مرکز منطقه ای اطلاع رساني علوم و فناوري - The optimum line source for the best mean-square approximation to a given radiation pattern

Abstract :

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 $\\gamma$ . 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 $P(frac{\\pi a}{\\lambda } \\sin \\theta) = \\sum \\min{n=0} \\max {\\infty } a_{n}S_{0n}(c,\\sin \\theta)$ , and the resulting optimum aperture distribution by $a(frac{2x}{a}) = \\sum \\min{n=0} \\max {\\infty } frac {\\pi i^{-n}} {R_{0n}^{(1)}(c,1)} a_{n}S_{0n}(c,frac{2x}{a})$ , where the eigenfunctions $S_{0n}(c, \\eta)$ of the finite Fourier transform are the angular prolate spheroidal wave functions, $R_{0n}^{(1)}(c, 1)$ are the radial prolate spheroidal wave functions evaluated at unity, $a$ is the aperture length, $c=\\pi a/\\lambda$ and the expansion coefficients $a_{n}$ are $a_{n} = frac{b_{n}}{1+\\mu(\\lambda _{n}^{-1}-\\gamma )}$ ; $b_{n}$ are the expansion coefficients of the given radiation pattern, the eigenvalues $\\lambda _{n}$ are $(2c/\\pi) [R_{0n}^{(1)} (c, 1)]^{2}$ , $\\mu$ is a unique positive number satisfying ${\\infty \\atop {\\sum \\atop n=0}} {{k_n|b_{n}|^{2} \\over {\\lambda}_{n}^{-1} - \\gamma } \\over \\left( \\mu + {1 \\over {\\gamma}_{n}^{-1} - \\gamma} \\right)^{2}} = 0$ , and $k_{n}$ is the normalization factor for the eigenfunctions on (-1, 1). The pattern approximation is determined largely by the first $(2a/\\lambda )+1$ terms of its expansion, beyond which the expansion converges quickly for practical values of the superdirective ratio.