On optimal beamforming for noise reduction and interference rejection

In this paper, we study the performance of the minimum variance distortionless response (MVDR) and linearly constrained minimum variance (LCMV) noise reduction filters when a source of interference and ambient noise coexist with the target signal. We demonstrate that both filters are related as we decompose the MVDR filter into the LCMV and a matched filter (MVDR solution in the absence of interference). Both components are properly weighted to achieve maximum interference-plus-noise reduction at each frequency bin. Furthermore, we elaborate new closed-form expressions for the signal-to-interference ratio (SIR) and signal-to-noise ratio (SNR) at the output of the LCMV, MVDR, and matched filters. These expressions theoretically prove that a tradeoff between noise reduction and interference rejection has to be made. In fact, the total removal of the interference may severely amplify the output ambient noise. Conversely, totally focussing on noise reduction leads to increased level of residual interference. The proposed study is finally supported by numerical examples.