A New Design Methodology for Modular Broadband Arrays Based on Fractal Tilings

Open Access
Author:
Kuhirun, Waroth
Graduate Program:
Electrical Engineering
Degree:
Doctor of Philosophy
Document Type:
Dissertation
Date of Defense:
June 13, 2003
Committee Members:
  • Douglas Henry Werner, Committee Chair
  • Raj Mittra, Committee Member
  • James Kenneth Breakall, Committee Member
  • Pingjuan Li Werner, Committee Member
  • Brian Lewis Weiner, Committee Member
Keywords:
  • Fractal
  • Broadband Arrays
  • Irregular
  • Fractile
Abstract:
In this thesis, a new and innovative technique based on the theory of fractal tilings is introduced for the design of modular broadband arrays. These arrays are unique in the sense that they possess a fractal boundary contour that tiles the plane without gaps or overlaps. The first of these new array configurations that will be considered is directly related to the family of space-filling and self-avoiding fractals known as Peano-Gosper curves. The elements of the fractal array are uniformly distributed along a Peano-Gosper curve, which leads to a planar array configuration with parallelogram cells that is bounded by a closed Koch curve. These unique properties are exploited in order to develop a design methodology for deterministic arrays that have no grating lobes even when the minimum spacing between elements is increased to at least one-wavelength. This leads to a class of arrays that are relatively broadband when compared to more conventional periodic planar arrays with square or rectangular cells and regular boundary contours. This type of fractal array differs fundamentally from other types of fractal array configurations that have been studied previously that have regular boundaries with elements distributed in a fractal pattern on the interior of the array. An efficient iterative procedure for calculating the radiation patterns of these Peano-Gosper fractal arrays to arbitrary stage of growth P is also introduced in this thesis. Moreover, we note that Peano-Gosper arrays are self-similar since they may be formed in an iterative fashion such that the array at stage P is composed of seven identical stage P-1 sub-arrays (i.e., they consist of arrays of arrays). This lends itself to a convenient modular architecture whereby each of these sub-arrays could be individually controlled. In other words, the unique arrangement of tiles forms sub-arrays that could be designed to support simultaneous multibeam and multifrequency operation. Finally, several other examples of fractal tilings that lead to broadband array configurations will be considered including terdragon and 6-terdragon arrays. This thesis also introduces several new self-scalable arrays that can be generated by repeated application of a ring subarray generator, including pentagonal, octagonal, and honeycomb arrays. These arrays have the advantage that they can be recursively generated, allowing development of rapid algorithms for calculating their radiation patterns. They are also shown to possess relatively low sidelobe levels. Lastly, the radiation characteristics of some basic three-dimensional volumetric fractal arrays generated using concentric sphere subarrays will be briefly considered.