Derivation of excitation coefficients for Chebyshev arrays

An arrangement of isotropic elements, uniformly spaced at a half wavelength or greater and suitably excited, was used by Dolph to obtain optimum radiation patterns for linear broadside arrays. Riblet generalized the technique and extended it to include arrays of odd numbers of elements with uniform spacing of less than a half wavelength. DuHamel demonstrated that the calculation of the excitation coefficients for an array of 2n + 1 elements is equivalent to that of determining the coefficients bm in the expression Tn(ax + b) = ¿m=0n bmTm(x) (n > 0), where Tm(X) = cos (m cos¿1 x), |x| ¿ 1, m ¿ 0, and a and b are constants. An expression of simpler and more convenient form than that of DuHamel´s formulae has been provided by Salzer, and rederived rather simply by Brown. It is the purpose of the paper to present two alternative formulae for bm. The first is recursive, consisting of a single summation with coefficients involving the sum of only two binomial coefficients, and of simpler form than that of Salzer. The second, also a single summation of reasonably simple quantities, is similar to a recently announced formulation by Brown, except that all the coefficients bm are obtainable from the one expression.