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The self-complementary antenna structure was originated and its constant-impedance property was discovered by the author in 1948. He pursued investigations of this type of antenna for many years, and he attained many extensions of the principle of self-complementarity, from the simplest planar structure to various other cases. In parallel with these studies, extensive developmental investigations of extremely broadband antennas have been carried out in Japan, based on this principle. This article succinctly describes a long history of these studies on self-complementary antennas, including the background of its origination. In connection with the extremely broadband property of this type of antenna, the non-constant-impedance property of incorrectly arranged log-periodic antennas is clearly shown, based on the results of experiments. This experimental fact indicates that the log-periodic shape in an antenna's structure does not guarantee a broadband property for the antenna. Most of experimental details and all of the theoretical treatments are omitted from this article.