Robust adaptive beamforming via coprime coarray interpolation

Robust Adaptive Beamforming via Coprime Coarray Interpolation

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Robust Adaptive Beamforming via Coprime Coarray Interpolation Zhi Zheng, Tong Yang, Wen-Qin Wang, Shunsheng Zhang PII: DOI: Reference:

S0165-1684(19)30431-1 https://doi.org/10.1016/j.sigpro.2019.107382 SIGPRO 107382

To appear in:

Signal Processing

Received date: Revised date: Accepted date:

31 March 2019 29 September 2019 15 November 2019

Please cite this article as: Zhi Zheng, Tong Yang, Wen-Qin Wang, Shunsheng Zhang, Robust Adaptive Beamforming via Coprime Coarray Interpolation, Signal Processing (2019), doi: https://doi.org/10.1016/j.sigpro.2019.107382

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Highlights • A robust adaptive beamforming scheme using coprime arrays is presented. • The array aperture loss via coarray interpolation can be avoided. • The INCM reconstruction is significantly enhanced. • The output SINR of the beamformer at high SNRs is improved.

1

Robust Adaptive Beamforming via Coprime Coarray Interpolation✩ Zhi Zheng*a,b , Tong Yanga,, Wen-Qin Wanga,, Shunsheng Zhangc, a School

of Information and Communication Engineering, University of Electronic Science and Technology of China, Chengdu 611731, China b Institute of Electronic and Information Engineering of UESTC in Guangdong, Dongguan 523808, China c Research Institute Electronic Science and Technology, University of Electronic Science and Technology of China, Chengdu 611731, China

Abstract Recently, the coprime array has aroused wide concern due to its capability of achieving enhanced degrees of freedom (DOF) and increased array aperture compared to uniform linear arrays (ULAs). However, it experiences a certain DOF and aperture loss due to the existence of “holes” in the difference coarray. These drawbacks are adverse to enhance the performance of direction-of-arrival (DOA) estimation and adaptive beamforming. In this paper, we propose a new robust adaptive beamforming algorithm with the coprime array. The proposed algorithm yields the beamforming coefficients by estimating the steering vector of the signal-of-interest (SOI) and reconstructing the interference-plus-noise covariance matrix (INCM). Unlike previous techniques, we estimate the DOAs of the SOI and interferences using an interpolated virtual ULA, and further derive their steering vectors and powers. With the high-precision DOA estimates, the steering vectors and powers can be more accurately estimated. The proposed algorithm avoids the coarray aperture loss and enhances the accuracy of INCM reconstruction. Numerical results indicate that the proposed algorithm outperforms the existing approaches in high SNR regions. ✩ Fully

documented templates are available in the elsarticle package on CTAN. author: Zhi Zheng Email addresses: [email protected] (Zhi Zheng*), [email protected] (Tong Yang), [email protected] (Wen-Qin Wang), zhangss [email protected] (Shunsheng Zhang) ∗ Corresponding

Preprint submitted to Signal Processing

November 18, 2019

Keywords: Robust adaptive beamforming, coprime array, degrees of freedom (DOF), difference coarray, interpolation.

1. Introduction Adaptive beamforming has always been considered as a fundamental technique in array signal processing because it is extensively applied in radar [1], sonar [2], communications [3], medical imaging [4], radio astronomy [5] and 5

other areas [6]. The standard Capon beamformer (SCB) is an optimal spatial filter, which has remarkable resolution and interference rejection capability [7]. However, it is quite sensitive to the steering vector mismatch of the signal-of-interest (SOI), resulting from some non-ideal factors like wavefront distortion, local scattering or look direction errors. In the presence of these fac-

10

tors, the performance of the SCB will significantly degrade, especially when the SOI component is contained in the training snapshots. To enhance the robustness of the SCB, various robust beamforming techniques have been suggested during the past decades. Their categories mainly include: diagonal loading method [8, 9, 10, 11, 12, 13], eigenspace-based method [14, 15, 16], uncertainty

15

set-based method [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], and covariance matrix reconstruction-based method [30, 31, 32, 33, 34, 35, 36, 37]. It is generally known that the degrees of freedom (DOF) and array aperture are critical to improving the performance of adaptive beamforming. They determine the spatial resolution and interference rejection capability. Currently, most

20

of adaptive beamformers consider conventional uniform linear arrays (ULAs), where the inter-sensor spacing is within half a wavelength to avoid spatial aliasing. However, the DOF and array aperture of a conventional ULA are only linear to the number of sensors. To increase the DOF and array aperture of the conventional ULA, more physical sensors are required, thus increasing the hard-

25

ware cost and computational complexity [38]. Fortunately, nonuniform linear arrays (NLAs) provide an effective solution to these problems. By generating the difference coarray, a NLA can achieve a number of DOF sensors that is

3

much greater than that of physical sensors. In the past few years, nested [39] and coprime [40] arrays, as two represen30

tative NLAs, have attracted considerable interest because they not only can achieve increased DOF and array aperture compared to traditional ULAs, but also have closed-form expressions for the sensor positions. Compared with the nested array, the coprime array further reduces the mutual coupling effects between sensors. However, the prototype coprime array offers more less DOF than

35

the nested array with the same number of sensors. To overcome this shortcoming, an extended coprime array is presented in [41] by doubling the number of sensors in one ULA. By using 2M + N − 1 physical sensors, the extended coprime array can achieve 2M N + 2N − 1 consecutive coarray lags. Recently, many direction-of-arrival (DOA) estimation techniques using co-

40

prime arrays have been suggested [42, 43, 44, 45, 46, 47]. However, coprimearray-based adaptive beamforming techniques are seldom studied. Gu et al. [48] proposed a coprime array adaptive beamforming based on compressive sensing (CS), where a compressive matrix is used to transform the virtual array signal to a signal with physical array dimension. The CS-based method exploits

45

an enhanced DOF, but it suffers from serious performance loss because only a part of virtual sensors is used in compressive sampling and the compressed signal still contains the SOI component. By using the Capon spatial spectrum of the coprime coarray, the interference-plus-noise covariance matrix (INCM) is reconstructed by integration or summation, and two robust beamformers are

50

then derived [49]. However, the coarray Capon spectrum is imperfect and two derived beamformers suffer different degrees of performance loss compared to the optimal SINR. In [50], a robust and efficient algorithm for coprime array beamforming was proposed. This algorithm reconstructs the INCM by estimating the DOAs and powers of the incoming signals. But it employs coprime

55

subarrays to perform DOA estimation, the number of achievable DOF is limited by the sensor number of subarrays. In [51], a coprime array robust beamforming using covariance matrix reconstruction was suggested. But this method requires the knowledge about the angular sector in which each interference is located. 4

Moreover, the prototype coprime array is used in this method, and thus a lower 60

number of DOF is available compared to the preceding methods. The abovementioned methods only utilize the consecutive ULA segment in the difference coarray. Thus, a certain DOF or aperture loss is unavoidable. That degrades the performance of the adaptive beamformer to some extent. More recently, the coarray interpolation technique has been used for DOA

65

estimation [52, 53, 54, 55, 56]. This technique fully exploits all the coarray sensors, and thus can avoid DOF and array aperture loss. Inspired by the idea, in this paper we propose a new robust adaptive beamforming algorithm based on coprime arrays. Unlike any existing technique, the proposed algorithm reconstructs the INCM of the coprime array based on an interpolated virtual

70

ULA. Our algorithm increases the available aperture of the coprime array, and thus can reconstruct the INCM more accurately. Numerical results verify the performance of the proposed algorithm. Our major contributions are listed as follows: 1. We estimate the steering vectors and powers of the SOI and interferences

75

via coprime coarray interpolation. 2. We enhance the INCM reconstruction with the high-precision estimates of steering vectors and powers. 3. We analyze the computational complexity of the proposed algorithm, and investigate its advantages over existing techniques by numerical examples.

80

The remainder of this paper is organized as follows. The coprime array signal model and problem background are described in Section 2. A robust adaptive beamforming algorithm with the coprime coarray is presented in Section 3. Numerical examples are provided in Section 4. Finally, conclusions are drawn in Section 5.

85

Throughout this paper, we use bold lower-case (upper-case) characters to represent vectors (matrices). In particular, I and 0 stand for the identity matrix and zero matrix with appropriate dimension. The superscripts ∗, T and H denote the complex conjugate, transpose and conjugate transpose, respectively. 5

E{·} is the statistical expectation, k · kF denotes the Frobenius norm, k · k∗ 90

denotes the nuclear norm, rank(·) denotes the rank of a matrix, vec(·) represents the vectorization operator, and diag(a) denotes a diagonal matrix that uses the elements of a as its diagonal elements. A  0 means that A is positive definite. [x]i stands for the ith element of a vector x, hxin denotes the value of the signal

x at the index location n, and hRin1 ,n2 = E[hxin1 hxi∗n2 ]. |A| represents the

95

cardinality of a set A. The symbol ⊗ denotes the Kronecker product. 2. Coprime Array Signal Model and Background

We consider the coprime array configuration as depicted in Fig. 1, which consists of two sparsely-spaced ULAs. The first ULA has 2M sensors with intersensor spacing N d, and the second ULA has N sensors with inter-sensor spacing M d, where M and N are coprime integers satisfying M < N . Without loss of generality, d is chosen to be λ/2, where λ is the carrier wavelength. According to the property of coprime integers, there is no overlap between two sparse ULAs except the commonly referred one. Consequently, there are 2M +N −1 physical sensors in the coprime array [41]. For coprime arrays, the sensor positions are denoted by pi d, where pi belongs to an integer set: S = {pi , i = 1, 2, . . . , 2M + N − 1} [ = {M n, 0 ≤ n ≤ N − 1} {N m, 0 ≤ m ≤ 2M − 1}.

(1)

Assume that one narrowband SOI and L narrowband interferences impinge on the array from distinct directions θ0 , θ1 , . . . , θL . The array observed vector at time instant k can be modeled as x(k) = xs (k) + xi (k) + xn (k) where xs (k) = a0 s0 (k), xi (k) =

PL

l=1

(2)

al sl (k), and xn (k) ∈ CM ×1 are statisti-

cally independent components of the SOI, interferences, and sensor noise, respec6

Nd

Nd

Nd

Ă 0

1 Md

Md

2

Ă M-1

M

2M-1

Ă 0

1

N-1

2 (a)

Nd

Nd

Nd Ă

Md

Ă

Md (b)

Figure 1: The coprime array configuration. (a) A coprime pair of sparsely-spaced ULAs. (b) Aligned coprime array geometry.

tively. sl (k) stands for the lth source signal waveform, and al = a(θl ) ∈ CM ×1

is the corresponding steering vector. The elements of xn (k) ∼ CN (0, σn2 I) are assumed to be independent and identically distributed (i.i.d.) additive white Gaussian noise. The steering vector for the lth source is expressed as  T 2π 2π a(θl ) = 1, ej λ p2 d sin θl , . . . , ej λ p2M +N −1 d sin θl .

(3)

The adaptive beamformer output is given by y(k) = wH x(k),

(4)

where w = [w1 , . . . , wM ]T ∈ CM ×1 is the beamformer weight vector. The output signal-to-interference-plus-noise ratio (SINR) of the coprime array is defined as SINR =

σ02 |wH a0 |2 wH Ri+n w

(5)

where σ02 = E{|s0 (k)|2 } is the SOI power, and Ri+n ∈ C(2M +N −1)×(2M +N −1)

7

denotes the INCM, given by  Ri+n = E (xi (k) + xn (k))(xi (k) + xn (k))H =

L X

2 σl2 al aH l + σn I,

(6)

l=1

where σl2 stands for the lth interference power. To maximize the output SINR, the output interference-plus-noise power should be minimized while maintaining a distortionless response toward the SOI. Consequently, the maximization of (5) is equivalent to the following minimization problem: min wH Ri+n w

s.t.

w

wH a0 = 1,

(7)

which results in the Capon beamformer wopt =

R−1 i+n a0 −1 aH 0 Ri+n a0

.

(8)

Since the exact Ri+n is unavailable in practice, it is usually replaced by the following sample covariance matrix (SCM) K X ˆ = 1 x(k)xH (k), R K

(9)

k=1

where K denotes the number of training snapshots. Then the beamformer weight vector (8) can be rewritten as wSMI =

ˆ −1 a0 R , H ˆ −1 a0 a0 R

(10)

which is known as the sample matrix inversion (SMI) beamformer. The SMI 100

beamformer is sensitive to limited samples and steering vector mismatches, especially when the SOI appears in the training snapshots [17, 30].

8

3. Proposed Algorithm In this section, we develop a robust adaptive beamforming algorithm with the coprime array. Unlike previous techniques, we reconstructs the INCM via 105

coarray interpolation. Therefore, our scheme can obtain more accurate INCM than the existing techniques. ˆ yields the following vector Vectorizing R ˆ = Bp + σ 2 1n z = vec(R) n

(11)

where B = [b(θ0 ), b(θ1 ), . . . , b(θL )] with b(θl ) = a∗ (θl )⊗a(θl ) for l = 0, 1, . . . , L, 2 T ] , and 1n = vec(I) = [eT1 , eT2 , . . . , eT2M +N −1 ]T with ei being p = [σ02 , σ12 , . . . , σL

a column vector of all zeros except a 1 at the ith position. The vector z can be 110

viewed as a single snapshot data from the equivalent signal vector p, and σn2 1n represents a deterministic vector. The distinct rows of B act like the manifold of a longer array whose sensor positions are represented by the distinct integers in the set D = {n1 − n2 |n1 , n2 ∈ S}. Removing the repeated rows from the vector z and sorting them to get a new vector z1 given by hz1 im =

1 |T(m)|

X

(n1 ,n2 )∈T(m)

ˆ n ,n , m ∈ D hRi 1 2

(12)

where the set T(m) collects all pairs (n1 , n2 ) contributing to the coarray index m, i.e., T(m) = {(n1 , n2 ) ∈ S2 |n1 − n2 = m}, m ∈ D.

(13)

Based on the property of coprime integers, there only exists a consecutive virtual ULA between −(M N + M − 1)d and (M N + M − 1)d in the difference coarray [52]: D = {±(M n − N m)} 9

(14)

with 0 ≤ n ≤ N − 1 and 0 ≤ m ≤ 2M − 1. That is, there are some “holes” in the 115

difference coarray D. Therefore, we cannot directly exploit the coarray output

z1 when spatial smoothing or Toeplitz step is utilized to derive the full-rank covariance matrix of the virtual array. Inspired by the idea of coarray interpolation [52, 53, 54, 55, 56], we construct ¯1 ∈ C4M N −2N +1 , which can be initialized as an interpolated signal vector z h¯ z1 in =

 hz1 in , n ∈ D,  0,

(15)

n∈V−D

¯1 , where the signals corresponding to the coprime coarray D are included in z while those corresponding to the interpolated sensors are set to zeros. The ¯1 can be viewed as a single snapshot data received by a interpolated vector z longer ULA consisting of 4M N − 2N + 1 sensors, whose sensor positions are given by the integer set: V = {m| min(D) ≤ m ≤ max(D)}.

(16)

¯1 is also Hermitian symmetric, i.e., Since z1 is Hermitian symmetric [57], z h¯ z1 in = h¯ z1 i∗−n ,

(17)

[¯ z1 ]i = [¯ z1 ]∗4M N −2N +2−i

(18)

where n ∈ [0, 2M N − N ] and i ∈ [1, 4M N − 2N + 1].

+

¯1 , we can form a Toeplitz matrix RV ∈ C|V From the vector z

10

|×|V+ |

expressed

as 

[¯ z1 ]|V+ |

  z1 ]|V+ |+1  [¯ RV =   ..  .  [¯ z1 ]2|V+ |−1

[¯ z1 ]|V+ |−1

···

[¯ z1 ] 1

[¯ z1 ]|V+ | .. .

···

[¯ z1 ] 2

..

[¯ z1 ]2|V+ |−2

.

.. .

···

[¯ z1 ]|V+ |

       

(19)

where V+ = {n|n ∈ V, n ≥ 0} is the non-negative part of V. Clearly, the matrix RV contains some zero elements. Therefore, we need to recover the missing correlation information in RV . Since the desired covariance matrix is Hermitian Toeplitz and owns low-rank property, the covariance matrix of the interpolated ULA can be recovered by solving the minimization problem [52]: min

¯ V ∈C|V+ |×|V+ | R

¯ V) rank(R

¯V = R ¯H s.t. R V ¯ V in ,n = hRV in ,n hR 1 2 1 2 n1 , n2 ∈ V+ , n1 − n2 ∈ D. 120

(20)

¯ V is a Hermitian Toeplitz matrix, and The constraints in (20) can ensure that R ¯ V. the known correlation information on D, is entirely contained in R Clearly, (20) is a nonconvex NP-hard problem. By using the nuclear norm minimization, (20) can be reformulated as [52]: min

¯ V ∈C|V+ |×|V+ | R

¯ V k∗ kR

¯V = R ¯H s.t. R V ¯ V in ,n = hRV in ,n hR 1 2 1 2 n1 , n2 ∈ V+ , n1 − n2 ∈ D.

(21)

The optimization problem (21) is convex and can be efficiently solved by using

11

interior point methods. ¯ V yields Performing eigenvalue decomposition (EVD) of R ¯ V = U s Σs U H + U n Σn U H R s n where Σs ∈ C(L+1)×(L+1) and Σn ∈ C(|V

125

+

|−(L+1))×(|V+ |−(L+1))

(22) are the diagonal

matrices containing the (L+1) largest and (|V+ |−(L+1)) smallest eigenvalues of ¯ V , respectively, Us ∈ C|V+ |×(L+1) and Un ∈ C|V+ |×(|V+ |−(L+1)) are composed R

¯ V corresponding to the (L+1) largest and (|V+ |−(L+1)) of the eigenvectors of R smallest eigenvalues, respectively. Then, we create the MUSIC spatial spectrum as PˆMUSIC (θ) =

1 dH (θ)U

H n Un d(θ)

, ∀ θ ∈ [−90◦ , 90◦ ]

(23)

where d(θ) denotes the steering vector of the interpolated virtual ULA and has the following general form h iT 2π 2π d(θ) = 1, ej λ d sin θ , . . . , ej λ (2M N −N )d sin θ .

(24)

From the spatial spectrum, we can obtain multiple peaks whose locations 130

may correspond to the signal DOAs. As the INCM-reconstruction-based methods [30, 31, 32, 33, 34, 35, 36, 37, 49, 50, 51], we assume that Θ is the angular ¯ is the out-of-sector of Θ in the whole spatial sector in which the SOI lies and Θ domain. Therefore, we can identify the DOAs of the SOI and interferences. Specifically, the maximum peak in Θ determines the estimate of the SOI DOA,

135

¯ correspond to the estimates of the interferθˆ0 , while the L maximum peaks in Θ ence DOAs, θˆ1 , . . . , θˆL . Further, the steering vectors of the SOI and interferences can be estimated as a(θˆ0 ), a(θˆ1 ), . . . , a(θˆL ), respectively. Defining the direction vector θˆ = {θˆ0 , θˆ1 , . . . , θˆL }, we can formulate the least

12

squares (LS) problem: H ˆ 2 ˆ ˆ ˆ −σ min kR ˆn2 I − A(θ)diag{p( θ)}A (θ)kF ˆ p(θ)

ˆ 0 s.t. p(θ)

(25)

ˆ = {P (θˆ0 ), P (θˆ1 ), . . . , P (θˆL )} represents the power distribution on where p(θ)

ˆ = {a(θˆ0 ), a(θˆ1 ), . . . , a(θˆL )} ∈ θˆ with P (θˆl ) being the lth power estimate, A(θ) 140

C(2M +N −1)×(L+1) is the corresponding array manifold matrix, and σ ˆn2 is the

noise power estimate, which is usually considered as the minimum eigenvalue of ˆ R. In practice, the interference powers are generally greater than zero. Therefore, we may neglect the inequality constraint in (25). Then, the solution to (25) can be expressed as ˆ = (GH G)−1 GH r p(θ)

(26)

ˆ −σ where G = {vec(a(θˆ0 )aH (θˆ0 )), . . . , vec(a(θˆL )aH (θˆL ))}, and r = vec(R ˆn2 I). Now we reconstruct the INCM of the coprime array as ˆ i+n = R

L X

P (θˆl )a(θˆl )aH (θˆl ) + σ ˆn2 I.

(27)

l=1

ˆ i+n and the estimated SOI steering Substituting the reconstructed INCM R vector a(θˆ0 ) into (8), we can get the weight vector of the proposed beamformer expressed as

wpro =

ˆ −1 a(θˆ0 ) R i+n . H ˆ ˆ −1 a(θˆ0 ) a (θ0 )R

(28)

i+n

Next, we analyze the computational complexity of the proposed algorithm. 145

Its major computations involve peak searching, high-dimensional EVD operation, as well as power estimation. The complexity required by peak searching is O((2M N − N + 1)2 P ), where P is the number of grid points in the an13

¯ V requires a complexity of gular domain. In addition, the EVD operation of R O((2M N −N +1)3 ). While the power estimation needs to calculate the pseudo150

inverse of a (2M + N − 1)2 × (L + 1) matrix, which requires a complexity of O((2M +N −1)2 (L+1)2 ). Consequently, the overall complexity of the proposed

algorithm is approximately O(max((2M N − N + 1)2 P, (2M N − N + 1)3 )).

For clarity, the main steps of the proposed algorithm are summarized as follows: 155

ˆ to get the coarray output z, remove the reStep 1) Vectorize the SCM R peated rows from z and rearrange them to yield a new vector z1 . ¯1 using (15). Step 2) Construct an interpolated signal vector z ¯1 . Step 3) Construct a Toeplitz matrix RV using the interpolated vector z Step 4) Formulate the nuclear norm minimization problem (21), solve the

160

¯ V. optimization problem (21) to obtain R ¯ V to obtain Un , and create the MUSIC spatial Step 5) Perform EVD on R spectrum using (23). Step 6) Estimate the signal DOAs via peak searching, and derive corresponding steering vectors.

165

Step 7) Formulate the LS problem (25), and solve this problem to obtain ˆ the power estimate vector p(θ). ˆ i+n via (27). Step 8) Reconstruct the INCM R ˆ i+n and the estimated SOI Step 9) Substitute the reconstructed INCM R steering vector a(θˆ0 ) into (8) to get the proposed beamformer (28).

170

4. Numerical Examples Numerical examples are presented to investigate the performance of the proposed algorithm. In all the examples, we consider a coprime array with coprime integers M = 3 and N = 5, which means the coprime array consists of 2M + N − 1 = 10 sensors. One SOI is assumed to impinge on the array from the

175

direction θ0 = 0◦ , while three interferences are presumed to arrive at the array from the directions θ1 = −30◦ , θ2 = 30◦ and θ3 = 45◦ , respectively. The INR in 14

each sensor is equal to 30 dB in all simulations. The input SNR is fixed at 20 dB (except the figures where the SNR varies), while the number of snapshots is set as K = 500 (except the figures where the number of snapshots varies). For each 180

figure, 500 Monte-Carlo trials are performed. The proposed beamformer (28) is compared to the following beamformers: i) integration beamformer [49]; ii) summation beamformer [49]; iii) CS-based beamformer [48]; iv) reconstructionbased beamformer [30]; v) worst-case-based beamformer [17]. For the proposed beamformer and the beamformers of [30, 49], we assume the angular sector of

185

¯ = [−90◦ , −5◦ ) ∪ (5◦ , 90◦ ]. The factor the SOI to be Θ = [−5◦ , 5◦ ], and thus, Θ ε is set to ε = 0.2(M + N − 1) in the worst-case-based beamformer. The Matlab CVX toolbox [58] is used for solving these optimization problems in these compared methods. 4.1. Example 1: Exactly Known Signal Steering Vector

190

In the first example, we focus on the case where the steering vectors corresponding to the incoming signals are exactly known. Note that even in this case, the SOI component contained in the training snapshots can still cause the performance deterioration of the adaptive beamformer [17]. The output SINR of the tested beamformers versus SNR is shown in Fig. 2, from which we ob-

195

serve that the proposed beamformer achieves nearly optimal SINR from -30 dB to 30 dB and outperforms all the compared beamformers at high SNRs. The worst-case-based beamformer [17] also provides nearly optimal performance at low SNRs, but suffers significant performance degradation at high SNRs because they are designed based on the SCM where the SOI component is not thoroughly

200

removed. The proposed beamformer outperforms the remaining beamformers because it not only fundamentally eliminates the SOI component, but also is based on a more precise INCM reconstruction. The output SINR of the investigated beamformers against the number of snapshots is plotted in Fig. 3. It is observed that the proposed beamformer always performs better than the other

205

beamformers tested when the snapshots K varies from 100 to 1000.

15

40 Optimal SINR Proposed Integration Summation CS Reconstruction Worst-Case

30

Output SINR (dB)

20 10 0 -10 -20 -30 -40 -30

-20

-10

0 SNR (dB)

10

20

30

Figure 2: Output SINR versus SNR at K = 500 and INR = 30 dB. First example.

30

Output SINR (dB)

25

Optimal SINR Proposed Integration Summation CS Reconstruction Worst-Case

20

15

10

5

0

0

100 200 300 400 500 600 700 800 900 1000 Number of snapshots

Figure 3: Output SINR versus the number of snapshots K at SNR = 20 dB and INR = 30 dB. First example.

16

4.2. Example 2: Mismatch Due to Signal Look Direction Error In the second example, we examine the impact of signal look direction error on performance of the beamformers. The random DOA error of the incoming signals are uniformly distributed in [−4◦ , 4◦ ]. More precisely, the actual DOA of 210

the SOI is evenly distributed in [−4◦ , 4◦ ], and the DOAs of three interferences are uniformly distributed in [−34◦ , −26◦ ], [26◦ , 34◦ ] and [41◦ , 49◦ ], respectively. Note that the DOAs of the incoming signals vary in each Monte-Carlo trial while remaining unchanged over snapshots. Fig. 4 displays the output SINR of the tested beamformers versus SNR. It is observed that the proposed beamformer

215

and the summation beamformer [49] achieve good performance across a wide range of SNR, while the worst-case-based beamformer [17] performs well only at low SNRs. Furthermore, the proposed beamformer outperforms the summation beamformer [49] because the former achieves more accurate INCM reconstruction than the latter via coarray interpolation. The output SINR of the examined beamformers against the number of snapshots is illustrated in Fig. 5. It can be seen that our proposed beamformer achieves better performance than its rivals in the range from 100 to 1000 snapshots. 40 30 20 Output SINR (dB)

220

10

Optimal SINR Proposed Integration Summation CS Reconstruction Worst-Case

0 -10 -20 -30 -40 -30

-20

-10

0 SNR (dB)

10

20

30

Figure 4: Output SINR versus SNR at K = 500 and INR = 30 dB. Second example.

17

30

Output SINR (dB)

25

Optimal SINR Proposed Integration Summation CS Reconstruction Worst-Case

20

15

10

5

0

0

100 200 300 400 500 600 700 800 900 1000 Number of snapshots

Figure 5: Output SINR versus the number of snapshots K at SNR = 20 dB and INR = 30 dB. Second example.

4.3. Example 3: Mismatch Due to Coherent Local Scattering In the third example, the SOI steering vector is distorted by coherent local scattering effects and can be expressed as

a0 = p +

4 X

ejψp a(θp )

(29)

p=1

where p refers to the direct path while a(θp ) (p = 1, 2, 3, 4) is corresponding 225

to the scattered paths. The angles θp (p = 1, 2, 3, 4) are independently uniformly distributed in [θ0 − 2◦ , θ0 + 2◦ ] in each trial. The phase parameters ψp (p = 1, 2, 3, 4) are independently and uniformly drawn from the interval [0, 2π] in each trial. Note that θp and ψp vary from trial to trial while remaining fixed from snapshot to snapshot. Fig. 6 demonstrates the output SINR of

230

the tested beamformers versus the SNR. Note that in this example, the SNR is defined by collecting all the scattering paths. As can be seen from Fig. 6, our proposed beamformer and the worst-case-based beamformer [17] exhibit better

18

performance than the remaining beamformers at low SNRs, while our proposed beamformer and the summation beamformer [49] are superior to the other beam235

formers in high SNR regions. Furthermore, the proposed beamformer yields the best performance among all the beamformers compared here at high SNRs. The output SINR of the tested beamformers against the number of snapshots is shown in Fig. 7. We observe clearly that the proposed beamformer offers higher output SINR than its competitors across a wide range of snapshots. 40 30

Output SINR (dB)

20 10

Optimal SINR Proposed Integration Summation CS Reconstruction Worst-Case

0 -10 -20 -30 -40 -30

-20

-10

0 SNR (dB)

10

20

30

Figure 6: Output SINR versus SNR at K = 500 and INR = 30 dB. Third example.

240

4.4. Example 4: Mismatch Due to Incoherent Local Scattering In the fourth example, we consider the case of incoherent local scattering. The SOI is assumed to has a time-varying spatial signature and its steering vector is modelled as [17]:

a(k) = s0 (k)a(θ0 ) +

4 X

sp (k)a(θp )

(30)

p=1

where sp (k) (p = 0, 1, 2, 3, 4) are i.i.d. zero-mean complex Gaussian random variables independently drawn from a random generator N (0, 1). The angles 19

30

Output SINR (dB)

25 Optimal SINR Proposed Integration Summation CS Reconstruction Worst-Case

20 15 10 5 0

0

100 200 300 400 500 600 700 800 900 1000 Number of snapshots

Figure 7: Output SINR versus the number of snapshots K at SNR = 20 dB and INR = 30 dB. Third example.

θp (p = 0, 1, 2, 3, 4) are independently drawn in each Monte-Carlo trial from a Gaussian random generator N (θ0 , 4◦ ). Note that θp varies from trial to trial while remains unchanged from snapshot to snapshot. In the meantime, the random variables sp (k) vary both from trial to trial and from snapshot to snapshot. In the case of incoherent local scattering, the source covariance matrix Rs is no longer a rank-one matrix, and the output SINR should become [59]: SINR =

wH Rs w . wH Ri+n w

(31)

The SINR (31) can be maximized by [59]: wopt = P{R−1 i+n Rs }

(32)

where P{·} represents to compute the principal eigenvector of a matrix. Fig. 8 depicts the output SINR of the tested beamformers versus SNR. It is observed that the proposed beamformer and the summation beamformer [49] achieve good performance across a wide range of SNR, while the worst-case-based beam-

20

245

former [17] suffers significant performance degradation in high SNR regions. We also notice that the output SINR of the proposed beamformer is closer to the optimal value as compared to that of the summation beamformer [49]. The output SINR of the tested beamformers against the number of snapshots is demonstrated in Fig. 9, from which we see that the proposed beamformer achieves

250

performance advantage over the remaining beamformers across a wide range of snapshots.

60

Output SINR (dB)

40

Optimal SINR Proposed Integration Summation CS Reconstruction Worst-Case

20

0

-20

-40 -30

-20

-10

0 SNR (dB)

10

20

30

Figure 8: Output SINR versus SNR at K = 500 and INR = 30 dB. Fourth example.

5. Conclusion In this paper, a new robust adaptive beamforming algorithm based on coprime arrays is introduced. Unlike these previous techniques, the proposed 255

algorithm increases the available aperture of the coprime array via coarray interpolation, and thus can achieve a more precise INCM reconstruction. Compared with the existing approaches, our scheme is more robust in the presence of the non-ideal factors like look direction error, coherent local scattering, incoherent local scattering. Numerical results demonstrate that the output SINR of the

21

40 35 Optimal SINR Proposed Integration Summation CS Reconstruction Worst-Case

Output SINR (dB)

30 25 20 15 10 5 0

0

100 200 300 400 500 600 700 800 900 1000 Number of snapshots

Figure 9: Output SINR versus the number of snapshots K at SNR = 20 dB and INR = 30 dB. Fourth example.

260

proposed beamformer is closest to the optimal value among all the beamformers tested in high SNR regions.

Conflict of Interest Dear Dr. Pesavento, No conflict of interest exits in the publication of this manuscript, and manuscript 265

is approved by all authors for publication. I would like to declare on behalf of my co-authors that the work described is original research that has not been published previously, and not under consideration for publication elsewhere, in whole or in part. All the authors listed have approved the manuscript that is enclosed. If you have any queries, please dont hesitate to contact me.

270

Thank you and best regards. Yours sincerely, Zhi Zheng Email: [email protected] Tel: +86-28-61831105 22

275

Acknowledgment This work is supported in part by the National Natural Science Foundation of China under Grant 61671122, by the Applied Basic Research Program of Sichuan Province under Grant 2019YJ0191, by the Sichuan Science and Tech280

nology Program under Grant 2018RZ0141, by the Key Project of Sichuan Education Department of China under Grant 18ZA0221, by the Natural Science Foundation of Guangdong Province under Grant 2018A0303130064, and by the Fundamental Research Funds for the Central Universities of China under Grant 2672018ZYGX2018J003.

285

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