Robust adaptive beamforming for large-scale arrays

Robust adaptive beamforming for large-scale arrays

ARTICLE IN PRESS Signal Processing 90 (2010) 165–172 Contents lists available at ScienceDirect Signal Processing journal homepage: www.elsevier.com/...

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ARTICLE IN PRESS Signal Processing 90 (2010) 165–172

Contents lists available at ScienceDirect

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

Robust adaptive beamforming for large-scale arrays Fei Huang , Weixing Sheng, Xiaofeng Ma, Wei Wang Millimeter Wave Technology Laboratory, School of Electronic Engineering and Optoelectronic Technique, Nanjing University of Science and Technology, Nanjing 210094, China

a r t i c l e in fo

abstract

Article history: Received 19 August 2008 Received in revised form 29 March 2009 Accepted 8 June 2009 Available online 12 June 2009

For a large-scale adaptive array, heavy computational load and high-rate data transmission are two challenges in the implementation of an adaptive digital beamforming system. Moreover, the large-scale array becomes extremely sensitive to array imperfections. First, based on a restructured recursive linearly constrained minimum variance algorithm and a gradient-based optimization method, a new robust recursive linearly constrained minimum variance (RRLCMV) algorithm is proposed in this paper. The computational load of the RRLCMV algorithm is on the order of o(N), which is less than that of the conventional gradient-based robust adaptive algorithm. Then, a new efficient parallel robust recursive linearly constrained minimum variance (PRRLCMV) adaptive algorithm is proposed by appropriately partitioning the RRLCMV algorithm into a number of operational modules. It can be easily executed in a distributed-parallel-processing fashion, sequentially and in parallel. As a result, the PRRLCMV algorithm provides an effective solution that can alleviate the bottleneck of high-rate data transmission and reduce the computational cost. Finally, an implementation scheme of the PRRLCMV algorithm based on a distributed-parallel-processing system is also proposed. The simulation results demonstrate that the new PRRLCMV algorithm can significantly reduce the degradation due to various array errors. & 2009 Elsevier B.V. All rights reserved.

Keywords: Large-scale array Robust adaptive beamforming Linearly constrained minimum variance

1. Introduction The adaptive array processing technique is widely used in radar, communications, sonar, acoustics, and medicine. It has received considerable attention in the past decades. In an adaptive array processor, adaptive beamforming is an effective scheme for reconstructing the source signals from the acquired data of a sensor or antenna array. To obtain a high spatial resolution and good beamforming performance, an antenna array with a large number of antenna elements should be used. For a large-scale adaptive array, computational burden and high-rate data transmission are two bottlenecks in the implementation of an adaptive beamforming system. Many techniques have been proposed to alleviate these bottlenecks. Among

 Corresponding author.

E-mail address: [email protected] (F. Huang). 0165-1684/$ - see front matter & 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.sigpro.2009.06.006

them, a widely considered one is the partially adaptive processing technique [1–7], which utilizes a fraction of the available adaptive dimensions of an array for adaptation. This results in reducing the required computational load and increasing the convergence speed. Many partially adaptive processing methods have been proposed. There are three main architectures [2,3]: the reduced-rank minimum variance beamformer (RR-MVB), the reducedrank generalized sidelobe canceller (RR-GSC), and the subarray beamformer. In order to obtain good array performance, a proper rank-reducing transformation should be chosen for most of these methods. In using the RR-MVB and RR-GSC methods [2–5], to find the appropriate rankreducing matrix, one usually has to obtain the eigenvectors and eigenvalues of the observation data covariance matrix, and the required computational load of that is on the order of o((NJ)3) (N is the number of array elements, J is the number of the linear or derivative constraints of the GSC beamformer). Some sub-array processing based

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techniques have been presented in [6,7]. Each of the subarrays adjusts its own adaptive weights independently. The sub-array processor can achieve similar array performance with the original array beamformer provided that all the interference signals lie outside the mainlobe of the smallest sub-array. To achieve this condition, the number of elements of the smallest sub-array should be greater than the amount of signal sources, which limits further reduction of the computational load. Moreover, the subarray processing techniques cannot be applied to adaptive beamforming with multiple linear or derivative constraints. A new efficient technique based on the recursive least square (ERLS) algorithm has been presented in [8]. Unlike traditional sub-array adaptive beamforming techniques, in this technique each of the sub-arrays adjusts its own adaptive weights with the data it receives, as well as with the intermediate results from other sub-arrays. This technique does not cause the beamformer to suffer any loss in the degree of freedom for interference suppression. Furthermore, the number of elements of the smallest subarray can be less than the amount of signal sources. With proper decomposition of the optimal weight vector, the ERLS method can lead to good beamforming performance. However, when the number of the sub-arrays is large, the ERLS algorithm has a greater computational load than that of the traditional recursive least square adaptive algorithm. In [9], a parallel linearly constrained minimum variance (PLCMV) algorithm has been proposed. This algorithm requires less computational load than that of the ERLS method while keeping the same degrees of freedom. The linearly constrained minimum variance (LCMV) beamformer, on which the PLCMV algorithm in [9] is based, is a popular beamforming technique [10–12]. In this method, weights are chosen to minimize the output power subject to a linear constraint known as desired signal direction constraint, which ensures the array response from a specific direction. The constraint is determined based on the a priori knowledge of the directional vector associated with the desired source signal, and thus the beamforming performance of this method can be quite sensitive to the perturbation of this vector. Consequently, the steering vector error strictly limits the performance of the PLCMV beamformers in practice. When using large-scale arrays to obtain high resolution capability, the array becomes more sensitive to practical imperfections [1]. Various array imperfections, such as direction-of-arrival mismatch and array geometry error, can cause the steering vector error. To alleviate this problem, multiple linear constraints [13], derivative constraints [14,15], have been introduced in array processor. Any additional constraints will cause the array processor to suffer a loss in the degree of freedom for interference rejection. It has been shown that these constraints belong to the class of covariance matrix tapering approaches [16,17]. The weight norm constraint method has been proposed to restrict the extreme growth of the norm of array weights to obtain robust performance. It belongs to the class of diagonal loading method [18]. Diagonal loading [19] has been a popular approach to

improve the robustness of the beamformer. However, for most of these methods, determining the level of diagonal loading remains an open problem. A gradient-based optimization algorithm has been proposed for searching the optimal phases of the steering vector [20,21]. In this method, the array processor does not suffer any loss in the degree of freedom. In this paper, the gradient-based optimization method is restructured into a parallel algorithm after some mathematical manipulations, which is described in Section 3. The parallel gradient-based optimization method can be used in a distributedparallel-processing system to alleviate the bottlenecks of high-rate data transmission and heavy computational burden of large-scale arrays. In this paper, in order to achieve the robustness of beamformers, and to alleviate the bottlenecks of high-rate data transmission and heavy computational load of largescale arrays, a new efficient parallel robust recursive linearly constrained minimum variance (PRRLCMV) adaptive beamforming algorithm is proposed. First, based on the restructured recursive LCMV (RLCMV) algorithm, a new efficient robust RLCMV (RRLCMV) adaptive beamforming algorithm is proposed, in which the gradientbased optimization method in [20,21] is used to search out the actual steering vector of the desired signal. The computational load of the RRLCMV algorithm is on the order of o(N), which is less than the traditional robust adaptive beamforming algorithm based on the gradientbased optimization method [20,21]. Then, the PRRLCMV adaptive algorithm is proposed. In the PRRLCMV algorithm, an effective approach is developed to partition the RRLCMV algorithm into parallel operational modules. Each operational module adjusts its own adaptive weight vector for a subsection with the data it receives and a small amount of data (intermediate results) from other subsections. Although some steps of the PRRLCMV algorithm have to be processed sequentially, most computation of this algorithm can be executed in parallel. The important property of the proposed PRRLCMV algorithm is that some steps of it for different subsections can be computed in parallel for each iteration, which is described in Section 4. It is suitable to be applied in a distributed-parallel-processing system to reduce the computational load and to solve the bottleneck of highrate data transmission for a large-scale adaptive array. The implementation scheme of the proposed PRRLCMV adaptive beamforming algorithm based on a distributedparallel-processing system is also proposed in this paper. The simulation results show that the PRRLCMV algorithm can effectively estimate the actual steering vector and achieve a high signal-to-interference plus noise ratio (SINR). The PRRLCMV algorithm can be used to deal with real time digital beamforming for large-scale adaptive arrays. This paper is organized as follows. In Section 2, the PLCMV adaptive algorithm is reviewed. In Section 3, the new proposed RRLCMV and PRRLCMV adaptive algorithm is discussed in detail. The implementation scheme of the proposed PRRLCMV adaptive beamforming algorithm based on a distributed-parallel-processing system is introduced in Section 4. Some numerical studies are

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presented in Section 5 to illustrate the effectiveness of the proposed PRRLCMV algorithm. Finally, a brief conclusion is given in Section 6. 2. The PLCMV adaptive beamforming algorithm 2.1. Signal model Consider a uniform linear array comprising N isotropic antenna elements with half wavelength spacing. Assume that K (KoN) uncorrelated narrowband and far field signals impinge on the array. The N  1 dimensional received signal vector present on the antenna elements at time t is denoted by xðtÞ ¼

K1 X

si ðtÞaðyi Þ þ nðtÞ

(1)

i¼0

where aðyi Þ ¼ ½1; ejp sin yi ; . . . ; ejpðN1Þ sin yi T is the steering vector corresponding to the direction of arrival yi , si ðtÞ is the complex waveform of the ith signal, and nðtÞ is an additive white Gaussian noise. In this paper, the sources and noise are assumed to be statistically uncorrelated. The N  N received data covariance matrix R is given by R ¼ E½xðtÞxH ðtÞ

(2)

where the superscript H denotes the complex conjugate transpose. In practice applications, R is replaced by a finite sample covariance matrix L X b1 R xðtÞxH ðtÞ L t¼1

(3)

where L is the training sample size. 2.2. The PLCMV adaptive algorithm In the LCMV adaptive algorithm, the optimal weight vector w is obtained by minimizing the array output power subject to J linear or derivative constraints as follows [10–12]: Minimize E½jyðtÞj2  ¼ wH Rw

Subject to C H w ¼ f

(4)

where y(t) ¼ wHx(t) is the output of the array, C the N  J constraint matrix, and f the J  1 response vector. The optimal solution for (4) is given by w¼R

1

H 1

CðC R



1

f

extension of the work of [11] in two aspects. First, the complex weight vectors (as against real weight discussed in [11]) are treated in [12]. Second, the look direction is chosen as any direction (not only perpendicular to the line of antennas but also other directions). In this paper, beamformers for narrowband signals are considered. The assumption is that signals impinging on the array are narrowband signals and there is only one constraint in the desired signal direction (J ¼ 1, f ¼ 1). The C matrix is defined as C ¼ ½1; ejp sin y0 ; . . . ; ejpðN1Þ sin y0 T , which is a special case of the C matrix in [12], and it has the same form as the C matrix in [20]. Computational load and high-rate data transmission are two problems for the implementation of the LCMV beamforming algorithm for large-scale antenna arrays. In [9], the PLCMV algorithm has been proposed to alleviate these drawbacks. The weight updated equation of the recursive LCMV algorithm is restructured as [9] wðk þ 1Þ ¼ ½I  SCC H ½wðkÞ  myH ðkÞxðkÞ þ SC ¼ wðkÞ  SCDðkÞ  myH ðkÞxðkÞ þ myH ðkÞSCQ ðkÞ þ SC ¼ wðkÞ þ ðS½1  DðkÞ þ my ðkÞQ ðkÞÞC  my ðkÞxðkÞ (8) where S ¼ (CHC)1, which is a constant that can be obtained before updating and will not change during the update, and D(k) ¼ CHw(k), Q(k) ¼ CHx(k), y(k) ¼ wH(k)x(k), my(k) ¼ myH(k). In the PLCMV algorithm, the N  1 dimensional received signal vector x(k) is partitioned into M subsections: xðkÞ ¼ ½X T1 ðkÞ; X T2 ðkÞ; . . . ; X TM ðkÞT

1

(6)

C H R1 C

The recursive LCMV algorithm can be described as follows [11,12]: H

wðk þ 1Þ ¼ P½wðkÞ  mxðkÞy ðkÞ þ F;

k ¼ 1; 2; 3 . . .

C ¼ ½C T1 ; C T2 ; . . . ; C TM T

(10)

wðkÞ ¼ ½wT1 ðkÞ; wT2 ðkÞ; . . . ; wTM ðkÞT

(11)

Having partitioned the vectors, the PLCMV algorithm can be summarized as wi ðk þ 1Þ ¼ wi ðkÞ þ SDQ ðkÞC i  my ðkÞX i ðkÞ;

i ¼ 1; 2; . . . ; M (12)

SDQ ðkÞ ¼ S½1  DðkÞ þ my ðkÞQ ðkÞ

(13)

where S¼

M X

!1 Si

M X

¼

i¼1

Q ðkÞ ¼

yðkÞ ¼

M X

M X i¼1

!1 CH i Ci

(14)

CH i X i ðkÞ

(15)

wH i ðkÞX i ðkÞ

(16)

i¼1

Q i ðkÞ ¼

i¼1

(7)

where m is a small, positive, step size parameter, k is an iteration index, P ¼ I–C(CHC)1CH, and F ¼ C(CHC)1f. The recursive LCMV algorithm in [12] can be regarded as an

(9)

where Xi(k) is an Ni  1 dimensional vector for i ¼ 1,2,y,M, Ni is the number of array elements of the PM ith subsection, i¼1 N i ¼ N, and M is the number of subsections. Also, C and w(k) is correspondingly partitioned as

(5)

If a constraint of unit gain in the desired signal direction is imposed (J ¼ 1, f ¼ 1), then (5) becomes w ¼ R1 CðC H R1 CÞ1 . And the array mean output power of the desired signal is Ps ¼

167

M X i¼1

yi ðkÞ ¼

M X i¼1

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DðkÞ ¼

F. Huang et al. / Signal Processing 90 (2010) 165–172 M X

Di ðkÞ ¼

i¼1

M X

CH i wi ðkÞ

(17)

i¼1

Cðy0 ; DFk Þ ¼ diagfejDf1 ðkÞ ; . . . ; ejDfN ðkÞ gCðy0 ; DF0 Þ

H

my ðkÞ ¼ my ðkÞ.

(18)

3. Robust adaptive beamforming for large-scale arrays The PLCMV algorithm can be easily implemented in a parallel DSP or FPGA processing system due to some steps of this algorithm that can be executed in parallel. It can be used to deal with real time adaptive beamforming for large-scale arrays. If there is no mismatch between the presumed steering vector and the actual direction vector, which means the presumed C is perfect, the array processor can output the desired signal without distortion and suppress the interference and noise simultaneously. In practice applications, the exact C is unavailable. Therefore, we use the presumed C instead of the actual C in the PLCMV algorithm. Due to the existence of array imperfections, some underlying assumptions on adaptive arrays can be invalid, resulting in a mismatch between the presumed C and the actual C. The mismatch between them will degrade the performance of the PLCMV beamforming algorithm. The array imperfections, such as direction-of-arrival mismatch and array geometry error, can be modeled as generalized array phase errors in the steering vector [20,21]. As a consequence of the multiple errors, the actual steering vector of the desired signal can be expressed as Cðy0 ; DFÞ ¼ ½ejw0 t1 ðy0 ÞþjDf1 ; . . . ; ejw0 tN ðy0 ÞþjDfN T

(19)

where w0 ¼ 2pf 0 , f 0 is the frequency of the desired signal, tj ðy0 Þ ¼ dj sin y0 =c, for j ¼ 1,y,N, denotes the ideal steering delay corresponding to the assumed target direction

y0 of an ideal array, dj is the distance between the jth and the first antenna, and c denotes the velocity of the light. Dfj , for j ¼ 1,y,N, denotes the generalized phase error of the array processor. DF is an N  1 vector which is expressed as DF ¼ ½Df1 ; . . . ; Dfi ; . . . ; DfN T . The key issue of this robust beamforming algorithm is to estimate DF by maximizing the array output power of the desired signal [20,21], which in turn can be expressed as an optimization problem: min P 0S ¼ min C H ðy0 ; DFÞR1 Cðy0 ; DFÞ DF

desired signal obtained in the (k1)th iteration which can be expressed as

DF

(20)

Next, the gradient search method is used to iteratively adjust the phase error vector DF  @P0 ðkÞ (21) DFðk þ 1Þ ¼ DFðkÞ  m0 s  @DF DF¼DFðkÞ

where k is an iteration index, m0 is a small, positive, step size parameter, and @P 0s ðkÞ[email protected] is the gradient of P 0s with respect to DF. The gradient vector @P0s ðkÞ[email protected] can be given by @P0s ðkÞ 1 ¼ 2RefC H DF ðy0 ; DFk ÞR Cðy0 ; DFk Þg @DF

(22)

where Refg denotes the operator to get the real part of the complex vector, Cðy0 ; DFk Þ is the steering vector of the

(23)

where Cðy0 ; DF0 Þ is the initial steering vector, and diagfg is the operator used to expand the vector into a diagonal matrix. Cðy0 ; DFk Þ is denoted as CðkÞ hereafter, and Cðy0 ; DF0 Þ is denoted as Cð0Þ. C DF ðy0 ; DFk Þ ¼ jdiagfCðkÞg denotes the gradient of CðkÞ with respect to DF, and C DF ðy0 ; DFk Þ is denoted as C DF ðkÞ hereafter. Eq. (21) can be expressed approximately as @P0s ðkÞ  2RefC H DF ðkÞwðkÞg @DF

(24)

since the weight vector w can be expressed as wðkÞ ¼ aR1 CðkÞ H

(25) 1

where a ¼ C ðkÞR CðkÞ is a constant. First, based on the restructured recursive LCMV algorithm and the gradient-based optimization method introduced above, a new robust recursive LCMV algorithm is proposed. In the RRLCMV adaptive algorithm, the gradient-based optimization method shown in Eqs. (19)–(25) is used to search out the actual steering vector to improve the robust performance of LCMV, and the restructured recursive LCMV algorithm shown in Eq. (8) is used to obtain the optimum algorithm. The computational load of the RRLCMV algorithm is on the order of o(N), which is less than that of the traditional robust adaptive algorithm based on the gradient-based optimization method [20,21]. Then, in order to realize real time processing for large-scale arrays, the parallel robust recursive LCMV beamforming algorithm is proposed, which effectively partitions the RRLCMV adaptive algorithm into a number of operational modules. In the proposed PRRLCMV algorithm, the vector space of DFðkÞ is partitioned into a number of subsections:

DFðkÞ ¼ ½DFT1 ðkÞ; DFT2 ðkÞ; . . . ; DFTM ðkÞ

(26)

where M is the number of subsections, DFi ðkÞ is an Ni  1 P data vector and M i¼1 N i ¼ N. CðkÞ can be correspondingly partitioned as CðkÞ ¼ ½C T1 ðkÞ; C T2 ðkÞ; . . . ; C TM ðkÞT with C i ðkÞ ¼ diagfejDFi ðkÞ gC i ð0Þ

for i ¼ 1; 2; . . . ; M

(27)

is correspondingly partitioned as And C DF ðkÞ where C DF ðkÞ ¼ ½ðC DF ÞT1 ðkÞ; ðC DF ÞT2 ðkÞ; . . . ; ðC DF ÞTM ðkÞT , ðC DF Þi ðkÞ ¼ jdiagfC i ðkÞg for i ¼ 1,2,y,M. Furthermore, a legitimate way to partition @P 0s ðkÞ[email protected] needs to be determined. As @P 0s ðkÞ[email protected] can be approximately expressed as @P0s ðkÞ[email protected]  2RefC H DF ðkÞwðkÞg, it can be partitioned as   @P0s ðkÞ @P1 ðkÞ @P 2 ðkÞ @PM ðkÞ (28) ¼ ; ;...; @DF @DF @DF @DF with @Pi ðkÞ ¼ 2RefðC DF ÞH i ðkÞwi ðkÞg @DF

ði ¼ 1; 2; . . . ; MÞ

(29)

where ðC DF Þi ðkÞ can be achieved in the (k1)th iteration, and wi ðkÞ can also be obtained in the (k1)th iteration by the PLCMV algorithm which was described in Section 2,

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with the except that Eqs. (12)–(14) of the PLCMV algorithm are revised as wi ðk þ 1Þ ¼ wi ðkÞ þ SDQ ðkÞC i ðkÞ  my ðkÞX i ðkÞ

ði ¼ 1; 2; . . . ; MÞ

(30) SDQ ðkÞ ¼ SðkÞ½1  DðkÞ þ my ðkÞQ ðkÞ

SðkÞ ¼

M X i¼1

!1 Si ðkÞ

¼

M X

(31) !1

CH i ðkÞC i ðkÞ

(32)

ði ¼ 1; 2; . . . ; MÞ (33)

where k is an iteration index and M the number of subsections. The updated steering vector of the desired signal Ci(k+1) for each subsection can be obtained by substituting DFi ðk þ 1Þ into Eq. (27) for i ¼ 1,2,y,M. The proposed PRRLCMV adaptive beamforming algorithm is now summarized as follows: Initialize w(1) ¼ 0, DFð1Þ ¼ 0, and Cð0Þ ¼ Cðy0 ; 0Þ; Step 1. Update Si(k), Qi(k), yi(k) , Di(k) for i ¼ 1,2,y,M by Eqs. (32) and (15)–(17), respectively. Step 2. Obtain SDQ(k) by Eq. (31). Step 3. Update the weight vectors wi(k+1) using Eq. (30) for i ¼ 1,2,y,M. Step 4. Estimate gradient vectors @P i ðkÞ[email protected] by Eq. (29) for i ¼ 1,2,y,M. Step 5. Update the phase error vectors DFi ðk þ 1Þ by Eq. (33) for i ¼ 1,2,y,M. Step 6. Update the steering vectors C i ðk þ 1Þby equation (27) for i ¼ 1,2,y,M. Step 7. Replace the nominal steering vectors by the updated steering vectors in step 6.Repeat step 1 through step 7 until jjDfðk þ 1Þ  DfðkÞjjo, where e is the error tolerance. The optimal weight vector can be constructed from the weight vectors of subsections by Eq. (11). Assume the number of subsections is M, and the weighting vector wi of the ith subsection is an Ni  1 vector for i ¼ 1,2,y,M. The number of the complex multiplications required for finding Si(k), Qi(k), yi(k) and Di(k) in step 1 is about 4Ni for i ¼ 1,2,y,M. Step 2 requires 3 complex multiplications to obtain the SDQ(k) value. Moreover, step 3 requires about 2Ni complex multiplications to obtain each wi(k+1) by Eq. (30) for i ¼ 1,2,y,M. In step 4, to obtain @P i ðkÞ[email protected] by Eq. (29) approximately 2Ni complex multiplications for i ¼ 1, 2,y,M are needed. It requires Ni complex multiplications to find DFi ðk þ 1Þ by Eq. (33) in step 5 for i ¼ 1, 2,y,M. Moreover, in step 6 it requires about Ni complex multiplications to obtain the steering vector Ci by using Eq. (27) for i ¼ 1, 2,y,M. Hence, the total number of the complex multiplications required for computing the optimal weight vector w is given by M X i¼1

which is on the order of o(N) whereas the computational load of the robust algorithm based on the gradient method proposed in [20] are about o(N3), and the robust method in [21] is on the order of o(N2). There is no difference in computation load between the PRRLCMV and RRLCMV adaptive beamforming algorithm. Compared with the RRLCMV algorithm, the PRRLCMV algorithm can be applied to a highly efficient distributed-parallelprocessing system to achieve high computational efficiency.

i¼1

Consequently, Eq. (21) can be given by  @P ðkÞ DFi ðk þ 1Þ ¼ DFi ðkÞ  m0 i  @DF DFi ¼DFi ðkÞ

CMpu ¼

169

6N i þ 4N þ 3 ¼ 10N þ 3

(34)

4. Parallel processing scheme of the proposed PRRLCMV adaptive algorithm The proposed PRRLCMV adaptive algorithm is a hybrid algorithm. Steps 1 and 3–6 of this algorithm for different subsections can be processed in parallel for each iteration. Moreover, steps 4–6 can be processed in parallel with step1 in each subsection. But steps 4–6 are processed sequentially. Also, to obtain the sub-weight vectors in step 3 for each iteration, the information from other subsections (SDQ(k)) is required. Although some steps of the PRRLCMV algorithm have to be processed sequentially, most computation of this algorithm can be executed in parallel. Due to steps 1 and 3–6 for different subsections of the PRRLCMV algorithm can be computed in parallel for each iteration, the computation of different subsections in these steps can be distributed to different computing nodes. Fig. 1 shows the flowchart of the proposed PRRLCMV algorithm. It can be seen that the PRRLCMV algorithm can be mapped onto a highly efficient distributed-parallel-processing system. In Fig. 1, each block represents a computing node. In the flowchart, the received data of each subsection are distributed to a computing node. The input signal of the ith subsection block is X i ðkÞ in each iteration for i ¼ 1,2,y,M. In the first snapshot, yi(k), Si ðkÞ, Di ðkÞ, and Q i ðkÞ for i ¼ 1,2,y,M are obtained in each subsection block in parallel. Then, the results that are obtained in subsection blocks are transmitted to Block 2 computing node. Moreover, in the first snapshot, @P i ðkÞ[email protected] can be computed in the ith subsection block for i ¼ 1,2,y,M, in parallel. In the second snapshot, SDQ(k) and my ðkÞ are obtained in Block 2 and then are transmitted back to the subsection blocks. Also, DFi ðk þ 1Þ can be achieved by Eq. (33) in the ith subsection block for i ¼ 1,2,y,M, in the second snapshot. In the third snapshot, C i ðk þ 1Þ can be achieved by Eq. (27) in the ith subsection block for i ¼ 1,2,y,M. Similarly, wi(k+1) can be obtained by Eq. (30) in subsection blocks in the third snapshot, and can then be transmitted to Block 2 where i ¼ 1,2,y,M. Finally, In Block 2, w(k+1) is obtained by Eq. (11) in the fourth snapshot. Moreover, in the fourth snapshot, each section block can go on with the next iteration. This implementation scheme of the proposed PRRLCMV adaptive beamforming algorithm needs only three snapshots for each iteration. Steps 1 and 3–6 in subsection blocks can be performed in parallel for each iteration, and steps 4–6 can be processed in parallel with

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Fig. 1. The Parallel processing scheme of the proposed PRRLCMV adaptive algorithm.

step 1 in each subsection block. The PRRLCMV adaptive algorithm is suitable to be applied in a distributedparallel-processing system to achieve high computational efficiency. It can be used for real time digital beamforming for large-scale antenna arrays. 5. Numerical results In this section, several simulation studies are carried out to evaluate the performance of the proposed PRRLCMV adaptive algorithm. The studied array is a uniform linear array with the number of array elements N and half wavelength for interelement spacing. Assume the number of array elements of each subsection are equal, this means Ni ¼ Nj, and i, j ¼ 1,2,y,M where iaj, Ni is the number of elements for the ith subsection, and M is the number of subsections. The initial w(1) is a zero vector. In this study, the non-directional noise is assumed to be a spatially white Gaussian noise with unit covariance. It is worth mentioning that the m and m0 have to be chosen properly. The larger the m and m0 values are, the faster the convergence is. But the larger the m and m0 values are, the more severe the jitter of the signal-tointerference plus noise ratio curve is. Both m and m0 are empirically determined to be appropriate for each example. Two kinds of uncertainty in the array steering vector are considered. One is the well-studied steering direction error. The other is the array geometry error. In this simulation study, all the simulation results are obtained via 50 Monte-Carlo runs.

5.1. Performance evaluation: steering direction error In the first experiment, the effect of the number of elements for each subsection on array performance is considered when the direction-of-arrival mismatch is present. For this case the number of elements in the array is N ¼ 16. The incident angle of the desired signal is 01, and the presumed steering angle is 51. Signal-to-noise ratio is equal to 0 dB. An interference signal is present with incident angle of 201, and interference-to-noise ratio is equal to 10 dB. The empirically found values of m and m0 are 0.004 and 0.013, respectively. To obtain the results, 2500 iterations steps have been taken. Fig. 2 shows the array beampatterns formed by the PRRLCM algorithm when steering vector error existed with different subsection sizes (Ni ¼ 8, Ni ¼ 4, and Ni ¼ 1). The solid curve denotes the beampattern with Ni ¼ 4. The plus signs denote the beampattern with Ni ¼ 1, and the circles denote the beampattern with Ni ¼ 8. Fig. 2 shows that all the plus signs and circles are on the solid curve. This indicates that the PRRLCMV algorithm has the same beam performance with different subsection sizes. The output SINRs for Ni ¼ 4, Ni ¼ 8, and Ni ¼ 1 are all 11.7 dB, showing the same SINR performance. Simulation results show that the PRRLCMV algorithm can reduce the degradation effect due to steering angle error, regardless how an array is partitioned. In the second example, the effect of the direction-ofarrival mismatch on array output SINR is investigated. Consider the case in which the number of array elements

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Fig. 2. Performance comparison of the PRRLCMV algorithm with different subsection sizes while the steering direction error existed. Fig. 4. Comparison of the beampatterns among the SLCMV algorithm with ASV, the ERLS algorithm (Ni ¼ 1) with ESV and the PRRLCMV algorithm (Ni ¼ 1) with ESV.

Fig. 3. Comparison of the output SINR among the SLCMV algorithm with actual steering vector (ASV), the ERLS algorithm (Ni ¼ 1) with erroneous steering vector (ESV) and the PRRLCMV algorithm (Ni ¼ 1) with ESV. Fig. 5. Performance comparison of the PRRLCMV algorithm with different subsection sizes when array geometry error is present.

is N ¼ 12. The incident angle of the desired signal is 01 while the steering angle is 51, i.e., the steering angle error is 51. Signal-to-noise ratio is 0 dB. Two interference signals are present with incident angles of 101 and 201. Both interference-to-noise ratios are 10 dB. The values of m and m0 are 0.0029 and 0.0072, respectively, and these values are empirically found to be appropriate. Fig. 3 shows that the output SINRs of the ERLS algorithm (Ni ¼ 1) with erroneous steering vector (ESV), the PRRLCMV algorithm (Ni ¼ 1) with erroneous steering vector (ESV) and the standard LCMV(SLCMV) algorithm with actual steering vector (ASV) are 4.3, 10.4, and 10.8 dB, respectively. The array beampatterns of the three cases are shown in Fig. 4. Then, Figs. 3 and 4 demonstrate that the PRRLCMV algorithm can solve the problem of steering angle error in adaptive array beamforming. And it works well even when Ni ¼ 1, which demonstrates that the number of elements of the smallest subsection can be less than the number of signal sources.

5.2. Performance evaluation: array geometry error In the third example, we investigate how the subsection size affects array performance when the array geometry error is present. In this example, the number of whole array elements is N ¼ 16. The incident angle of the desired signal is 101. Signal-to-noise ratio is 0 dB. Two interference signals are present with incident angles of 401 and 201. Both interference-to-noise ratios are 10 dB. The variance of array geometry error is (0.05l)2. The empirically found values of m and m0 are 0.0016 and 0.006, respectively. The array beampatterns shown in Fig. 5 are obtained by the PRRLCMV algorithm with Ni ¼ 8, Ni ¼ 4, and Ni ¼ 1. The asterisks denote the beampattern with Ni ¼ 1, and the circles denote the beampattern with Ni ¼ 8. Fig. 5 shows that all the asterisks and circles are on the solid curve which is the beampattern with Ni ¼ 4.

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and alleviate the bottleneck of high-rate data transmission. Simulation results show that the PRRLCMV algorithm can also reduce the degradation of the array performance caused by array imperfections. It is expected that the proposed algorithm will be used in adaptive beamforming systems to deal with real time array signal processing for large-scale arrays.

References

Fig. 6. Comparison of the output SINR of the PLCMV (Ni ¼ 1) and PRRLCMV (Ni ¼ 1) algorithm when array geometry error is present.

It indicates that the beam performance of the PRRLCMV algorithm does not vary with different decompositions of an array when the array geometry error is present. The output SINRs of the three cases are all equal to 11.5 dB, while the ideal output SINR is 11.9 dB. In the fourth example, the influence of the array geometry error on array output SINR is considered. The antenna array parameters in this example are the same as those used in example three. Fig. 6 shows that the output SINR of the PLCMV algorithm (Ni ¼ 1) is 3.7 dB and that of the PRRLCMV algorithm (Ni ¼ 1) is 11.5 dB, whereas the ideal output SINR is 11.9 dB. Simulation results indicate that the PRRLCMV algorithm can also reduce the degradation caused by the array geometry error, and that the number of elements of the smallest subsection can be less than the number of signal sources. 6. Conclusions This paper has presented an efficient parallel robust adaptive beamforming method for large-scale arrays in the presence of array imperfections, including steering direction error, array geometry error, etc. First, based on a restructured recursive linear constraint minimum variance algorithm and a gradient-based optimization method, a new robust recursive LCMV algorithm is proposed. The computational load of the RRLCMV algorithm is on the order of o(N), which is less than that of the traditional robust adaptive algorithm based on the gradient method [20,21]. Then, in order to realize real time adaptive beamforming processing, a new efficient parallel RRLCMV adaptive algorithm for implementing the RRLCMV adaptive algorithm is proposed, which effectively partitions the RRLCMV adaptive algorithm into a number of operational modules. The PRRLCMV adaptive algorithm can be easily implemented in a distributed-parallel-processing system due to its parallel property. The proposed PRRLCMV method can significantly reduce computational workload

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