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Solutions are presented for the optimal electric field waveforms radiated by an arbitrary ultrawideband (UWB) antenna. Optimization criteria include maximization of the electric field amplitude at a particular time and location, or maximization of energy density over a specified time interval at a particular location. Assuming bandpass signals, constraints are placed on the total radiated energy, the Q of the antenna, and the size of the antenna. The solution is developed using a spherical mode expansion of the fields radiated by an arbitrary antenna enclosed by a spherical mathematical surface, and optimized using variational methods. A closed-form result is obtained for the case of amplitude maximization, while an integral equation must be solved numerically for the case of energy maximization in a time interval. An interesting result from these solutions is that the shapes of the optimal radiated field waveforms are largely independent of the size of the antenna. The solutions also indicate that the antenna characteristics that provide optimum field amplitude or energy in the transient case are identical to those associated with maximum gain in the CW case.