This paper introduces a new optimization criterion for the design of microphone arrays, and derives an optimum filter based on this criterion. The algorithm computes two separate correlation matrices for the signal: one for when only background noise is present, and one for when both noise and signal are present. A filter is then computed based on these matrices, optimizing the proposed weighted mean-square error criterion. A blockrecursive version of the algorithm is presented, using LMS-like adaptation of the multichannel filters, with a computational complexity under 40 MIPS for a typical application with four microphones. Simulation results with typical office noise show improvements of up to 20 dB in signal-to-noise ratio, even in low-noise environments.