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Distributed detection and information fusion have received recent research interest due to the success of emerging wireless sensor network (WSN) technologies. For the problem of distributed detection in WSNs under energy constraints, a weak signal model in the canonical parallel fusion scheme with additive non-Gaussian noises and fading channels is considered. To solve this problem in the Neyman-Pearson setting, a unified asymptotic fusion rule generalizing the maximum ratio combiner (MRC) fusion rule is proposed. Explicit formulas for the threshold and detection probability applicable for wide classes of fading channels and noise distributions are written out. Both asymptotic analysis and Monte Carlo modeling are used to examine the performance of the proposed detection fusion rule.