Robust adaptive beamforming via subspace for interference covariance matrix reconstruction

Robust adaptive beamforming via subspace for interference covariance matrix reconstruction

Signal Processing 167 (2020) 107289 Contents lists available at ScienceDirect Signal Processing journal homepage: www.elsevier.com/locate/sigpro Ro...

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Signal Processing 167 (2020) 107289

Contents lists available at ScienceDirect

Signal Processing journal homepage: www.elsevier.com/locate/sigpro

Robust adaptive beamforming via subspace for interference covariance matrix reconstruction Xingyu Zhu a,b, Xu Xu a,b, Zhongfu Ye a,b,∗ a b

Department of Electronic Engineering and Information Science, University of Science and Technology of China, Hefei, Anhui 230027, China National Engineering Laboratory for Speech and Language Information Processing, Hefei, Anhui 230027, China

a r t i c l e

i n f o

Article history: Received 25 January 2019 Revised 12 August 2019 Accepted 11 September 2019 Available online 12 September 2019 Keywords: Robust adaptive beamforming Covariance matrix reconstruction Interference subspace Orthogonality Steering vector estimation

a b s t r a c t Adaptive beamforming may cause performance degradation when model mismatch errors exist. In this paper, we have developed subspace methods for robust adaptive beamforming (RAB). The two proposed methods utilize the orthogonality of subspace to reconstruct the interference covariance matrix (ICM). Above all, the steering vector (SV) of desired signal in proposed methods is estimated from the desired signal covariance matrix, where desired signal covariance matrix is reconstructed based on the modified Capon spatial power spectrum estimator and it contains the less useless components. In the first proposed method, the ICM is reconstructed from the projected snapshots. Through projection, the desired signal in snapshots is eliminated and the interference components is retained to reconstruct the ICM. Based on the proposed-1 method, the peaks of spatial power spectrum corresponding to the reconstructed ICM indicate the nominal interference SVs which are used in the second proposed method. In the second proposed method, we reconstruct the ICM by each interference SV and corresponding power, and all interference SVs are exploited by solving a quadratic convex problem, where it depends on the orthogonality between interference SVs and interference subspace. The proposed-2 method is the extension of proposed-1 and it can achieve the better performance. Simulation results demonstrate that the two proposed methods are robust against types of mismatch to achieve well performance. © 2019 Elsevier B.V. All rights reserved.

1. Introduction Adaptive beamforming has been widely applied in wireless communication, radar, sonar, speech signal processing and so on [1–3]. As a spatial filtering technique, adaptive beamforming is employed to receive the desired signal from certain direction and suppress the interference and noise [4,5]. However, traditional adaptive beamformer [6] is sensitive to mismatch errors which would cause severe performance degradation. Therefore, various robust adaptive beamforming (RAB) algorithms have been developed in the past years. Diagonal loading (DL) [7–11] technology is one of the popular methods to improve the robustness, which is derived to add a scaled identity matrix on sample covariance matrix or impose an additional quadratic constraint on the steering vector (SV), while it’s difficult to choose the optimal DL factor.

∗ Corresponding author at: Department of Electronic Engineering and Information Science, Technology Building in West Campus No. 443, Huangshan Road, Shushan, Dis Hefei 230027, China. E-mail address: [email protected] (Z. Ye).

https://doi.org/10.1016/j.sigpro.2019.107289 0165-1684/© 2019 Elsevier B.V. All rights reserved.

The Eigenspace-based algorithms [12–14] are implemented by projecting the nominal SV onto the signal-plus-interference subspace to eliminate mismatch errors. The modified subspace projection methods [15,16] utilize the correlations between the nominal SV and the eigenvector of sample covariance matrix to span the more precise signal-plus-interference subspace. In [17], the authors propose a criterion to search the basis vector of signal subspace among the eigenvectors of the sample covariance matrix. However, all these methods are restricted at low signal-to-noise ratio (SNR), where the signal subspace may be corroded by the noise subspace. Uncertain-set based technology is another type of beamforming methods [9,18–21]. They make explicit use of uncertainty set to constrain the SV, which has been proven to be equivalent to the DL approach. However, there is no clear criterion to decide the norm bound of SV as well. Besides, most of these methods need to solve a optimization problem which results in high computational complexity. The above methods mainly focus on the SV estimation of desired signal or sample covariance matrix processing without reconstruction. Although these methods have improved the robustness of adaptive beamforming, they share the weakness that the

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X. Zhu, X. Xu and Z. Ye / Signal Processing 167 (2020) 107289

performance of beamforming would degrade dramatically in the high input SNR, which caused by the desired signal components existing in sample covariance matrix. In recent years, interferenceplus-noise covariance matrix (INCM) reconstruction methods [22–32] have become attractive since they can achieve nearly optimal performance across a wide range of SNR. These methods aim to reconstruct interference covariance matrix (ICM) or INCM instead of using sample covariance matrix. The INCM reconstruction method is first proposed in [22], where it’s based on the Capon spectral estimator integrated over a region separated from the desired signal direction. The integration in [22] can be regarded as linear, and the method in [23] modifies the linear integration area into annular volume with high computation complexity. In [24], a sparse INCM reconstruction algorithm has been proposed which makes explicit use of sparsity of source distribution, but it still can not cope with array calibration errors. The [25] proposes a spatial power sampling method to reconstruct the INCM, however it’s sensitive to the number of array sensors. The main ideas of [26,28] are the same, where they utilize the same way based on [33] to search for the SV lying in the intersection of two subspaces. The difference is that the method in [26] estimates the desired signal SV while the method in [28] estimates the interference SVs. In [26], the authors estimate the desired signal SV from the two subspaces spanned by the sample covariance and reconstructed desired signal covariance matrix, respectively. The method in [28] aims to search for the interference SV from the intersection of two subspaces as well, and one of the subspaces is spanned by the reconstructed INCM in [22]. Most of the above methods are based on the uniform linear array (ULA), and some of RAB methods [27,29,30] are designed for the coprime array which can offer the better spatial resolution with increased degrees of freedom. The method in [27] estimates the spatial spectrum on the virtual array according to the coprime array received signals, where the virtual ULA has more virtual sensors than the number of physical sensors. The methods in [29,30] use the pair of decomposed coprime subarrays to reconstruct the ICM. However, with the same number of sensors, the coprime array would cost the higher computational complexity compared with ULA. The abovediscussed INCM methods can be classified as three types. The first category improves the sample covariance matrix or remove the desired signal components [25,26]. The second category employs the Capon power spectrum to integrate over the specific angular regions [22,23], and the third type reconstructs INCM based on each interference SV, corresponding power and noise power [24,27–32]. In this paper, we devise two RAB methods based on the orthogonality of subspace and SVs. The proposed two methods reconstruct the ICM and noise covariance matrix respectively to obain the INCM finally. At first, the estimated SV of desired signal is estimated from the dominant eigenvector of the reconstructed desired signal covariance matrix, and it’s utilized in two proposed methods. In the first proposed method, we span the interference subspace based on the array geometry, and the spanned subspace is proved to be orthogonal to the desired signal SV. Depending on this property, we project the received snapshots onto the subspace to derive the reconstructed ICM. The peaks of spatial power spectrum distribution corresponding to the reconstructed ICM indicate the impinging directions of interference. From the directions, we can obtain the corresponding nominal SVs. In the second proposed method, the ICM is reconstructed by the estimated SVs of interference and associated powers to further improve the accuracy, where the estimated interference SVs are exploited by solving a quadratic convex optimization problem from the orthogonality between the nominal SVs and interference subspace. The main contributions of this paper are summarized as follows:





We develop two subspace methods to reconstruct the ICM. The first proposed method eliminates the desired signal components from received snapshots to reconstruct the ICM through subspace projection, which achieves well performance with low computational complexity. Based on the reconstructed ICM in first method, we obtain the nominal SVs of interference. The second proposed method utilizes the orthogonality of interference subspace to optimize the nominal SVs to reconstruct the more precise ICM.

The rest of this paper is organized as follows. The signal model and necessary background about adaptive beamforming technology are introduced in Section 2. In Section 3, the proposed RAB methods are described in detail and the analysis of performance is performed. The simulation results are provided in Section 4. Finally, conclusions are drawn in Section 5. 2. Signal model and background Consider a ULA composed of M omnidirectional sensors that receive multiple uncorrelated narrowband signals from far-field sources. All signals are uncorrelated with noise. The M × 1 complex array observation at time k can be modeled as:

x ( k ) = xs ( k ) + xi ( k ) + xn ( k )

(1)

L

where xs (k ) = s(k )a0 , xi (k ) = l=1 sl (k )al and xn (k) represent for desired signal, interference and noise, respectively. s(k) stands for the desired signal waveform and a0 is corresponding SV. al is the l-th SV of interference and sl (k) denotes the associated interference waveform. xn (k) is complex Gaussian white noise with zero mean and fixed variance σn2 . For the signal imposed from θ , the SV is formulated as:



a ( θ ) = 1 , e − j 2π

dsinθ

λ

, · · · , e − j 2π

(M−1 )dsinθ λ

T

(2)

where λ is signal wavelength and d represents space distance between √ two adjacent sensors. Apparently, the Euclidean norm of SV is M, that is:

a22 = M

(3)

where  · 2 denotes the 2 norm. The output of beamformer is written as:

y ( k ) = wH x ( k )

(4) ]T

where w = [w1 , w2 , · · · , wM denotes the complex weight vector, (·)T and (·)H are the transpose and Hermitian transpose, respectively. The performance of beamformer is measured by the output signal-to-interference-plus-noise ratio (SINR) as follow:

SINR =

σs2 |wH a0 |2 wH Ri+ n w

(5)

where σs2 = E {|s(k )|2 } denotes the power of desired signal and E{·} stands for the expectation operator of stochastic variables. Ri+n is the theoretical INCM which is expressed as:

Ri+n = E {(xi (k ) + xn (k ))(xi (k ) + xn (k ))H } =

L 

σl2 al aHl + E {xn (k )xHn (k )}

l=1

= Ri + σn2 I

σl2

(6)

where = E {|sl and I denote the l-th interference power and identity matrix, respectively. The optimal weight vector is solved by maximizing the output SINR (5):

min wH Ri+n w w

( k )|2 }

subject to wH a0 = 1

(7)

X. Zhu, X. Xu and Z. Ye / Signal Processing 167 (2020) 107289

The above optimization problem is known as minimum variance distortionless respond (MVDR) beamformer and the solution is given by:

3

3. Proposed methods

where R is data covariance matrix expressed as:

The main idea of proposed methods is to employ the orthogonality property of spanned interference subspace to achieve the reconstruction of ICM from received data. The proposed-1 method utilizes the interference subspace to remove the desired signal components from received snapshots and retain the interference components through projection, where the interference subspace is spanned based on the array geometry Based on the first reconstructed ICM, we obtain the nominal SVs of interference. In proposed-2 method, we employ the orthogonality of interference subspace to optimize the nominal SVs of interference and the ICM is reconstructed by each interference SV together with powers.

R = E {x ( k )xH ( k )}

3.1. Desired signal SV estimation

R−1 a i+n 0 H a0 R−1 a i+n 0

wopt =

(8)

Replacing Ri+n with R doesn’t change the optimal output SINR which has been proved in [5]. Capon spatial power spectrum [34] is used as a power estimator over all directions:

P (θ ) =

=

σ

1 aH (θ )R−1 a(θ )

2 H s a0 a0

+

L 

σ

(9)

2 H l al al

+ E {xn ( k )

xH n

( k )}

l=1

= Rs + Ri + σn2 I = Rs + Ri+ n

(10)

In practical applications, R is unavailable and it is usually replaced by the sample covariance matrix:

ˆ = R

K 1 x ( k )xH ( k ) K

(11)

In the whole angular regions, the desired signal and interference are distributed sparsely due to the number of impinging signals is limited. Based on the low resolution finding methods [6,35,36], the observation regions can be divided into the desired signal region s and interference region i . The complement sector of s and i is defined as noise regions n . That is to say, s ∪ i ∪ n covers the whole regions. In the region n , we employ the Capon spatial power spectrum distribution to calculate the residual noise power:



k=1

ˆ converges where K is the number of snapshots. As K increases, R to R. The precise array structure is hard to obtain as well which means a0 is not accuracy. The compromise is to substitute a0 by the nominal SV a¯ 0 based on the known array structure, so the optimal weight vector (8) has become to the sample covariance inversion (SMI) beamformer:

wSMI

ˆ −1 a¯ 0 R = ˆ −1 a¯ 0 a¯ H R

(12)

0

and the corresponding Capon spatial power spectrum is written as:

Pˆ(θ ) =

1

(13)

ˆ −1 a¯ (θ ) a¯ H (θ )R

Most existed covariance matrix reconstruction methods are proposed based on (13) integrated in the specific regions and they all don’t take residual noise components into consideration, which will cause the reconstructed covariance matrix to be inaccurate. The existence of residual noise components in Capon power spectrum has been proved in [32], and the value of residual noise power has become M-th of actual noise. Assume that x(k) is comprised of complex Gaussian white noise and one signal impinging from θ i1 , no matter interference or desired signal. That is to say the data covariance matrix becomes to R = σ 2 a(θi1 )aH (θi1 ) + σn2 I, where σ 2 stands for the power of impinging signal, then equation (9) becomes to:

P (θ ) =

aH



(θ ) σ

1 2a

( θi 1 )

aH

( θi 1 ) + σ



2 −1 a nI

(14)

(θ )

When θ = θi1 , the above equation changes to:

P ( θi 1 ) =

=

1

 aH (θi1 ) σ 2 a(θi1 )aH (θi1 ) + σn2 I −1 a(θi1 ) σn2 M

+ σ2

From Eq. (15), we can see the residual noise components its relationship to the actual noise.

(15) σn2 M

σ

¯ n2



J 1  1 = , θ j ∈ n H ( θ )R −1 a J ˆ ¯ ¯ (θ j ) a j j=1

(16)

where θ j is the discrete sample point in n , J is the number of sample points. Based on the relationship of residual noise power in (15), the actual noise power is estimated as:

σˆ n2 = Mσ¯ n2

(17)

The σˆ n2 is estimated actual noise power instead of diagonal factor, and it’s utilized to reconstruct the noise covariance matrix. Based on σˆ n2 , we can express the practical power spectrum distribution of impinging signals:

1

P¯ (θ ) =

ˆ −1 a¯ (θ ) a¯ H (θ )R ˆ = P (θ ) − σ¯ n2

− σ¯ n2 (18)

Subsequently, the desired signal covariance matrix is reconstructed as:



ˆs = R =

s

P¯ (θ )a¯ (θ )a¯ H (θ )dθ ,



s



1 − σ¯ n2 a¯ (θ )a¯ H (θ )dθ H ˆ a¯ (θ )R−1 a¯ (θ )

(19)

s is the angular sector where the desired signal locates. It shoule be noted that the H ˆ1−1 − σ¯ n2 may be negative in s , which a¯ (θ )R a¯ (θ )

is conflicted to the value of power. So we only choose the positive values and ignore the negative values to guarantee positivity of (19). Besides, the part of σ¯ n2 a¯ (θ )a¯ H (θ ) is the residual noise comˆ s. ponents in integration, which would influence the accuracy of R ˆ s contains the less useless components. After eliminating this, the R The eigenvector corresponding to the largest eigenvalue covers ˆ s , which is regarded as the estimator of the most information of R the desired signal SV.

ˆs = R

M 

αn dn dHn

(20)

n=1

and

aˆ s =

√ M d1

(21)

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X. Zhu, X. Xu and Z. Ye / Signal Processing 167 (2020) 107289

where  = H = B1 BH is projection matrix. From Fig. 1, the value 1 of SVs corresponding to i doesn’t change after projection because the SVs corresponding to i can be linear expressed by the columns of B1 based on [17,21]. So we can have the following approximation:

H a¯ (θi ) ∼ = a¯ (θi ), θi ∈ i

(25)

The reason is of the unity column vectors in B1 , and each column of B1 is orthogonal to each other. Fig. 1 also shows that the larger the value N, the poorer the orthogonality between a¯ (θ ) and B1 . If we choose N = M, where B1 BH = BBH = I, then it doesn’t has any 1 orthogonality property. However, if the value of N is too small, the interference components would be destroyed. The choice of N is the trade-off between desired signal elimination and interference components preservation. Besides, the choice of N is related to the number of sensors, where the different number of sensors determines the different choice of N. Based on the Fig. 1, N is set N = 7 through experience in the following experiment examples. We utilize the property of (24) and (25) to eliminate the desired signal components from the received snapshots as follow: 2 ¯ Fig. 1. Values of B1 BH 1 a (θ )2 versus θ .

x˜ (k ) = H x(k ) = H (xs (k ) + xi (k ) + xn (k )) ∼ = H xi (k ) + H xn (k ) ∼ xi (k ) + H xn (k ) =

ˆ s arranged in dewhere α n , n = 1, 2, · · · , M are the eigenvalues of R scending order (i.e. α 1 ≥ α 2 ≥  ≥ α M ), dn is the eigenvector corresponding to α n .

Then we can calculate the projected sample covariance matrix as (11):

3.2. Subspace projection for ICM reconstruction Different from previous Capon power spectrum integration based methods, in this section, we propose the first effective ICM reconstruction method with lower complexity, which removes the desired signal components from the received data through subspace projection. Before calculating the sample covariance matrix, we receive each snapshot x(k) expressed in (1). Distinguished from the previous subspace projection methods, we utilize the subspace matrix to project on the each snapshot to eliminate the desired signal components. Based on the known array geometry, we can define the matrix given as:

=

i

a¯ (θ )a¯ H (θ )dθ

(22)

where i stands for the angular sector which contains the locations of interference, so  collects the spatial information of interference regions. We can regard that the dominant eigenvectors of  denote the interference subspace where the interference SVs lie. Then employ eigen-decomposition on  to choose the different subspaces:

 = B B H

(23)

where B = [b1 , b2 , · · · , bM ] = [B1 , B2 ] and  = diag{γ1 , γ2 , · · · , γM } denote unitary and diagonal matrices, respectively. bm stands for the eigenvector corresponding to γ m , m = 1, 2, · · · , M which are arranged in descending order. B1 contains N eigenvectors corresponding to N largest eigenvalues. We can demonstrate the orthogonality between subspace B1 2 ¯ and a¯ (θ )Fig. 1 shows the curve of B1 BH 1 a (θ )2 versus the angle θ with M = 10 sensors spaced half a wavelength apart. Assume that i = [−51◦ , −35◦ ] ∪ [19◦ , 35◦ ] and s = (−35◦ , 19◦ ). It can be seen 2 ¯ that the values of B1 BH 1 a (θ )2 in s is much smaller than that in i . So we can eliminate the desired signal components and retain the interference components through projecting onto the subspace spanned by B1 , and it can be expressed as following:

H a¯ 0 ∼ =0

(26)

(24)

˜ = R

K 1 x˜ (k )x˜ H (k ) K k=1

= H

K 1 x ( k )xH ( k ) K k=1

ˆ = H R

(27)

Employ the approximation property of (26), we can rewrite the above equation

˜ = R

K 1 x˜ (k )x˜ H (k ) K k=1

K 1 ∼ (xi (k ) + H xn (k ))(xi (k ) + H xn (k ))H = K k=1

∼ ˆ i + σˆ n2 H  =R K

(28)

( k )H

σˆ n2 H 

is the reconstructed ICM, k=1 xi (k )xi  is denoted by K1 Kk=1 (H xn (k ))(H xn (k ))H , and the cross component of xi (k) and xn (k) could be ignored. Combining the (27) and (28), we can obtain the reconstructed ICM: ˆi = where R

1 K

ˆ i + σˆ n2 H  H Rˆ  ∼ =R ˆi ∼ ˆ  − σˆ n2 H  R = H R

(29)

where σˆ n2 is the estimated noise power in (17), then we obtain the reconstructed INCM:

˜ i+n = R ˆ i + σˆ n2 I R ˜ − σˆ n2 H  + σˆ n2 I =R ˜ + σˆ n2 (I − H ) =R

(30)

using the (21) and (30), the optimal weight vector is computed as:

ˆ pro1 = w

˜ −1 aˆ s R i+n ˜ −1 ˆ s aˆ H s Ri+ n a

(31)

X. Zhu, X. Xu and Z. Ye / Signal Processing 167 (2020) 107289

5

have the constraint inequality:

UH (a¯l + e )22 < UH a¯ l 22 , l = 1, 2, · · · , Lˆ

(35)

Without loss of generality, the mismatch e of a¯ l is decomposed into orthogonal component e⊥ and parallel component e . While e is scaling vector and doesn’t influence SINR, which can be ignored, so the object is to solve the e⊥ . Inspired by the similar idea of [16,22,23,31], the solution of e⊥ can be formulated as following problem:

minimize

(a¯ l + e⊥ )H Rˆ −1 (a¯ l + e⊥ )

subject to

e =0 a¯ H l ⊥ UH (a¯ l + e⊥ )2  UH a¯ l 2

e⊥

Fig. 2. Comparison of Eqs. (13) and (32) with one 10 dB desired signal and two 10 dB interference at K = 30, M = 10.

We can examine the effectiveness of projection (26) by plotting ˜ i+n , replacing R ˜ i+n with R ˆ in (13), which the power spectrum of R is written as:

P˜(θ ) =

1 ˜ −1 a¯ (θ ) a¯ H (θ )R

(32)

The Eq. (36) is similar to that in [16,22,23,31]. The difference is that the quadratic convex optimization problem in [16,22,23,31] aims to obtain the SV of desired signal, while the (36) devotes to solving the SVs of interference. The convex optimization problem in (36) can be efficiently solved by convex optimization toolbox [37]. The optimization of a¯ lˆ is written as:

a˜ l = a¯ l + e⊥ ,

˘i = R

l = 1, 2, · · · , Lˆ

(33)

where Lˆ is the number of peaks searched. Compared with [31], the most difference is that Eq. (33) doesn’t contain the desired signal components, so the peaks of (33) don’t suffer from the dither of desired signal or the influence of some spurious peaks. Combining the known array geometry with the estimated directions, we get the nominal SVs of interference {a¯ (θ¯1 ), a¯ (θ¯2 ), · · · , a¯ (θ¯Lˆ )} = {a¯ 1 , a¯ 2 , · · · , a¯ Lˆ }. Obviously, these nominal SVs of interference are not precise in the presence of array geometry mismatch errors. The true SVs of interference al = a¯ l + e must lie in the interference subspace, where e is the mismatch vector, and we have spanned the interference subspace B1 in the proposed-1 method. So the complement subspace of B1 would be orthogonal to the true SVs of interference as follow:

U al = 0,

l = 1, 2, · · · , L − B1 BH 1.

Lˆ 

σ˜ (θ¯l )a˜ l a˜ Hl

=

Lˆ 

a˜ l a˜ H l

l=1

˜ −1 a¯ (θ¯l ) a¯ H (θ¯l )R i+n

(38)

Combing the estimated noise power σˆ n2 , we obtain the reconstructed ICM as follow:

˘ i+n = R ˘ i + σˆ n2 I R =

Lˆ 

a˜ l a˜ H l

l=1

˜ −1 a¯ (θ¯l ) a¯ H (θ¯l )R i+n

+ σˆ n2 I

(39)

Substituting (21) and (39) to (8), we get the optimal weight vector:

From Fig. 1, we can see that the peaks of (32) is no lower than that of (13), which means that the desired signal components are eliminated completely and the interference components are completely preserved. The locations of these peaks indicate the directions of impinging interference {θ¯1 , θ¯2 , · · · , θ¯Lˆ }, and the value of peaks approximately denote the powers of interference:

H

(37)

l=1

3.3. Subspace orthogonality for ICM reconstruction

1 , ˜ −1 a¯ (θ¯l ) a¯ H (θ¯l )R i+n

l = 1, 2, · · · , Lˆ

Based on the each optimized SV of interference and corresponding power, the more precise ICM is reconstructed as theoretical definition (6):

i+n

Fig. 2 shows the spatial power spectrum distribution based on (32). The desired signal is assumed impinging from θ0 = −5◦ , and two interference are from θ1 = −40◦ , θ2 = 30◦ , respectively. The other conditions are the same as those in Fig. 1. It could be seen that there are only two peaks denoting the interference in the curves of (32), and comparing to the curves of (13), the peak of desired signal disappear. So we can conclude that the proposed-1 subspace projection method can eliminate the desired signal components effectively from the received snapshots.

σ˜ (θ¯l ) =

(36)

(34)

where U = I For nominal SVs, the orthogonality of (34) is destroyed [16], which is replaced by UH a¯ l  > 0. Therefore, we

ˆ pro2 = w

˘ −1 aˆ s R i+n ˘ −1 ˆ s aˆ H s Ri+ n a

(40)

In our proposed two methods, the noise power estimation costs the computational complexity of O(max(M2 J + M2 K + M3 )) due to the sample covariance matrix calculation, matrix inversion and discrete point calculation in (16), respectively. Usually, J > K M, so the noise power estimation has the complexity of O(M2 J). The desired signal SV estimation denotes the complexity of O(M2 S + M3 ), where S is the number of the sampling points in s and is larger than M commonly. For ICM reconstruction, the proposed-1 method only needs multiplication of matrix. Consequently, the proposed-1 method has the complexity of O(max(M2 J, M2 S)) and it’s more efficient than [22,23] with O((M3.5 ). The proposed-2 method needs to solve the quadratically constrained quadratic programming (QCQP) problem for each interference SV which has the complexity of O(LM3.5 ) according to [22,23,31], and the peaks searching needs the complexity of O(M2 T) based on [31], where T is the number of searching points in i . In total, the proposed-2 method roughly has the complexity of O(max(M2 T, LM3.5 , M2 J, M2 S)). 4. Simulation results In this section, a ULA with M = 10 omnidirectional sensors spaced half a wavelength is considered. Assuming three signals impinging from the directions of θ0 = −5◦ , θ1 = −40◦ and θ2 = 30◦ ,

6

X. Zhu, X. Xu and Z. Ye / Signal Processing 167 (2020) 107289

Fig. 3. Output SINR versus input SNR in case of look direction error.

and the estimated directions are θ¯0 = −8◦ , θ¯1 = −43◦ , θ¯2 = 27◦ , respectively. The first signal is considered to be the desired signal and the remaining two signals are interference with 20 dB interference-to-noise-ratio (INR). The additive noise is modeled as the complex Gaussian temporally and spatially white process with zero mean and unit covariance. The angular sector of the desired signal is set as s = [θ¯0 − 8◦ , θ¯0 + 8◦ ], while the interference region is set to be i = 1 ∪ 2 = [θ¯1 − 8◦ , θ¯1 + 8◦ ] ∪ [θ¯2 − 8◦ , θ¯2 + 8]. Integral operations in this paper are replaced by discrete summation and all angular sectors are uniformly sampled to be discrete sectors with the same angular interval 0.1◦ . Each simulation results are based on 200 Monte-Carlo trials. The proposed-1 beamformer (31) and proposed-2 beamformer (40) are compared to the subspace intersection reconstruction beamformer (INCM-subspace) [28], the covariance matrix reconstruction beamformer of volume integration (INCM-volume) [23], the spatial power spectrum sampling beamformer (INCM-SPSS) [25], the interference covariance matrix reconstruction beamformer (INCM-linear) [22], the worstcase-based beamformer [18] and the eigen-based beamformer [16]. The N = 7 dominant eigenvectors of matrix B is employed for B1 both in proposed-1 method and proposed-2 methods. The energy percentage ρ is set as 0.9 both in [16,28]. The parameter√ = 0.3M is set in [18] and in [23] the parameter ε is set as 0.1. The α0 = 0◦ is used and δ = sin−1 (M/2 ) in [25]. CVX toolbox [37] is employed to solve the optimization problem.

Fig. 4. Deviations of different beamformers.

Fig. 5. Output SINR versus the number of snapshots in case of look direction error.

2 method achieves the best performance among all beamformers and proposed-1 method obtains the better performance than INCM-linear beamformer [22] and INCM-SPSS [25] beamformer. The results mean that the number of snapshots doesn’t affect the performance seriously for all tested beamformers.

4.1. Example 1: Mismatch due to look direction error In the first simulation example, the direction mismatch errors of all signals are considered to be randomly and uniformly distributed in [−4◦ , 4◦ ]. Fig. 3 demonstrates the output SINR of tested methods versus the input SNR in the condition of K = 30 snapshots. Fig. 4 displays the deviations of the proposed methods and INCM methods [22,23,25,28]. The proposed-1 method outperforms the INCM-linear beamformer [22] and INCM-SPSS [25] bemformer, and it’s better than INCM-volume beamformer [23] in the low input SNR with the lower computational complexity. The proposed-2 method and INCM-subspace beamformer [28] almost get the optimal performance because they reconstruct the ICM as theoretical definition. Fig. 5 depicts the output SINR of tested beamformers versus the number of snapshot for the fixed SNR = 20 dB. No matter in large or small number of snapshots, the proposed-

4.2. Example 2: Mismatch due to amplitude and phase perturbations error In this simulation example, we examine the influence of amplitude and phase perturbations on the performance of beamformers. It’s supposed that the amplitude and phase error of each sensor are drawn from the random generator N(1, 0.12 ) and N(1, (0.25π )2 ) respectively. Fig. 6 describes the output SINR curves versus the input SNR for the number of snapshots K = 30. When input SNR is less than 5dB, the [16,18] beamformers outperform INCM beamformers [22,23,28] and proposed methods. When the SNR is more than 10dB, the proposed-2 method and INCM-subspace beamformer [28] get the best performance. Fig. 7 shows the deviations of the proposed methods and INCM beamformers [22,23,25,28]. The proposed-1 achieves better performance

X. Zhu, X. Xu and Z. Ye / Signal Processing 167 (2020) 107289

Fig. 6. Output SINR versus input SNR in case of amplitude and phase perturbations.

Fig. 8. Output SINR versus the number of snapshots in case of amplitude and phase perturbations.

Fig. 7. Deviations of different beamformers.

than INCM-linear beamformer [22] and INCM-SPSS beamformer [25] while it performs a little bit worse than INCM-volume beamformer [23]. The slight deviation between proposed-2 method and INCM-subspace beamformer [28] means the proposed-2 method provides the higher output SINR especially when the input SNR is lower than 5 dB. Fig. 8 corresponds to the performance curves versus the number of snapshots at SNR = 20 dB and the results are similar to those in the first example. The proposed-2 method reaches the best performance in the small number of snapshots, and the proposed-1 method also get the same performance with INCMvolume beamformer [23]. While the output SINR of all tested methods are lower than the optimal value.

4.3. Example 3: Mismatch due to sensors location error In the third simulation example, we consider the sensors location error d¯i , i = 1, 2, · · · , M which are uniformly distributed in the interval [−0.1, 0.1] which is measured in wavelength. Based on the known array geometry, the positions of all sensors are expressed

7

Fig. 9. Output SINR versus input SNR in case of sensors location error.

as:

d¯ = [0 + d¯1 , d + d¯2 , · · · , (M − 1 )d + d¯M ]

(41)

then the actual SV is modeled as:

a (θ ) = ϒ a (θ )



= ϒ 1 , e − j 2π

dsinθ

λ

, · · · , e − j 2π

(M−1 )dsinθ λ



T

(42)

where ϒ is the diagonal matrix:

  d¯1 sinθ d¯2 sinθ d¯M sinθ ϒ = diag e− j2π λ , e− j2π λ , · · · , e− j2π λ

(43)

Fig. 9 displays the output SINR of beamformers versus the input SNR under the condition of the number of snapshots K = 30. The proposed methods, INCM-volume beamformer [23] and INCMsubspace beamformer [28] nearly achieve the same performance while [16,18] beamformers obtain the higher output SINR when SNR is lower than 5 dB. Fig. 10 depicts the deviations between proposed methods and INCM methods. It can be observed that the proposed-1 method achieves better performance than INCM-linear

8

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Fig. 10. Deviations of different beamformers. Fig. 12. Output SINR versus input SNR in case of SV random error.

Fig. 11. Output SINR versus the number of snapshots in case of sensors location error.

beamformer [22] and almost gets the same output SINR as INCMvolume beamformer [23]. The deviation between proposed-2 and INCM-subspace [28] is greater than zero which means that the proposed-2 method obtains the higher output SINR. Fig. 11 shows the output SINR of the tested beamformers against the number of snapshots for the fixed SNR = 20 dB. The results demonstrate that the number of snapshots doesn’t affect the output SINR of tested beamformers seriously, and the proposed methods and [23,28] beamformers almost get the same performance.

4.4. Example 4: mismatch due to SV random error In this simulation example, we assume that the SVs of desired signal and interference are randomly distributed in an uncertainty set, which is modeled as:

al = a¯ l + el ,

l = 0, 1, · · · , L

(44)

where a¯ l denotes the nominal SV corresponding to the direction θ¯l , and el represents the random error vector which is expressed

Fig. 13. Deviations of different beamformers.

as:

ε 

el = √ l e jφ0 , e jφ1 , · · · , e jφM−1 M l

l

l

T (45)

where ε l is√the norm of el and it’s uniformly distributed in the l , m = 0, 1, · · · , M − 1 stands for phase of the interval [0, 0.3]. φm random error vector el , which is independently and uniformly distributed in the interval [0, 2π ). The model in (44) is comprehensive which is considered to contain lots of errors, such as direction error, calibration error and so on. Fig. 12 describes the output SINR versus the input SNR for the fixed number of snapshots K = 30. Fig. 13 illustrates the deviations between proposed methods and INCM beamformers. It’s clearly seen that the proposed-1 method outperforms the INCM-linear beamformer [22] and is a little bit worse than INCM-volume beamformer [23] with the lower computational complexity. When SNR is less than 0 dB, the traditional beamformers [16,18] performs better. Fig 14 shows the SINR versus the number of snapshots for the fixed SNR = 20 dB. The proposed methods enjoy the obvious improvements compared with INCMlinear beamformer [22], and the proposed-2 method obtains highest output SINR of all tested bemaformers.

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Fig. 14. Output SINR versus the number of snapshots in case of SV random error.

9

Fig. 15. Output SINR versus input SNR in case of incoherent local scattering error.

4.5. Example 5: Mismatch due to incoherent local scattering error In the fifth example simulation, we analysis effect of the incoherent local scattering error on the output SINR. The desired signal is assumed to have the time-varying signature which is modeled as:

as ( k ) = s0 ( k )a0 +

4 



s p (k )a¯ (θ p )

(46)

p=1



where a0 stands for the direct path, whereas a¯ (θ p ) corresponds to

the incoherently scattering path. The directions of arrival θ p , p = 1, 2, 3, 4 are are independently and randomly drawn from the Gaussian generator N(θ 0 , 4◦ ) and s p (k ), p = 0, 1, 2, 3, 4 independently and randomly drawn from the random generator N(0, 1). The model of (46) is consistent to the case of incoherent local scattering in [18,22,38,39]. In this scenario, we know that the norm square of SV as (k )22 keeps changing from snapshot to snapshot [22], so we have normalized the as (k) based on the number of scattering sources. Besides, under this assumption, the desired signal covariance matrix is no longer rank-one matrix and the output SINR is expressed in a more general form:

SINRopt =

wH Rs w wH Ri+ n w

Fig. 16. Deviations of different beamformers.

(47)

which is maximized by weight vector [18]:

wopt = P {R−1 R} i+n s

(48)

where P denotes the prime eigenvector corresponding to the matrix. Fig. 15 illustrates the output SINR curves versus the input SNR for the fixed number of snapshots K = 30. It can been seen that the proposed-2 method almost achieves the optimal performance. Fig. 16 depicts the deviations between proposed methods and INCM beamformers. The proposed-1 method obtains the bigger output SINR than INCM-linear beamformer [22] no matter in high or low input SNR. The curves of deviation of INCM-volume [23] and INCM-subspace [28] are close to each other, and the proposed-2 methods reaches the better performance than INCMvolume [23] beamformer and INCM-subspace beamformer [28] as long as the input SNR is larger than -5 dB. The output performance of tested beamformers versus the number of snapshots for the fixed SNR = 20 dB is drawn in Fig. 17. No matter in small or large number of snapshots, the proposed methods are robust enough in this case.

Fig. 17. Output SINR versus the number of snapshots in case of incoherent local scattering error.

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5. Conclusion In this paper, we develop subspace methods for RAB. The proposed methods utilize the orthogonality of interference subspace to reconstruct the ICM. The proposed-1 method employs the interference subspace to project onto the received snapshots, which effectively eliminates desired signal components and achieve the better performance. This method is robust against types of mismatch errors with lower computational complexity compared with [22,23]. The spatial power spectrum corresponding to the reconstructed ICM in the first proposed method indicates the nominal SVs of interference. The second proposed method employs the orthogonality property of the interference subspace to optimize the nominal SVs to derive the more precise reconstructed ICM as theoretical definition form. Simulations results demonstrate that the proposed methods are robust enough to against various mismatch errors, where the proposed-1 method can achieve well performance with lower computational complexity and proposed2 method almost obtains the best performance of all the tested beamformers. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgement This work is supported by the National Natural Science Foundation of China (No. 61671418) and the advanced research fund of University of Science and Technology of China. Supplementary material Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.sigpro.2019.107289. References [1] Y. Kaneda, J. Ohga, Adaptive microphone-array system for noise reduction, IEEE Trans. Acoust. Speech Signal Process. 34 (6) (1986) 1391–1400. [2] L.C. Godara, Application of antenna arrays to mobile communications. ii. beam-forming and direction-of-arrival considerations, Proc. IEEE 85 (8) (1997) 1195–1245. [3] A.B. Gershman, E. Nemeth, J.F. Bohme, Experimental performance of adaptive beamforming in a sonar environment with a towed array and moving interfering sources, IEEE Trans. Signal Process. 48 (1) (20 0 0) 246–250. [4] I.S. Reed, J.D. Mallett, L.E. Brennan, Rapid convergence rate in adaptive arrays, IEEE Trans. Aerospace Electron. Syst. (6) (1974) 853–863. [5] H.L. Van Trees, Optimum Array Processing: Part IV of Detection, Estimation, and Modulation Theory, John Wiley and Sons, 2004. [6] J. Capon, High-resolution frequency-wavenumber spectrum analysis, Proc. IEEE 57 (8) (1969) 1408–1418. [7] H. Cox, R. Zeskind, M. Owen, Robust adaptive beamforming, IEEE Trans. Acoust. Speech Signal Process. 35 (10) (1987) 1365–1376. [8] B.D. Carlson, Covariance matrix estimation errors and diagonal loading in adaptive arrays, IEEE Trans. Aerospace Electron. Syst. 24 (4) (1988) 397– 401. [9] J. Li, P. Stoica, Z. Wang, On robust capon beamforming and diagonal loading, IEEE Trans. Signal Process. 51 (7) (2003) 1702–1715. [10] A. Elnashar, S.M. Elnoubi, H.A. El-Mikati, Further study on robust adaptive beamforming with optimum diagonal loading, IEEE Trans. Antennas Propagat. 54 (12) (2006) 3647–3658.

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