Widely linear MVDR beamformers for the reception of an unknown signal corrupted by noncircular interferences

For nonstationary observations, potentially secondorder (SO) noncircular, the SO optimal complex filters are time variant (TV) and, under some conditions of noncircularity, widely linear (WL). Moreover, for applications such as spectrum monitoring or passive listening, the sources' waveforms are unknown and no training sequence or spreading code is a priori available. In this context, this paper aims at introducing the time invariant (TI) WL minimum variance distortionless response (MVDR) beamformer for the optimal reception of an unknown signal, whose waveform is unknown but whose steering vector is known, corrupted by potentially noncircular interferences. Its properties, performance, and adaptive implementation in noncircular contexts are analyzed in this paper. This optimal beamformer is shown to always improve, in the steady state, the performance of the well-known Capon's beamformer for noncircular interferences. Moreover, it should be noticed that this optimal beamformer allows the processing of up to 2(N - 1) rectilinear interferences from an array of N sensors. Finally, at the end of this paper, a TV extension of this TI WL MVDR beamformer is presented to process SO noncircular interferences having a nonnull carrier residue or frequency offset.