On the synthesis of fractal radiation patterns

The fundamental relationship between self-similar, that is, fractal, arrays and their ability to generate radiation patterns which possess fractal features is examined in this paper. The theoretical foundation and design procedures are developed for using fractal arrays to synthesize fractal radiation patterns having certain desired characteristics. A family of functions, known as generalized Weierstrass functions, are shown to play a pivotal role in the theory of fractal radiation pattern 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 symmetrically spaced linear array can be expressed in terms of a Weierstrass partial sum (band-limited Weierstrass function) for an appropriate choice of array element spacings and excitations. The resulting fractal radiation patterns from these arrays possess structure over a finite range of scales. This range of scales can be controlled by the number of elements in the array. For a fixed array geometry, the fractal dimension of the radiation pattern may be varied by changing the array current distribution. A general and highly flexible synthesis technique is introduced 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. It is shown that this synthesis procedure results in fractal arrays which are composed of a sequence of self-similar uniformly spaced linear subarrays. Finally, a synthesis technique for application to continuous line sources is presented which also makes use of Fourier-Weierstrass expansions.