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A simplified derivation of the Fourier coefficients for Chebyshev patterns

A simplified derivation of the Fourier coefficients for Chebyshev patterns

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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.

Inspec keywords: antenna theory

Subjects: Antennas

References

    1. 1)
      • H.E. Salzer . Note on the Fourier Coefficients for Chebyshev Patterns. Proc. I.E.E., Part C
    2. 2)
      • R.H. DuHamel . Optimum Patterns for Endfire Arrays. Proceedings of the Institute of Radio Engineers
    3. 3)
      • A. Erdélyi . (1953) , Higher Transcendental Functions.
    4. 4)
      • W. Gröbner , N. Hofreiter . (1950) , Integraltafel—Zweiter Teil—Bestimmte lntegrale.
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