Third-Order Volterra MVDR Beamforming for Non-Gaussian and Potentially Non-Circular Interference Cancellation

Abstract

Linear beamformers are optimal, in a mean square (MS) sense, when the signal of interest (SOI) and observations are jointly Gaussian and circular. Otherwise, linear beamformers become sub-optimal. When the SOI and observations are zero-mean, jointly Gaussian and non-circular, optimal beamformers become widely linear (WL). They become non-linear with a structure depending on the unknown joint probability distribution of the SOI and observations when the latter are jointly non-Gaussian, assumption which is very common in radiocommunications. In this context, the paper aims at introducing, for small-scale systems, third-order Volterra minimum variance distortionless response (MVDR) beamformers, for the reception of a SOI, whose waveform is unknown but whose steering vector is known, corrupted by non-Gaussian and potentially non-circular interference, omnipresent in practical situations. Properties, performance, complexity and adaptive implementation of these beamformers in the presence of non-Gaussian and potentially non-circular interference are analyzed in this paper. These new beamformers are shown to always improve, in the steady state, the performance of Capon beamformer for non-gaussian/circular interference, whereas some of them improve the performance of the WL MVDR beamformer for non-Gaussian/non-circular interference. These new beamformers open new perspectives for spectrum monitoring of non-Gaussian signals and for radiocommunication networks using such signals.

On the sensitivity of third-order Volterra MVDR beamformers to interference-pulse shaping filter

“…We briefly recall the principle and structure of the third order Volterra MVDR beamformers introduced in [11]. These beamformers consist in estimating s(t) from x(t) without any knowledge on the distribution 50 of s(t) and n(t).…”

“…x(t) depending on the probability distribution of the total noise only. As this distribution is unknown in practice, we have proposed in [11] to approximate this MVDR beamformer by third-order Volterra MVDR beamformers. The output of a third-order Volterra beamformer is defined by…”

“…To impose no distorsion on s(t) at the output y(t), a set of linear constraints on w, on the form C H w = f , has been imposed in [11]…”

“…In the more general case of non- 15 Gaussian observations, omnipresent in practice, optimal beamformers become non-linear with a structure depending on the unknown joint probability distribution of the SOI and observations. In this context, thirdorder Volterra MVDR beamformers have been proposed recently [11] for small-scale systems, to improve the performance of Capon beamformer in the presence of non-Gaussian and potentially non-circular interference, omnipresent in practical situations. High performance gains have been reported for Bernoulli-based 20 impulsive interference and for BPSK and QPSK interference having a square pulse shaping filter in particular.…”

“…Section 2 introduces the observation model, recalls the structure and implementation of the third-order 25 Volterra MVDR beamformers introduced in [11] and formulates the problem addressed in this paper. Section 3 presents, for a discrete time implementation of the previous beamformers, the asymptotic second order (SO), fourth order (FO) and sixth order (SIO) time-averaged statistics of an interference having an arbitrary pulse shaping filter.…”