Elsevier

Signal Processing

Volume 104, November 2014, Pages 167-173
Signal Processing

Fast communication
Robust widely linear beamforming based on spatial spectrum of noncircularity coefficient☆

https://doi.org/10.1016/j.sigpro.2014.03.043Get rights and content

Highlights

  • •

    We propose a robust widely linear (WL) beamforming algorithm for receiving noncircular signals.

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    The spatial spectrum of noncircularity coefficient is firstly defined in the context of the optimal WL MVDR beamformer.

  • •

    The conjugated interference-plus-noise covariance matrix is reconstructed.

  • •

    The extended steering vector is corrected by using the augmented interference-plus-noise covariance matrix.

Abstract

In this communication a robust widely linear (WL) beamforming algorithm for the reception of a noncircular signal is proposed. The spatial spectrum of noncircular coefficient is firstly introduced in the context of the optimal WL minimum variance distortionless response beamformer. The augmented interference-plus-noise covariance matrix is reconstructed instead of the augmented sample covariance matrix. The proposed algorithm not only outperforms the conventional robust adaptive beamforming techniques, but also has a better performance against the model mismatch than the previous robust WL beamforming algorithms. Numerical simulations verify the effectiveness of the proposed algorithm.

Introduction

Conventional beamforming techniques have mainly considered a linear complex filter for stationary observations, whose complex envelopes have been proved to be necessarily second-order (SO) circular [1]. A well-known optimal beamformer is the minimum variance distortionless response (MVDR) beamformer proposed by Capon [2]. Nevertheless, in the areas of radio communication or satellite communication, the signals are often second-order (SO) noncircular and nonstationary, such as binary phase-shift keying (BPSK), amplitude-shift keying (ASK), amplitude modulated (AM) and unbalanced quaternary phase shift keying (UQPSK). For this class of signals, the conventional MVDR beamformer is shown to be suboptimal and the optimal complex filters become widely linear (WL) [3].

The optimal WL MVDR beamformer was proposed by Chevalier et al. [4] and its super performance was analyzed in [5] and [6]. The optimal WL MVDR beamformer is based on the assumption that the time-averaged SO noncircularity coefficient, which is called noncircularity coefficient for simplicity in this communication, and the steering vector of the desired signal are known precisely as a priori. However, in many applications, the noncircularity coefficient is unavailable and the steering vector is not known exactly. The latest work about the robust WL MVDR beamfoming is presented by Wang et al. in [7]. In his work, the noncircularity coefficient is assumed to be an inaccurate value and independent with the steering vector [7]. However, this algorithm is sensitive to the large mismatch of noncircularity coefficient. Subsequently, Xu et al. estimated the noncircularity coefficient of the desired signal, which makes the optimal WL beamformer available in practical applications [8]. But this estimator only achieves optimal performance by using the exact steering vector. Although the diagonal loading technique is used in Xu’s work, it is only effective for suppressing the perturbations of the estimation error of the noncircularity coefficient.

Recently, Gu et al. presented a new robust beamforming algorithm based on interference-plus-noise covariance matrix (INCM) reconstruction technique [9]. This robust beamforming algorithm attains a better performance over a wide range of signal-to-interference-plus-noise ratios (SINRs). Sequentially, Li et al. exploited the cyclostationarity of interferences to reconstructed INCM, which needs no information of the array structure [10]. In this communication, we develop this covariance matrix reconstruction technique in area of the WL beamformer. A spatial spectrum of the noncircularity coefficient is firstly defined to reconstruct conjugated INCM. Subsequently, the corrected steering vector is obtained by using Gu’s method 2 and then used to calculate the noncircularity coefficient of the desired signal. At last, the obtained extended steering vector is further more corrected by maximizing the output power of the WL beamformer under the constraint that the corrected extended steering vector does not converge to any interference.

Section snippets

Signal model and data statistics

Let x(t) be an N×1 complex vector representing the digitized data received by an array of N narrowband (NB) sensors. x(t) can be modeled asx(t)=s(t)a+v(t),where s(t), a, v(t) are the desired signal, steering vector, and total interference-plus-noise vector, respectively. The noise is assumed to be circularly symmetric Gaussian white process with zero-mean.

The SO statistics of the noncircular observation x(t) are defined byRx≜〈E[x(t)x(t)H]〉=πsaaH+R,Cx≜〈E[x(t)x(t)T]〉=πsγsaaT+C,where 〈⋅〉 denotes

The spatial spectrum of the noncircularity coefficient

The noncircularity coefficient estimator in (15) only shows the noncircularity property of the desired noncircular signal. We can define the spatial spectrums of noncircularity rate and phase to depict noncircularity properties of the noncircular signals from different directions.

Denote d(θ) to be the steering vector associated with a hypothetical direction θ based on the known array structure and substitute a by d(θ) in (15). We have the noncircular coefficient according to θ, i.e.,γ^(θ)=−d(θ)H

Simulation

In our simulations, a ULA with N=10 omnidirectional sensors spaced half a wavelength is used. The additive noise is modeled as complex circularly symmetric Gaussian zero-mean spatially and temporally white process. The desired signal is assumed to be from the presumed direction θs=5° with the noncircularity phase 60°. Two interfering sources are assumed to be from the direction −50° and 30°, with the noncircularity phase −120° and −150°, respectively. The interference-to-noise ratio (INR) in

Conclusion

In this communication, a robust WL beamforming algorithm based on the spatial spectrum of noncirculairty coefficient is proposed. The augmented INCM is firstly reconstructed and then the presumed extended steering vector is corrected to obtain a better performance. The simulation results demonstrate that the proposed WL beamforming algorithm achieves a better performance than other tested algorithms.

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This work is supported by the Science and Technology Plan Project of Anhui Province of China.

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