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One way for an antenna to achieve an ultra-wideband (UWB) performance is to employ a log-periodic array (LPA), which usually involves many scaled elements cascaded one after another. If an LPA is infinite at both ends (infinitely small at one end and extended to infinitely large at the other end), it is obvious that the input impedance of the array is periodical over frequencies. However, for practical antennas, infinite LPAs have to be truncated. The purely periodical performance will not hold anymore. On the other hand, UWB LPA antennas are often very large in terms of the wavelength at the highest operating frequency, which makes numerical simulation very time consuming. The author presents a theoretical analysis of the periodicity of the input impedance of a general finite LPA. New periodicity formulas are verified by examples of the Eleven antenna - a folded dipole LPA, with simulated and measured data. By using the new periodicity formula, the input impedance of a large LPA antenna at higher frequencies can be predicted by its values at lower frequencies, which leads to an efficient calculation when a numerical simulation is employed, and helps to have an efficient design of large LPA antennas.