A Two-Parameter Family of Weights for Nonrecursive Digital Filters and Antennas
We derive analytically a two-parameter family of weights for use in finite duration nonrecursive digital filters and in finite aperture antennas. This family of weights is based on the Gegenbauer orthogonal polynomials, which are a generalization of both Legendre and Chebyshev polynomials. It is shown that one parameter controls the main lobewidth and the other parameter controls the sidelobe taper. For a fixed main lobewidth, it is observed that the Gegenbauer weights can achieve a dramatic decrease in sidelobes "far removed" from the main lobe in exchange for a "small" increase in the first sidelobe adjacent to the main lobe. The Gegenbauer weights are derived first for discretely sampled apertures and filters. An appropriate limit is then taken to produce the Gegenbauer weighting function for continuously sampled apertures and filters. The continuous Gegenbauer weighting function contains the Kaiser-Bessel function as a special case. It is thus established that the Kaiser-Bessel function is implicitly based on Chebyshev polynomials of the second kind. Furthermore, the Dolph-Chebyshev/van der Maas weights are a limiting case of the discrete/continuous Gegenbauer weights.