Linear filtering for noise reduction and interference rejection

We study the linearly constrained minimum variance (LCMV) and the minimum variance distortionless response (MVDR) filters when multiple interferers and unknown (ambient) noise coexist with a target speech signal. Precisely, the LCMV is designed to remove all the interference signals while preserving the desired speech and attempting to reduce the ambient noise components. The MVDR is simply formulated such that the overall ambient-noise-plus-interference are reduced while satisfying a distortionless constraint. We provide simplified expressions for both beamformers and show their relationship. Furthermore, we underline the limitations of the LCMV when the ambient noise is present. When the latter is absent, we also prove that the MVDR degenerates to the LCMV. Numerical examples are provided to support our study.