ON THE SYNTHESIS OF FRACTAL RADIATION-PATTERNS

The fundamental relationship between self-similar, that is, fractal, a rrays and their ability to generate radiation patterns which possess f ractal features is examined in this paper. The theoretical foundation and design procedures are developed for using fractal arrays to synthe size fractal radiation patterns having certain desired characteristics . A family of functions, known as generalized Weierstrass functions, a re shown to play a pivotal role in the theory of fractal radiation pat tern synthesis. These functions are everywhere continuous but nowhere differentiable and exhibit fractal behavior at all scales. It will be demonstrated that the array factor for a nonuniformly but symmetricall y spaced linear array can be expressed in terms of a Weierstrass parti al sum (band-limited Weierstrass function) for an appropriate choice o f array element spacings and excitations. The resulting fractal radiat ion patterns from these arrays possess structure over a finite range o f scales. This range of scales can be controlled by the number of elem ents in the array. For a fixed array geometry, the fractal dimension o f the radiation pattern may be varied by changing the array current di stribution. A general and highly flexible synthesis technique is intro duced which is based on the theory of Fourier-Weierstrass expansions. One of the appealing attributes of this synthesis technique is that it provides the freedom to select an appropriate generating function, in addition to the dimension, for a desired fractal radiation pattern. I t is shown that this synthesis procedure results in fractal arrays whi ch are composed of a sequence of self-similar uniformly spaced linear subarrays. Finally, a synthesis technique for application to continuou s line sources is presented which also makes use of Fourier-Weierstras s expansions.