Sidelobe Reduction of the Ring Array for Use in Circularly Symmetric Imaging Systems

Abstract

The ring or annulus array offers certain attractive properties for use in circularly symmetric imaging systems. Its excitation function is defined by

$$ \text{i}\left( \text{r},\phi \right)\,=\,\text{a}{{\text{e}}^{\text{j}\phi }}\sum\limits_{\text{n=0}}^{\text{n=N-1}}{\text{ }\!\!\delta\!\!\text{ }\left( \phi \,-\,\frac{\text{2 }\!\!\pi\!\!\text{ n}}{\text{N}} \right)} $$

(1)

ae^jφis a continuous annulus of radius a while the second term is the sum of Dirac delta-functions that are nonzero at discrete angles 2πn/N (N is the number of elements in the array and n = 0,1,N-1). Its attractive properties are:

• 1. The underlying circular symmetry of the ring results in nearly circular symmetry on the radiation sphere, i.e., except for the discreteness of the array the radiation pattern is only a function of the polar angle θ and not the azimuthal angle φ. This property is generally advantageous in a scanning or imaging system in which there is no preferred azimuth.

• 2. Although the angular spacing of elements on the ring is periodic, as given in (1), the apparent spacing from any direction of view other than the zenith is uniform. This nonuniformity of apparent element spacing permits thinning the array, i.e., mean interelement spacing > λ/2, where λ is the wavelength, without introducing grating lobes (à la the interferometer) [*]. This property is advantageous because it permits a larger aperture and therefore a finer resolving power from a fixed system cost. The reasoning is as follows: given a limited number N of elements and accompanying circuits associated with a given system cost, the size L of the square array having λ/2 element spacing is $\text{ }\!\!\lambda\!\!\text{ }\frac{\sqrt{\text{N}}}{\text{2}}$ The available beamwidth Δθ always is about λ/L, which equals $\frac{\text{2}}{\sqrt{\text{N}}}$, and is limited, therefore, by N. Furthermore, the resolving power improves only as N^1/2. A bilinear crossed array of length L has a smaller beam cross-section: 2L = Nλ/2 and Δθ ≃ 4/N. The periodic ring array is still better because the λ/2 spacing limitation is removed.

• 3. Because of the circular symmetry, beamforming and scanning phase shifts and/or time delays need only be calculated for a single angular interval 2π/N. The same set of values can be made to pertain to every sector by rotating the angular frame of reference (in the computer or signal processor) in steps of 2π/N. This property can result in a significant reduction in the data handling requirement in a digital processing system.

• 4. The beamwidth of a ring array is theoretically smallest for a given size aperture for which the requirement of circular symmetry is imposed. This is because the elements are disperses most widely from the center subject to the given condition. This property leads to the finest available resolving power from a circularly symmetric aperture of fixed size.

This work was principally supported by Interspec, Inc. and by the Office of Naval Research, the latter under Contract No. N00014-79-C-0505.