Elsevier

Signal Processing

Volume 122, May 2016, Pages 123-128
Signal Processing

Fast communication
Widely linear minimum dispersion beamforming for sub-Gaussian noncircular signals

https://doi.org/10.1016/j.sigpro.2015.10.012Get rights and content

Highlights

  • We propose two WL-MD based beamforming for sub-Gaussian noncircular signals.

  • The proposed beamformers have good performance even in high SNR and are able to process more signals than sensors.

  • The SOCP formulations are given to solve the proposed beamformers.

Abstract

We propose two widely linear minimum dispersion based beamforming for sub-Gaussian noncircular signals: widely linear minimum dispersion distortionless response (WL-MDDR) and widely linear quadratically constrained minimum dispersion (WL-QCMD). In the nonideal conditions, taking full advantages of noncircularity and sub-Gaussian properties of signals, the proposed algorithms are shown to achieve good performance even in high signal-to-noise ratio and process more signals than the number of sensors. The WL-MDDR beamformer is designed when the information about the steering vector and noncircularity coefficient of the desired signal is precise, while the WL-QCMD beamformer is robust against arbitrary errors in the steering vector and noncircularity coefficient. Numerical simulations verify the effectiveness of the proposed algorithms.

Introduction

Beamforming is a fundamental technique in array signal processing, and widely used in radar, sonar and wireless communications [1]. Conventional beamforming techniques, such as minimum variance distortionless response (MVDR), are often based on the minimum variance (MV) criterion [2], [3], which is statistically optimal only when the true covariance matrix and steering vector of the desired signal are available. However, due to the nonideal conditions, such as signal direction error, uncalibrated array, the finite number of snapshots and so on, this conclusion is not always true in practical applications. It is only suitable for a Gaussian signal. The reason is that the first- and second-order statistics of a Gaussian distribution contain all necessary statistical information.

But many real-world signals are non-Gaussian [4], [5], which can be classified into sub-Gaussian and super-Gaussian based on the kurtosis [6], [7]. For a random stationary signal s(k) with zeros-mean, the kurtosis is defined as κ(s(k))=(E[|s(k)|4]2E[|s(k)|2]2|E[s(k)2]|2)/σs4 [8], where σs2 denotes the variance of s(k) and E[] is the expectation operator. If s(k) is Gaussian, κ(s(k)) is equal to zero. While the sign of the kurtosis determines the signal is sub-Gaussian or super-Gaussian, which corresponds the negative or positive. Sub-Gaussian signals are often arisen in wireless communication, radar and sonar [9], [10], such as phase shift keying (PSK), quadrature amplitude modulation (QAM), and pulse amplitude modulation (PAM). In this case, the higher-order statistics can be utilized to improve the beamformer performance [11], [12], [13]. In [11], a minimum dispersion (MD) criterion is proposed to minimize the lp-norm (p1) of the array output, and for sub-Gaussian signals, the performance can be improved significantly with p>2. On the other hand, signals are often second-order (SO) noncircular and nonstationary in radio communication, such as binary phase-shift keying (BPSK), amplitude-shift keying (ASK), and unbalanced quaternary phase shift keying (UQPSK) signals. For this class of signals, the widely linear (WL) beamformer [14] is developed to exploit the noncircularity. Moreover, the WL-MVDR beamformer [15] is shown a better performance than the conventional beamformers. In [16], the optimal WL-MVDR beamformer, which leads to a further performance improvement, is proposed by exploiting the noncircularity of the desired signal [17], [18].

Thus, for the class of sub-Gaussian signals encountered, which also exhibit noncircularity, such as BPSK, UQPSK and PAM signals, we propose two WL-MD based beamforming: widely linear minimum dispersion distortionless response (WL-MDDR) beamforming and widely linear quadratically constrained minimum dispersion (WL-QCMD) beamforming. The WL-MDDR beamformer is designed where we exactly known the steering vector (SV) and noncircularity coefficient of the desired signal. In the case of SV error and imprecise noncircularity coefficient, the WL-QCMD beamforming is developed to improve the robustness against the errors. The main contributions of our work contain: fully use the noncircularity and sub-Gaussian properties of signals, reserve the advantages of good performance at high signal-to-noise ratio (SNR) and be able to process more signals than the number of sensors.

Section snippets

Problem formulation

Considering an array of N sensors to receive narrowband signals, the array output x(k)N×1 is modeled asx(k)=s(k)a+v(k)where s(k), a, v(k) are the desired signal, SV, and total interference-plus-noise vector, respectively. We assume that x(k) is noncircular and nonstationary, and the additive noise is Gaussian white process with zero-mean.

Proposed algorithms

In this section, we assume that the desired signal is noncircular and sub-Gaussian. Based on the aforementioned introduction, we consider two scenarios: one is that the SV and noncircularity coefficient of the desired signal is exactly known, which leads to the WL-MDDR beamformer; the other is the case that errors exist in the SV and noncircularity coefficient, where the WL-QCMD beamformer is proposed to gain the robustness against the errors.

Simulation results

In our simulations, a uniform linear array with N=4 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 and two interferences are all BPSK signals, where the values of the kurtosis are equal to −2. The desired signal comes from the direction 0° with the noncircularity phase 30°. The two interferences come from 30° and 40°, with the noncircularity

Conclusion

In this communication, considering the sub-Gaussian noncircular signals, two WL-MD based beamformers have been proposed based on the MD criterion for two different scenarios, where the noncircularity and sub-Gaussianity of signals are in full used to improve the beamformer performence. Without model errors, the WL-MDDR beamformer is designed by minimizing the l-norm based on the exact ESV of the desired signal. Furthermore, considering the errors in the SV and noncircularity coefficient in

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