Chained-Function Filter Synthesis Based on the Legendre Polynomials

Abstract

A new kind of filter called Legendre chained-function (LCF) filter with characteristic function given by the product of low-degree Legendre orthogonal polynomials (seed functions) is studied in this paper. LCF filter magnitude response exhibits unequal ripple level in the passband compared to Chebyshev chained-function filter with identical edge ripple factor at the passband edge. A proper combination of seed functions, i.e., a product of them, is used to control the maximum ripple in the passband, which affects the return loss, selectivity and a group delay characteristics of a filter. By selecting one of the possible combinations of seed functions (thus obtaining various degrees of freedom in filter design), filters with improved performances compared to traditional approximation techniques can be obtained. The degrees of freedom increase if the degree of filter increases. Compared to existing Chebyshev chained-function (CCF) filters, whose performances are also presented, and Butterworth function filters as a special case of both LCF and CCF filters, the new family of LCF filters has many advantages. A table summarizing properties of CCF and LCF filters is given for design purpose.