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Robust widely linear beamforming using estimation of extended covariance matrix and steering vector
EURASIP Journal on Wireless Communications and Networking volume 2020, Article number: 205 (2020)
Abstract
The distribution of the received signals in many array processing applications is noncircular. Although optimal widely linear beamformer (WLB) can provide the best performance for noncircular received signals, its performance degrades severely under model mismatches in practical applications. As a remedy, we propose a robust WLB by using precise reconstruction of extended interferenceplusnoise covariance matrix (EINCM) and lowcomplexity estimation of extended desired signal steering vector (EDSSV). We propose to first determine the steering vectors, powers, and noncircularity coefficients of all signals and the noise power. In contrast to the previous reconstruction methods using the integration over a wide angular sector, we reconstruct the interferenceplusnoise covariance matrix (INCM) and the pseudo INCM accurately according to their definitions. By using INCM and pseudo INCM, we can precisely reconstruct the EINCM. We propose to estimate the EDSSV by intersecting two extended subspaces, which are respectively formed by eigendecomposing the extended sample covariance matrix and the extended desired signal covariance matrix. Unlike the convex optimization methods, the proposed EDSSV estimation does not require any optimization programming and yields a solution with closed expression in low computational complexity. Simulation results show that the proposed robust WLB provides near optimal performance under several model mismatch cases.
Introduction
Adaptive beamforming aims at extracting desired signal (DS) while suppressing interferences and noise and is a fundamental technique in array signal processing [1–5]. Traditional adaptive beamformers mostly focus on secondorder circular signals with stationary observations, such as minimum variance distortionless response (MVDR) beamformer and linearly constrained minimum variance (LCMV) beamformer [6–10]. The secondorder noncircular and nonstationary signals frequently appear in the fields of radio communication and satellite communication, such as unbalanced quaternary phase shift keying (UQPSK), amplitude modulated (AM), amplitudeshift keying (ASK), and binary phaseshift keying (BPSK) signals [11–13]. However, these beamformers become suboptimal when they encounter noncircular signals. In such cases, the optimal beamforming shall be as widely linear (WL) [14–20].
A variety of widely linear beamformers (WLBs) have been developed in the past decade to exploit the noncircularity of noncircular signals. In [21], a WL MVDR beamformer is introduced which outperforms the traditional MVDR beamformer for noncircular interferences. However, the beamformer in [21] ignores the noncircularity of the DS. As a result, this beamformer cannot fully exploit the noncircularity information which becomes suboptimal. To fully utilize the noncircularity of DS, an optimal WL MVDR is proposed in [22] where more DS components are retrieved through the orthogonal decomposition of the conjugate DS. The optimal WL MVDR beamformer has better performance than the WL MVDR beamformer and its excellent performance is further analyzed in [23, 24]. However, the optimal WL MVDR beamformer relies on the predefined noncircularity coefficient and the exact desired signal steering vector (DSSV), which are not available in practical applications. Many unideal factors will lead to the DSSV mismatches, such as imperfect array calibration, wavefront distortion, local scattering, and look direction error. The mismatch of noncircularity coefficient is often caused by phase offset, frequency offset, and partial waveform information of DS. The WL MVDR beamformer will suffer serious performance degradation due to these mismatches.
The robust WLBs have been proposed to improve the robustness against various mismatches. A robust method for the optimal WL MVDR beamformer is proposed in [25] to combat the mismatches of the noncircularity coefficient and the DSSV. However, this robust WLB is sensitive to the large mismatch of noncircularity coefficient. A noncircularity coefficient estimator is proposed in [26] by only using the noncircular DSSV. However, this robust WLB relies on the exact DSSV and it is not effective in DSSV mismatch case. The authors in [27] extend the robust Capon beamformer to the generalized case with noncircular DS and noncircular interferences. Although this WLB is robust against the errors in steering vector, sample covariance matrix (SCM), and DS noncircularity coefficient, its performance degrades greatly in high signaltonoise ratios (SNRs). In [28], the authors propose two WL minimum dispersion beamformers by fully utilizing the noncircularity and subGaussianity of signals to improve the beamforming performance. A class of DS noncircularity coefficient estimators for WLB are proposed in [29], which employ the algebraic structure of the extended covariance matrix in different viewpoints. These methods have excellent performance in different scenarios of low SNR and few numbers of snapshots. In [30], a spatial spectrum of noncircularity coefficient (SSNC) is estimated and the extended interferenceplusnoise covariance matrix (EINCM) is reconstructed based on the Capon power and noncircularity coefficient spectra. This robust WLB has robustness against look direction error and steering vector mismatch. However, it requires the precise array sensor geometry. A robust WLB is proposed in [31] which prevents the extended desired signal steering vector (EDSSV) from converging to the interferences based on a projection constraint (PC). Three robust WLBs are proposed in [32] by reconstructing the EINCM via modifying the SSNC and by estimating the EDSSV using three different estimators. The EINCM outperforms the extended SCM because the EINCM reduces the DS selfnulling effects that are usually caused by the extended SCM. The robust WLBs using EINCM achieve near optimal WL beamforming performance while the WLBs using extended SCM suffer from performance degradation at high SNRs. However, the existing EINCM reconstruction is still not precise because the exploited Capon spatial spectrum is sensitive to array perturbation and the integration angular sector is too wide. Moreover, the computational complexity of the existing EDSSV estimation is very high because it requires convex optimization programming.
In this paper, we propose a precise method for the EINCM reconstruction and an inexpensive and accurate method for the EDSSV estimation. Together, these methods lead to a robust WLB with significant performance improvement which is computationally efficient. We propose to compute the steering vectors of all signals by extending the iterative robust Capon beamformer (IRCB) and propose to estimate the powers of all signals by employing the covariance fitting approach. In our proposed method, the noncircularity coefficients of all signals are estimated by extending the DS noncircularity coefficient estimator and the noise power is estimated as the minimum eigenvalue of SCM. We propose to reconstruct the interferenceplusnoise covariance matrix (INCM) and the pseudo INCM accurately according to their definitions instead of the integration over interferenceplusnoise angular sector. These accurate INCM and pseudo INCM ensure that the EINCM is reconstructed precisely. We propose to estimate the EDSSV from the intersection of two extended subspaces. The first extended signalplusinterference subspace is formed by eigendecomposing the extended SCM, and the second extended DS subspace is constructed by eigendecomposing the extended DS covariance matrix. The estimated EDSSV has a closedform solution with low complexity, which avoids any optimization software. Simulation results indicate that the proposed WLB provides robust performance against several types of model mismatches.
The rest of this paper is arranged as follows. In Section 2, we describe the noncircular signal array model and introduce the knowledge of optimal WL MDVR beamformer. Section 3 presents the proposed robust WLB with EINCM reconstruction and EDSSV estimation. In Section 4, we carry out the numerical simulations to compare the performance of the proposed robust WLB with the existing WLBs. Finally, We make conclusion in Section 5. For simplicity, we put the definitions of the symbols used in the paper into Table 1 and add a short introduction of the abbreviations used in the paper into Table 2.
Signal model and optimal WL MVDR beamformer
We consider an array of N antennas receiving M narrowband signals. The array observation vector at the time index k can be modeled as
where a_{1} and s_{1}(k) are respectively the DSSV and the DS complex waveform, and \(\mathbf {v}(k)=\sum _{m=2}^{M}\mathbf {a}_{m}s_{m}(k)+\mathbf {n}(k)\) is the whole interferenceplusnoise vector. Here, \(\{\mathbf {a}_{m}\}_{m=2}^{M}\) and \(\{s_{m}(k)\}_{m=2}^{M}\) respectively denote the steering vectors and complex waveforms of interferences, and n(k) is the noise vector which is assumed to be a zeromean circularly symmetric Gaussian white process. The DS and interferences are potentially secondorder noncircular and statistically independent with each other. We denote the noncircularity coefficient of the mth signal as \(\gamma _{m}=\langle E[s_{m}(k)^{2}]\rangle /\sigma _{m}^{2}=\gamma _{m}e^{j\phi _{m}}\) for m=1,2,⋯,M, where \(\sigma _{m}^{2}=\langle E[s_{m}(k)^{2}]\rangle \) is the timeaveraged power, γ_{m} is the noncircularity rate with 0≤γ_{m}≤1, and ϕ_{m} is the noncircularity phase. Specially, γ_{1}=1 represents the rectilinear signal whose complex waveform is on a line. The secondorder statistics of the noncircular data x(k) are expressed as
where
Here, R_{x} and C_{x} are respectively the theoretical covariance matrix and theoretical pseudo covariance matrix of x(k),R_{v} and C_{v} are respectively the theoretical INCM and theoretical pseudo INCM, and \(\sigma _{n}^{2}\) is the noise power.
By stacking x(k) and its conjugate component, we define the extended observation vector as
where \(\mathbf {b}_{1}=\left [\mathbf {a}_{1}^{T},\mathbf {0}_{N}^{T}\right ]^{T}, \mathbf {c}_{1}=\left [\mathbf {0}_{N}^{T},\mathbf {a}_{1}^{H}\right ]^{T}\), and \(\breve {\mathbf {v}}(k)=\left [\mathbf {v}(k)^{T},\mathbf {v}(k)^{H}\right ]^{T}\). The extended covariance matrix of \(\breve {\mathbf {x}}(k)\) is denoted as
To further utilize the noncircularity of the DS, we orthogonally decompose s_{1}(k)^{∗} as
where \(\langle E\left [s_{1}(k)s_{1}^{\bot }(k)^{*}\right ]\rangle =0\) and \(\langle E[s_{1}^{\bot }(k)^{2}]\rangle =1\). In this way, we rewrite \(\breve {\mathbf {x}}(k)\) in (6) as
with
where \(\breve {\mathbf {a}}_{1}\) is the noncircular EDSSV and \(\breve {\mathbf {v}}_{\gamma }(k)\) is the global noise vector for \(\breve {\mathbf {x}}(k)\). The WLB output is denoted as
where \(\breve {\mathbf {w}}\) is the WLB weight vector. The optimal WL MVDR beamformer can be designed by solving [22]
where
denotes the theoretical EINCM. The solution of (13) is given by
The output signaltointerferenceplusnoise ratio (SINR) of a WLB is defined as
However, the theoretical \(\mathbf {R}_{\breve {v}_{\gamma }}\) and \(\breve {\mathbf {a}}_{1}\) are unfortunately not available in practice. In such cases, one may approximate \(\mathbf {R}_{\breve {v}_{\gamma }}\) as the following extended SCM
where \(\mathbf {R}_{\hat {x}}=(1/K)\sum _{k=1}^{K}\hat {\mathbf {x}}(k)\hat {\mathbf {x}}(k)^{H}\) and \(\mathbf {C}_{\hat {x}}=(1/K)\sum _{k=1}^{K}\hat {\mathbf {x}}(k)\hat {\mathbf {x}}(k)^{T}\) respectively represent the SCM and the pseudo SCM that are obtained using K observed snapshots. The unknown vector \(\breve {\mathbf {a}}_{1}\) is usually approximated by the presumed EDSSV \(\bar {\breve {\mathbf {a}}}_{1}\) with the exactly known DS noncircularity coefficient γ_{1}. To facilitate the implementation of the WLB, the authors in [26] propose to estimate the DS noncircularity coefficient as
where \(\mathbf {D}=\left (\mathbf {R}_{\hat {x}}\mathbf {C}_{\hat {x}}\mathbf {R}_{\hat {x}}^{*1}\mathbf {C}_{\hat {x}}^{*}\right)^{1}, \mathbf {E}=\mathbf {D}\mathbf {C}_{\hat {x}}\mathbf {R}_{\hat {x}}^{*1}\), and \(\bar {\delta }\) is the minimum eigenvalue of \(\mathbf {R}_{\hat {\breve {x}}}\).
Proposed robust WLB
From (15), we observe that the extended weight vector of a WLB is a function of the EINCM and the EDSSV. In this section, we use this dependency to design a lowcomplexity robust WLB by reconstructing precise EINCM and estimating accurate EDSSV.
EINCM reconstruction
According to (14), reconstructing the EINCM requires the INCM and pseudo INCM. From (4), the INCM is related to the powers and steering vectors of interferences and the noise power. From (5), the pseudo INCM is related to the powers, noncircularity coefficients, and steering vectors of interferences. Therefore, we should estimate the steering vectors of interferences first. Then, we estimate the powers and noncircularity coefficients of interferences and estimate the noise power. Finally, we reconstruct the INCM, pseudo INCM, and EINCM.
The rough directions of all signals can be easily determined by the beampattern nulling method [7] or the lowresolution directionofarrival estimation method [33]. We respectively denote the rough directions and angular sectors of all signals as \(\{\bar {\theta }_{m}\}_{m=1}^{M}\) and \(\{\Theta _{m}\}_{m=1}^{M}\). The steering vector corresponding to the signal with direction \(\bar {\theta }_{m}\) is denoted as \(\bar {\mathbf {a}}_{m}\) for m=1,2,⋯,M.
It is obvious that the steering vector mismatches exist in \(\left \{\bar {\mathbf {a}}_{m}\right \}_{m=1}^{M}\) due to the direction mismatches in \(\{\bar {\theta }_{m}\}_{m=1}^{M}\). Hence, we have to correct the steering vectors \(\left \{\bar {\mathbf {a}}_{m}\right \}_{m=1}^{M}\). In [34], the IRCB can only estimate the DSSV. Here, we extend the IRCB to not only correct the DSSV but also correct the interference steering vectors. We eigendecompose the SCM \(\mathbf {R}_{\hat {x}}\) as
where α_{1}≥α_{2}≥⋯≥α_{N} are the eigenvalues of \(\mathbf {R}_{\hat {x}}\) and g_{i} is the eigenvector associated with the eigenvalue α_{i}. G_{s}=[g_{1},g_{2},⋯,g_{M}] spans the signalplusinterference subspace and G_{n}=[g_{M+1},g_{M+2},⋯,g_{N}] spans the noise subspace. Γ_{s}=diag{α_{1},α_{2},⋯,α_{M}} and Γ_{n}=diag{α_{M+1},α_{M+1},⋯,α_{N}} are the diagonal matrices. We set the iterative initial values as
The iteration process of IRCB for the mth signal can be given by
where \(\beta _{m}^{i+1}\) is the adaptive uncertainty level of the mth signal in the (i+1)th iteration. The solution of (22) is expressed as
where η is the Lagrange multiplier which is obtained as the solution to the following equation
The purpose of (23) is to obtain a maximal invariant and to avoid the norm ambiguity. The iteration process in (21)–(23) will stop when \(\beta _{m}^{i}\beta _{m}^{i+1}\leq \varrho \), where ϱ is a threshold constant. After the whole iteration process is completed, we denote the final steering vector estimate of the mth signal as \(\hat {\mathbf {a}}_{m}\) for m=1,2,⋯,M.
By using the covariance fitting approach, the power of the mth signal is obtained from
According to (4), we reconstruct the INCM as
where α_{N} is the minimum eigenvalue of \(\mathbf {R}_{\hat {x}}\) which is treated as the estimated noise power. The noncircularity coefficient estimator (18) can only estimate the DS noncircularity coefficient. Here, we extend (18) to estimate the noncircularity coefficients of DS and interferences. By replacing a_{1} in (18) with \(\hat {\mathbf {a}}_{m}\), we calculate the noncircularity coefficient of the mth noncircular signal as
for m=1,2,⋯,M. According to (5), we reconstruct the pseudo INCM as
By using the reconstructed INCM \(\hat {\mathbf {R}}_{v}\) and the reconstructed pseudo INCM \(\hat {\mathbf {C}}_{v}\), we can reconstruct the EINCM as
EDSSV estimation
By substituting \(\hat {\mathbf {a}}_{1}\) and \(\hat {\gamma }_{1}\) into (10), we can compute the EDSSV as
However, the EDSSV \(\breve {\hat {\mathbf {a}}}_{1}\) may have relatively large error because both \(\hat {\mathbf {a}}_{1}\) and \(\hat {\gamma }_{1}\) have estimation errors. Hence, we have to further improve the accuracy of the EDSSV \(\breve {\hat {\mathbf {a}}}_{1}\). The extended SCM \(\mathbf {R}_{\hat {\breve {x}}}\) can be eigendecomposed as
where λ_{1}≥λ_{2}≥⋯≥λ_{2N} are the eigenvalues of \(\mathbf {R}_{\hat {\breve {x}}}, \mathbf {q}_{i}\) is the eigenvector associated with the eigenvalue λ_{i},Q=[q_{1},q_{2},⋯,q_{2N}] and Λ=diag{λ_{1},λ_{2},⋯,λ_{2N}}. In [35], the signalplusinterference subspace is formed by projecting the presumed DSSV onto the eigenvectors of SCM. Here, we focus on forming the extended signalplusinterference subspace. We project the EDSSV \(\breve {\hat {\mathbf {a}}}_{1}\) onto the q_{i},i=1,2,⋯,2N as
We sort \(\{f(i)\}_{i=1}^{2N}\) in descending order as f(2N)>f(2N−1)>⋯>f(1) and sort its corresponding eigenvectors as [q_{2N},q_{2N−1},⋯,q_{1}]. By choosing Q principal eigenvectors, the extended signalplusinterference subspace projection matrix is constructed as
where Q is the minimum integer satisfying the following relationship
where 0<ε<1 is a predefined constant. It is clear that the actual EDSSV should lie in the subspace spanned by the columns of F, which is denoted as
where π is the subspace coefficient vector.
By integrating over DS angular sector Θ_{1}, we construct the DS covariance matrix as
where p(θ) can be chosen as \(1/\left [\mathbf {a}(\theta)^{H}\mathbf {R}_{\hat {x}}^{1}\mathbf {a}(\theta)\right ]\) [36] or \(1/\left [\mathbf {a}(\theta)^{H}\mathbf {R}_{\hat {x}}^{2}\mathbf {a}(\theta)\\\right ]\) [37]. The corresponding pseudo DS covariance matrix is constructed as
where \(\bar {\gamma }(\theta)\) is obtained by replacing a_{1} with a(θ) in (18), which is represented as
By using R_{s} and C_{s}, the extended DS covariance matrix can be expressed as
We perform eigendecomposition on \(\breve {\mathbf {R}}_{s}\) as
where μ_{1}≥μ_{2}≥⋯≥μ_{2N} are the eigenvalues of \(\breve {\mathbf {R}}_{s}, \mathbf {u}_{l}\) is the eigenvector associated with the eigenvalue μ_{l}. U_{s}=[u_{1},u_{2},⋯,u_{U}] consists of U principal eigenvectors of \(\breve {\mathbf {R}}_{s}\), where U is the minimum integer satisfying
with a predefined constant 0<ζ<1. U_{n}=[u_{U+1},u_{U+2},⋯,u_{2N}] consists of the remaining eigenvectors. Ω_{s}=diag{μ_{1},μ_{2},⋯,μ_{U}} contains U largest eigenvalues and Ω_{n}=diag{μ_{U+1},μ_{U+2},⋯,μ_{2N}} contains the remaining eigenvalues. Obviously, the actual EDSSV also lies in the subspace spanned by the columns of U_{s}, which is given by
where ψ is the subspace coefficient vector.
From (36) and (43), we can conclude that the actual EDSSV should lie in the intersection of Π and Ψ, i.e. \(\breve {\mathbf {a}}_{1}\in \boldsymbol {\Pi }\cap \boldsymbol {\Psi }\). We can use the alternating projection approach to update \(\breve {\mathbf {a}}_{1}^{j+1}\) in the (j+1)th iteration as
where \(\breve {\mathbf {a}}_{1}^{0}=\breve {\hat {\mathbf {a}}}_{1}, \mathbf {L}_{a}=\mathbf {F}\mathbf {F}^{H}\) and \(\mathbf {L}_{b}=\mathbf {U}_{s}\mathbf {U}_{s}^{H}\). As \(j\to \infty, \breve {\mathbf {a}}_{1}^{j}\) should converge to the actual EDSSV [38–40]. The maximum eigenvalue of L_{a}L_{b} is one, which is proven as [41]
Therefore, the EDSSV can be computed as
where Υ(L_{a}L_{b}) represents the eigenvector associated with the maximum eigenvalue of L_{a}L_{b}. When we divide \(\breve {\mathbf {r}}\in \mathbb {C}^{2N\times 1}\) into two subvectors as \(\breve {\mathbf {r}}=\left [\mathbf {r}_{1}^{T},\mathbf {r}_{2}^{T}\right ]^{T}, \mathbf {r}_{1},\mathbf {r}_{2}\in \mathbb {C}^{N\times 1}\) should satisfy the special structure in (10), i.e.,\(\mathbf {r}_{2}=\gamma _{r}^{*}\mathbf {r}_{1}^{*}\). However, this relationship may not be satisfied because of the existence of error. To further correct \(\breve {\mathbf {r}}\), we have to solve the following problem
The solution of (47) is given by
In such case, \(\breve {\mathbf {r}}\) can be corrected as \(\hat {\breve {\mathbf {r}}}=\left [\mathbf {r}_{1}^{T},\hat {\gamma }_{r}^{*}\mathbf {r}_{1}^{H}\right ]^{T}\). Finally, the EDSSV is corrected as
Extended weight vector calculation
By combining the EINCM \(\hat {\mathbf {R}}_{\breve {v}_{\gamma }}\) and the EDSSV \(\hat {\breve {\hat {\mathbf {a}}}}_{1}\), the extended weight vector of the proposed robust WLB is calculated as
The proposed WL beamforming algorithm is summarized in Algorithm 1.
Computational complexity
In the proposed WLB, the main computational cost of EINCM reconstruction lies in the eigendecomposition of \(\mathbf {R}_{\hat {x}}\) with a complexity of \(\mathcal {O}(N^{3})\), and the EDSSV estimation has a complexity of \(\mathcal {O}(8N^{3})\) dominated by the eigendecomposition of \(\mathbf {R}_{\hat {\breve {x}}}\) from the standpoint of computational complexity. Therefore, the main computational complexity of the proposed WLB is \(\mathcal {O}(8N^{3})\). The main computational complexity of the WLSSNC beamformer [30] is \(\mathcal {O}\left (\text {max}\left \{JN^{2},N^{3.5}+(2N)^{3.5}\right \}\right)\), where EINCM reconstruction costs a complexity of \(\mathcal {O}(JN^{2})\) with J grid points in the complement angular sector of Θ_{1} and EDSSV estimation costs a complexity of \(\mathcal {O}\left (N^{3.5}+(2N)^{3.5}\right)\). The main computational cost of the WLPC beamformer [31] arises from solving the convex optimization problem with a complexity \(\mathcal {O}\left (\left (2N\right)^{3.5}\right)\). The computational complexity of the WL via iterative quadratically constrained quadratic programming (WLIQCQP) beamformer [32] is \(\mathcal {O}\left (\text {max}\left \{JN^{2},IN^{3.5}\right \}\right)\), where I is the number of iterations in [32]. As it can be seen, the proposed WLB has lower computational complexity than WLSSNC beamformer [30], WLPC beamformer [31], and WLIQCQP beamformer [32].
We provide a visual comparison of computational complexity by plotting the flops curves of WLSSNC beamformer [30], WLPC beamformer [31], WLIQCQP beamformer [32], and the proposed WLB. We set J to be 170 for the WLSSNC beamformer [30] and the WLIQCQP beamformer [32]. I=3 is selected for the WLIQCQP beamformer [32]. Figure 1 plots the flops curves of all the above WLBs versus the number of array antennas N ranging from 10 to 100. Obviously, we can see that the proposed WLB has the lowest computational complexity among the examined WLBs.
Simulation results
In this section, we perform simulation examples to examine the performance of the proposed robust WLB and compare it with other WLBs. We consider a uniform linear array (ULA) of N=10 omnidirectional sensors with interelement spacing of half a wavelength. Three BPSK signals including one desired signal and two interferences impinge on the considered array. The DS arrives at the considered array from \(\bar {\theta }_{1}=5^{\circ }\) with the noncircularity phase 60^{∘}. Two interferences come from − 30^{∘} and 60^{∘} with the noncircularity phases − 120^{∘} and 150^{∘}, respectively. The interferencetonoise ratio (INR) is set as 10 dB. The additive noise is a complex circularly symmetric Gaussian white process with zero mean and unit variance. Each point in the curves is an average of 200 Monte Carlo trials.
The proposed WLB is compared to the WLRCB [25], the noncircular robust Capon beamformer (NCRCB) [27], the WLSSNC beamformer [30], the WLPC beamformer [31], and the WLIQCQP beamformer [32]. For the proposed WLB, ϱ=10^{−5},ε=0.9 and ζ=0.7 are exploited. The values ε_{γ}=0.1 and ε_{a}=0.3N are used for the WLRCB beamformer [25]. \(\tilde {\varepsilon }=0.6\) is chosen for the NCRCB beamformer [27]. We set the DS angular sector as Θ_{1}=[0^{∘},10^{∘}] for the WLSSNC [30], the WLPC [31], the WLIQCQP [32], and the proposed WLB. ρ=0.9 is used for the WLPC beamformer [31]. ϖ is set to be 10^{−4} for the WLIQCQP beamformer [32].
Example 1
We investigate the effect of fixed look direction mismatch on the beamforming performance. The presumed DS direction is set as 5^{∘} while the actual DS direction is 2^{∘}, which means that the DS look direction mismatch is fixed at 3^{∘}. The output SINR of the aforementioned WLBs versus the input SNR with the fixed snapshots number K=100 is depicted in Fig. 2. It can be observed that the WLSSNC, the WLIQCQP and the proposed WLB have robust performance because these three WLBs remove the DS component from the extended SCM by reconstructing EINCM. The proposed WLB has the best performance among the tested WLBs because the proposed WLB reconstruct precise EINCM. Figure 3 shows the output SINR of the examined WLBs versus the number of snapshots with fixed SNR =20 dB. We can see that the proposed WLB always achieves near optimal performance and it outperforms the other WLBs.
Example 2
We consider the case of random look direction mismatch. We assume that the random look direction mismatch of the DS is distributed in [− 4^{∘},4^{∘}]. In such case, the actual DS direction is randomly produced from [1^{∘},9^{∘}]. It should be noted that the random look direction mismatch changes from run to run while keeping constant from snapshot to snapshot. Figures 4 and 5 display the output SINR versus the SNR with fixed snapshots number K=100 and versus the number of snapshots with fixed SNR =20 dB, respectively. It can be seen that the proposed WLB yields higher SINRs than that of the other beamformers, which demonstrates that the proposed WLB is robust against the random look direction mismatch. This is because the proposed WLB not only reconstructs precise EINCM but also estimates accurate EDSSV. The WLRCB and the NCRCB have performance degradation owing to the exploitation of extended SCM and the steering vector mismatch caused by random look direction mismatch.
Example 3
We simulate the scenario where the mismatch is caused by the coherent local scattering. Under this type of mismatch, the actual DSSV is given by
where \(\mathbf {a}(\bar {\theta }_{1})\)is the direct path with assumed direction \(\bar {\theta }_{1}\) and d(θ_{p})(p=1,2,3,4)are the coherently scattered paths. φ_{p}(p=1,2,3,4) are the scattered path phases that are randomly produced from [0,2π]. θ_{p}(p=1,2,3,4) are the scattered path directions that are independently produced from a random generator \(\mathcal {N}(3^{\circ },1^{\circ })\). Note that φ_{p} and θ_{p},p=1,2,3,4 change in each simulation run but all of them remain fixed over snapshots. The performance curves versus the input SNR with fixed snapshots number K=100 and versus the number of snapshots with fixed SNR =20 dB are depicted in Figs. 6 and 7, respectively. Compared with the optimal WL MVDR, the proposed WLB encounters some performance loss resulted from the influence of coherent local scattering. Nevertheless, the performance of the proposed WLB is still superior to the other WLBs because of the precise EINCM reconstruction and the accurate EDSSV estimation.
Example 4
We evaluate the beamforming performance in the presence of random steering vector mismatch. The actual DSSV is assumed to be randomly distributed in an uncertainty set, which is expressed by
where \(\mathbf {a}(\bar {\theta }_{1})\)denotes the presumed DSSV and ξ denotes the norm of random steering vector mismatch which is randomly produced from the interval \([0,\sqrt {0.5}]\) in each simulation run. φ_{p}(p=1,⋯,N) are the phases of the pth coordinate that are independently generated from the interval [0,2π] in each simulation run. Figures 8 and 9 plot the output SINR curves versus the input SNR with fixed snapshots number K=100 and versus the number of snapshots with fixed SNR =20 dB, respectively. We can observe that the output SINRs of the proposed WLB are close to that of the optimal WL MVDR in the whole range of input SNR and snapshots number, which means that the proposed WLB is effective for the noncircular signals. In addition, the proposed WLB performs better than the other WLBs, which illustrates that the proposed WLB can deal with the random steering vector mismatch.
Example 5
We take the mismatch caused by the wavefront distortion into consideration. We assume that the independentincrement phase distortions are accumulated by the entries of DSSV. The phase increments remain fixed in each simulation run that are independently produced from a random generator \(\mathcal {N}(0,0.04)\). The output SINR curves versus the input SNR with fixed snapshots number K=100 and versus the number of snapshots with fixed SNR =20 dB are plotted in Figs. 10 and 11, respectively. It can be found that the proposed WLB almost achieves the optimal beamforming performance, which demonstrates that the reconstruction of EINCM is precise and the estimation of EDSSV is accurate. In addition, the proposed WLB enjoys the best beamforming performance, which means that the proposed WLB can provide robustness against the wavefront distortion mismatch.
Conclusion
We proposed a robust WLB by combining the precise reconstruction of EINCM and the lowcomplexity estimation of EDSSV. By estimating the steering vectors, powers and noncircularity coefficients of all signals and the noise power, we propose to reconstruct the INCM and the pseudo INCM accurately using their definitions. These accurate estimates allow us to reconstruct the EINCM precisely. Moreover, we proposed to estimate the EDSSV from the intersection of two extended subspaces which has a closedfrom expression and requires lower computational complexity than the traditional convex optimization methods. Simulation results demonstrate that the proposed WLB can provide robust performance against fixed look direction, random look direction, coherent local scattering, random steering vector, and wavefront distortion mismatches.
Availability of data and materials
The datasets generated and analyzed are available from the corresponding author on reasonable request.
Abbreviations
 WLB:

Widely linear beamformer
 EINCM:

Extended interferenceplusnoise covariance matrix
 INCM:

Interferenceplusnoise covariance matrix
 EDSSV:

Extended desired signal steering vector
 DS:

Desired signal
 MVDR:

Minimum variance distortionless response
 LCMV:

Linearly constrained minimum variance
 UQPSK:

Unbalanced quaternary phase shift keying
 AM:

Amplitude modulated
 ASK:

Amplitudeshift keying
 BPSK:

Binary phaseshift keying
 WL:

Widely linear
 SNR:

Signaltonoise ratio
 DSSV:

Desired signal steering vector
 SCM:

Sample covariance matrix
 SSNC:

Spatial spectrum of noncircularity coefficient
 PC:

Projection constraint
 SINR:

Signaltointerferenceplusnoise ratio
 IRCB:

Iterative robust Capon beamformer
 ULA:

Uniform linear array
 INR:

Interferencetonoise ratio
 NCRCB:

Noncircular robust Capon beamformer
 IQCQP:

Iterative quadratically constrained quadratic programming
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Acknowledgements
This work was supported by the China Scholarship Council under Grant (No. 201806680057).
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This work was supported by the China Scholarship Council under Grant (No. 201806680057).
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Meng, Z., Zhou, W. & Gazor, S. Robust widely linear beamforming using estimation of extended covariance matrix and steering vector. J Wireless Com Network 2020, 205 (2020). https://doi.org/10.1186/s13638020018301
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Keywords
 Noncircular signal
 Robust widely linear beamforming
 Noncircularity coefficient
 Extended covariance matrix
 Subspace intersection
 Extended steering vector