Theory of unequally-spaced arrays

Although recently unequally-spaced arrays have been shown to be useful, the theory has not been fully developed, except for the use of matrices, computers, or the perturbation method. This paper presents a new approach to the unequally-spaced array problem. It is based on the use of Poisson's sum formula and the introduction of a new function, the "source position function." By appropriate transformation, the original radiation pattern is converted into a series of integrals, each of which is equivalent to the radiation from a continuous source distribution whose amplitude and phase distribution clearly exhibit the effects of the unequal spacings. By this method, it is possible to design unequally-spaced arrays which produce a desired radiation pattern. This method is effective in treating arrays of a large number of elements, and unequally-spaced arrays on a curved surface. Three examples are shown to illustrate the effectiveness of the method. The problem of sidelobe reduction for the array of uniform amplitude, which was attacked by Harrington, is treated by our method. A numerical example is shown for 25-db sidelobe level. Also, the problem of secondary beam suppression is attacked with the use of the Anger function. The interesting problem of azimuthal frequency scanning by means of an unequally-spaced circular array is also shown, using the method of stationary phase.