Christoffel-Darboux Formula Most Directly Applied in Generating Economical Linear Phase Low-Pass Digital FIR Filter Functions

Abstract

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.

Low complexity lowpass linear‐phase multiplierless FIR filter

“…As an example, the proposed FIR filter, for the concrete values of free real integer parameters, is designed by the proposed technique and its characteristics are illustrated. To verify the effectiveness of the proposed multiplierless linear-phase FIR filter, we have performed yet another experiment, comparing this filter with the already reported FIR filters in [1][2][3]. The latter class of filters may offer apparently an even lower spike in the passband and higher selectivity, confirming its superiority over other classical solutions.…”

“…Introduction: The three new classes of lowpass linear-phase FIR filter functions were described in our previous papers [1][2][3]. In addition, the new class of difference equations obtained using extremal properties of the Christoffel-Darboux formula for classical orthogonal continuous polynomials were presented.…”

“…It is verified that the proposed filter has high attenuation in the stopband of the filter. Comparison: To verify the effectiveness of the proposed filter function, we have performed one more experiment for the filter functions reported in [1][2][3]. The comparison for the same values of free integer parameters, N = 8 and K = 3, with the proposed filter function is performed in the lower part of the frequency range, and the obtained results are presented in Fig.…”

Ultra‐selective lowpass linear‐phase FIR filter function

“…Introduction: The extremal property of the Christoffel-Darboux formula for the continual orthogonal classical polynomial is described in [1]. In [2][3][4], the Christoffel-Darboux formula is applied in theory of continual filter functions given in an explicit form. The synthesis of linear-phase 1D FIR filters, with two free integer parameters, using the analytical method and direct mapping from the analogue into discrete domain, without error and remainders, is reported in [4].…”

“…In [2][3][4], the Christoffel-Darboux formula is applied in theory of continual filter functions given in an explicit form. The synthesis of linear-phase 1D FIR filters, with two free integer parameters, using the analytical method and direct mapping from the analogue into discrete domain, without error and remainders, is reported in [4]. The popular linear-phase cascaded integrator comb (CIC) filters are presented in [5].…”