Moment Synthesis of Array Factors With Nonuniform Spacing and Amplitude Parameters

A theory of nonuniform arrays based upon the classical moment problem is developed in this paper. We suppose that a given pattern can be represented as a Fourier‐Stieltjes transform of a cumulative current distribution over a finite (normalized) aperture interval, which we then approximate by a step‐function distribution whose “treads and risers” are determined as follows. The moments of the step‐distribution are equated to the moments of the continuous distribution belonging to the given pattern, producing a set of nonlinear equations for the positions (treads) and amplitudes (risers) of the steps. The former tum out to be roots of an orthogonal polynomial, of degree equal to number of array elements, belonging to the (continuous) cumulative distribution; the latter can then be found from a resulting set of linear equations and are the so‐called Christoffel numbers. An equivalent development, which helps to illuminate the basic ideas, is to approximate the kernel of the Fourier‐Stieltjes transform by a Lagrange interpolation polynomial. Both of these approaches will produce, in general, an array factor with both nonuniform spacings and amplitudes, and will be said to be optimum in a moment sense: i.e., the original distribution and its approximation agree in their first 2M moments if M is the number of elements used for the approximation. The equivalence of the moment and interpolation methods implies a mean‐square minimality for the polynomial factor of an error representation for the latter method. But the remaining factor of the error representation involves a parameter which cannot in general be determined explicitly; this error form is therefore of little use for large apertures (compared to the number of elements involved). Practical application of moment synthesis would therefore have to rely for an error assessment largely upon either statistical estimation or pattern computation. These facts notwithstanding, the moment method can be used to produce remarkable pattern approximations in the near main‐beam region, and, if the number of elements is sufficiently large, acceptable pattern behavior everywhere. Its chief drawback is that a large number of elements requires the computation of zeros of a orrespondingly large degree polynomial.