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We consider a decentralized detection problem in a power-constrained wireless sensor networks (WSNs), in which a number of sensor nodes collaborate to detect the presence of a deterministic vector signal. The signal to be detected is assumed known a priori. Each sensor conducts a local linear processing to convert its observations into one or multiple messages. The messages are conveyed to the fusion center (FC) by an uncoded amplify-and-forward scheme, where a global decision is made. Given a total network transmit power constraint, we investigate the optimal linear processing strategy for each sensor. Our study finds that the optimal linear precoder has the form of a matched filter. Depending on the channel characteristics, one or multiple versions of the filtered/compressed message should be reported to the FC.