Properties, performance and practical interest of the widely linear MMSE beamformer for nonrectilinear signals

Widely Linear (WL) Minimum Mean Square Error (MMSE) estimation has received a great interest these last 20 years for second order (SO) noncircular signals. In the context of radio communications networks, it has been shown in particular that WL MMSE receivers allow to implement Single Antenna Interference Cancellation (SAIC) of one rectilinear interference, such as Binary Phase Shift Keying (BPSK) or Amplitude Shift Keying (ASK) interference, or of quasi-rectilinear interference, such as Minimum Shift keying (MSK), Gaussian MSK (GMSK) or Offset Quadrature Amplitude Modulation (OQAM) interference, hence their great interest for Global System for Mobile Communications (GSM) cellular networks in particular. However, one may wonder whether WL MMSE receivers remain attractive for SO noncircular nonrectilinear interferences, not so scarce in practice. The purpose of this paper is mainly to answer to this important question by giving, in a self-contained and unified way, some new insights into the behavior, properties and performance of the WL MMSE beamformer in the presence of arbitrary noncircular signals and interference which are not necessarily rectilinear. It is shown in particular that, surprisingly, WL MMSE receivers lose their practical interest for strong interferences which are not rectilinear. This breakthrough thus generates a new open problem for the choice between linear and WL MMSE receiver corresponding to the detection of rectilinearity (and/or quasi-rectilinearity), instead of noncircularity, in a given noisy observation. Although this question is out of the scope of this paper, we finally propose preliminary tools based on blind source separation methods to solve this problem.