A Class of Optimal Polynomials Related to Electric Band-Pass Filters
Edwin Cohen, Henry Ruston · SIAM Journal on Applied Mathematics · 1974
A class of optimal polynomials is obtained satisfying constraints encountered in the design of electric band-pass filters having two levels of permissible attenuation in the pass band. Mathematically speaking, the x-axis is divided into six adjacent intervals, namely : (1) $( - \infty ,0)$ with no constraints; (2) $(0,x_{ - s} )$ and $(x_s ,\infty )$ with lower bound constraints ; (3) $(x_ - ,x_ + )$ with upper bound constraints ; and (4) $(x_{ - s} ,x_ - )$ and $(x_ + ,x_s )$, called the transition regions, with no constraints, where $0 < x_{ - s} < x_ - < 1 < x_ + < x_s < \infty $. For a given degree, necessary and sufficient conditions are determined for the optimal polynomial which minimizes the width of the transition regions. These conditions turn out to be sufficient to characterize the optimal polynomial uniquely. An algorithm is also presented for deriving the optimal polynomial. The sequence of polynomials obtained in the iterations is proved to converge uniformly to the sought polynomial.