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Signals captured by a set of microphones in a speech communication system are mixtures of desired and undesired signals and ambient noise. Existing beamformers can be divided into those that preserve or distort the desired signal. Beamformers that preserve the desired signal are, for example, the linearly constrained minimum variance (LCMV) beamformer that is supposed, ideally, to reject the undesired signal and reduce the ambient noise power, and the minimum variance distortionless response (MVDR) beamformer that reduces the interference-plus-noise power. The multichannel Wiener filter, on the other hand, reduces the interference-plus-noise power without preserving the desired signal. In this paper, a speech distortion and interference rejection constraint (SDIRC) beamformer is derived that minimizes the ambient noise power subject to specific constraints that allow a tradeoff between speech distortion and interference-plus-noise reduction on the one hand, and undesired signal and ambient noise reductions on the other hand. Closed-form expressions for the performance measures of the SDIRC beamformer are derived and the relations to the aforementioned beamformers are derived. The performance evaluation demonstrates the tradeoffs that can be made using the SDIRC beamformer.