A simplified derivation of the Fourier coefficients for Chebyshev patterns

In the design of linear arrays containing an odd number of elements with constant element spacing of less than a half wavelength, the mathematical problem reduces to that of finding explicitly the coefficients, bm, in the expansion where a and b are constants, n > 0, and Tm(x) is defined as cos (m arc cos x) for m ¿ 0. Such a problem was initially solved by DuHamel and later given by Salzer in a form more convenient for computation. The purpose of this paper is to give an alternative derivation of Salzer´s result, making use of the orthogonality properties the Chebyshev polynomials in order to obviate the fairly elaborate series manipulations required in the previous derivation.