The optimum line source for the best mean-square approximation to a given radiation pattern (1963)

An optimum aperture distribution for pattern shaping by a continuous line source of arbitrary length is derived in terms of the functions most natural to a least-squares fit: the eigenfunctions of the finite Fourier transform. It is expressed as an explicit function of Taylor's superdirective ratio <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\gamma</tex> . The new distribution produces the best mean-square approximation to a specified radiation pattern that is possible to obtain from an aperture of a given length for a given value of the superdirective ratio. The best mean-square pattern approximation is shown to be represented exactly by the orthogonal expansion <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">P(\frac{\pi a}{\lambda} \sin \theta) = \sum \min{n=0} \max{\infty} a_{n}S_{0n}(c,\sin \theta)</tex> , and the resulting optimum aperture distribution by <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a(\frac{2x}{a}) = \sum \min{n=0} \max{infty} \frac {\pi i^{-n}} {R_{0n}^{(1)}(c,1)} a_{n}S_{0n}(c,\frac{2x}{a})</tex> , where the eigenfunctions <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S_{0n}(c, \eta)</tex> of the finite Fourier transform are the angular prolate spheroidal wave functions, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R_{0n}^{(1)}(c, 1)</tex> are the radial prolate spheroidal wave functions evaluated at unity, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a</tex> is the aperture length, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c=\pi a/\lambda</tex> and the expansion coefficients <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a_{n}</tex> are <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a_{n} = \frac{b_{n}}{1+\mu(\lambda_{n}^{-1}-\gamma)}</tex> ; <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b_{n}</tex> are the expansion coefficients of the given radiation pattern, the eigenvalues <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\lambda_{n}</tex> are <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(2c/\pi) [R_{0n}^{(1)} (c, 1)]^{2}</tex> , <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\mu</tex> is a unique positive number satisfying <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\sum \min{n=0}\max{\infty}\frac{\frac{k_n}|b_{n}|^{2}} {\lambda_{n}^{-1}-\gamma}} {(\mu + \frac{1}{\gamma_{n}^{-1}-\gamma)^{2}}=0</tex> , and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k_{n}</tex> is the normalization factor for the eigenfunctions on (-1, 1). The pattern approximation is determined largely by the first <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(2a/\lambda)+1</tex> terms of its expansion, beyond which the expansion converges quickly for practical values of the superdirective ratio.