A generalized approach to the study of frequency-independent antennas is presented which relies on the recently developed theory of fractal geometry. It is demonstrated that this fractal geometric interpretation allows for the ability to characterize a much wider class of frequency-independent antennas. This includes radiating structures which are self-similar in the discrete sense, the smooth sense, and even the “rough” sense. The antenna configurations in this paper are all self-similar and have been parameterized in terms of a common similarity factor of τ. Finally, it is shown how this new theory of self-similar fractal radiators may be employed to develop a multiband linear array design methodology for which the directive gain is a log-periodic function of frequency.