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We consider distributed linearly constrained minimum variance (LCMV) beamforming in a wireless sensor network. Each node computes an LCMV beamformer with node-specific constraints, based on all sensor signals available in the network. A node has a local sensor array, and compresses its sensor signals to a signal with fewer channels, which is then shared with other nodes in the network. The compression rate depends inversely on the total number of linear constraints. Even though a significant compression is obtained, each node is able to generate the same outputs as a centralized LCMV beamformer, as if all sensor signals are available to every node. Since the distributed LCMV algorithm exploits a similar parametrization as previously developed distributed unconstrained MMSE signal estimation algorithms, it has similar dynamics and convergence properties. We provide simulation results to demonstrate the optimality and convergence of the algorithm.