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Signal design for maximizing the efficiency of the Neyman-Pearson detection procedure in randomly dispersive media is investigated. The medium is modeled as a randomly time-varying linear filter; by viewing the filter transfer function as a homogeneous random field on the time-frequency plane, a second-order theory results that relates various second-order measures of the time and frequency structures of input and output processes. A signal design strategy is developed that dictates transmitting signals that produce output processes with degrees of freedom possessing a signal-to-noise ratio (SNR) in the vicinity of 2. A distribution of signal energy in the output time-frequency plane that achieves the proper SNR for each degree of freedom is deduced and is used to infer constraints on input ambiguity functions that maximize detection efficiency. The general structure of efficient input signals for both high and low SNR is briefly discussed.