Computation of the far field from a nonuniformly spaced antenna elements using a least square method

The objective of this paper is to describe a least square method to compute the far field from a nonuniformly spaced antenna array. This is because the far field pattern of an antenna array is the Fourier transform of the current distribution developed on the antenna structure, This is equivalent to computing the spectrum from a nonuniformly spaced data set using a least squares method. The merit of analyzing a nonuniformly spaced data set is that one need not satisfy the Nyquist sampling theorem in the strict sense. This is acceptable as long as the average sampling rate for the nonuniformly spaced data satisfies the Nyquist sampling theorem. A Fourier based least squares method can easily be extended to deal with nonuniformly spaced data, however the disadvantage of this procedure is that it is highly computational time intensive. Hence in this approach a Hilbert transform based methodology is described to significantly reduce the computation time without sacrificing accuracy. Hence, in methodology, this method is an extension to the Lomb periodogram technique which deals with the computation of the spectrum from a nonuniformly based data set. The problem with the Lomb periodogram method is that it cannot differentiate between the positive and the negative frequencies of the spectrum even though it has various other advantages. For example the effect of bias in the processed data can be reduced significantly in addition for a nonuniformly spaced data. Also because of a nonuniformly sampled data it is difficult to correctly estimate the amplitude of the far field or equivalently the spectrum hence a two-step methodology will be presented where at the first stage one solves for the spectral components and then applies a least squares solution procedure to get the best amplitude of the spectrum or equivalently the complex amplitudes for the far field. The Hilbert transform methodology is applied to reduce the computation time by a factor of one half in the computation of the complex spectrum as the real and imaginary parts of a causal function are indeed related by the Hilbert transform.