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This paper investigates the problem of enhancing a single desired speech source from a mixture of signals in multispeaker environments. A beamformer structure is proposed which combines a fixed beamformer with postfiltering. In the first stage, the fixed multiobjective optimal beamformer is designed to spatially extract the desired source by suppressing all other undesired sources. In the second stage, a multichannel power spectral estimator is proposed and incorporated in the postfilter, thus enabling further suppression capability. The combined scheme exploits both spatial and spectral characteristics of the signals. Two new multichannel spectral estimation methods are proposed for the postfiltering using, respectively, inner product and joint diagonalization. Evaluations using recordings from a real-room environment show that the proposed beamformer offers a good interference suppression level whilst maintaining a low-distortion level of the desired source.