutions.