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We study the performance of partially coherent and noncoherent fusion rules in a distributed sensor network. We assume that sensors communicate their local (binary) decisions over a Rayleigh flat-fading channel to a central fusion center and treat separately the cases where the channel gain is known and unknown. Our analysis is motivated by the case that the channel signal to noise ratio (SNR) is low and the number of sensor observations is large. For partially coherent strategies, we assume that the residual phase error is described by the Tikhonov probability distribution function. We use the Central Limit Theorem to characterize the receiver operating characteristic (ROC) and the Deflection performance of this network. Our study is carried out using analytic techniques complemented by numerical simulations. Our main contribution is an analysis that allows the system designer to determine the level of phase asynchrony that must be present before noncoherent strategies outperform partially coherent ones.