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One of frequency translates of a common signal is sent over a fading channel and is received in additive white noise. The receiver decides which signal was sent using energy measurements only. It is shown that as is allowed to become very large, the performance depends only on the probability distribution of received energy per bit, and that this limiting behavior is the same as for optimum detection based on full knowledge of instantaneous channel properties. If sufficiently many independent copies of the channel can be achieved (i.e., independent diversity branches), the performance can be made to approach as closely as desired that of the nonfading infinite-bandwidth channel, independently of the exact channel statistics.