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Contents lists available at ScienceDirect

Signal Processing

Signal Processing 90 (2010) 165–172

0165-16

doi:10.1

� Cor

E-m

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 i n f o

Article history:

Received 19 August 2008

Received in revised form

29 March 2009

Accepted 8 June 2009Available online 12 June 2009

Keywords:

Large-scale array

Robust adaptive beamforming

Linearly constrained minimum variance

84/$ - see front matter & 2009 Elsevier B.V. A

016/j.sigpro.2009.06.006

responding author.

ail address: feihuang1206@hotmail.com (F. H

a b s t r a c t

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 implementa-

tion 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.

1. Introduction

The adaptive array processing technique is widely usedin radar, communications, sonar, acoustics, and medicine.It has received considerable attention in the past decades.In an adaptive array processor, adaptive beamforming isan effective scheme for reconstructing the source signalsfrom the acquired data of a sensor or antenna array. Toobtain a high spatial resolution and good beamformingperformance, an antenna array with a large number ofantenna elements should be used. For a large-scaleadaptive array, computational burden and high-rate datatransmission are two bottlenecks in the implementationof an adaptive beamforming system. Many techniqueshave been proposed to alleviate these bottlenecks. Among

ll rights reserved.

uang).

them, a widely considered one is the partially adaptiveprocessing technique [1–7], which utilizes a fraction ofthe available adaptive dimensions of an array for adapta-tion. This results in reducing the required computationalload and increasing the convergence speed. Many partiallyadaptive processing methods have been proposed. Thereare three main architectures [2,3]: the reduced-rankminimum variance beamformer (RR-MVB), the reduced-rank generalized sidelobe canceller (RR-GSC), and the sub-array beamformer. In order to obtain good array perfor-mance, a proper rank-reducing transformation should bechosen for most of these methods. In using the RR-MVBand RR-GSC methods [2–5], to find the appropriate rank-reducing matrix, one usually has to obtain the eigenvec-tors and eigenvalues of the observation data covariancematrix, and the required computational load of that is onthe order of o((N�J)3) (N is the number of array elements,J is the number of the linear or derivative constraints ofthe GSC beamformer). Some sub-array processing based

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techniques have been presented in [6,7]. Each of the sub-arrays adjusts its own adaptive weights independently.The sub-array processor can achieve similar array perfor-mance with the original array beamformer provided thatall the interference signals lie outside the mainlobe of thesmallest sub-array. To achieve this condition, the numberof elements of the smallest sub-array should be greaterthan the amount of signal sources, which limits furtherreduction of the computational load. Moreover, the sub-array processing techniques cannot be applied to adaptivebeamforming with multiple linear or derivative con-straints.

A new efficient technique based on the recursive leastsquare (ERLS) algorithm has been presented in [8]. Unliketraditional sub-array adaptive beamforming techniques,in this technique each of the sub-arrays adjusts its ownadaptive weights with the data it receives, as well as withthe intermediate results from other sub-arrays. Thistechnique does not cause the beamformer to suffer anyloss in the degree of freedom for interference suppression.Furthermore, the number of elements of the smallest sub-array can be less than the amount of signal sources. Withproper decomposition of the optimal weight vector, theERLS method can lead to good beamforming performance.However, when the number of the sub-arrays is large, theERLS algorithm has a greater computational load than thatof the traditional recursive least square adaptive algo-rithm. In [9], a parallel linearly constrained minimumvariance (PLCMV) algorithm has been proposed. Thisalgorithm requires less computational load than that ofthe ERLS method while keeping the same degrees offreedom.

The linearly constrained minimum variance (LCMV)beamformer, on which the PLCMV algorithm in [9] isbased, is a popular beamforming technique [10–12]. Inthis method, weights are chosen to minimize the outputpower subject to a linear constraint known as desiredsignal direction constraint, which ensures the arrayresponse from a specific direction. The constraint isdetermined based on the a priori knowledge of thedirectional vector associated with the desired sourcesignal, and thus the beamforming performance of thismethod can be quite sensitive to the perturbation of thisvector. Consequently, the steering vector error strictlylimits the performance of the PLCMV beamformers inpractice. When using large-scale arrays to obtain highresolution capability, the array becomes more sensitive topractical imperfections [1]. Various array imperfections,such as direction-of-arrival mismatch and array geometryerror, can cause the steering vector error. To alleviate thisproblem, multiple linear constraints [13], derivativeconstraints [14,15], have been introduced in array proces-sor. Any additional constraints will cause the arrayprocessor to suffer a loss in the degree of freedom forinterference rejection. It has been shown that theseconstraints belong to the class of covariance matrixtapering approaches [16,17]. The weight norm constraintmethod has been proposed to restrict the extreme growthof the norm of array weights to obtain robust perfor-mance. 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, formost of these methods, determining the level of diagonalloading remains an open problem. A gradient-basedoptimization algorithm has been proposed for searchingthe optimal phases of the steering vector [20,21]. In thismethod, the array processor does not suffer any loss in thedegree of freedom. In this paper, the gradient-basedoptimization method is restructured into a parallelalgorithm after some mathematical manipulations, whichis described in Section 3. The parallel gradient-basedoptimization method can be used in a distributed-parallel-processing system to alleviate the bottlenecks ofhigh-rate data transmission and heavy computationalburden of large-scale arrays.

In this paper, in order to achieve the robustness ofbeamformers, and to alleviate the bottlenecks of high-ratedata transmission and heavy computational load of large-scale arrays, a new efficient parallel robust recursivelinearly constrained minimum variance (PRRLCMV) adap-tive beamforming algorithm is proposed. First, based onthe restructured recursive LCMV (RLCMV) algorithm, anew efficient robust RLCMV (RRLCMV) adaptive beam-forming algorithm is proposed, in which the gradient-based optimization method in [20,21] is used to searchout the actual steering vector of the desired signal. Thecomputational load of the RRLCMV algorithm is on theorder of o(N), which is less than the traditional robustadaptive beamforming algorithm based on the gradient-based optimization method [20,21]. Then, the PRRLCMVadaptive algorithm is proposed. In the PRRLCMV algo-rithm, an effective approach is developed to partition theRRLCMV algorithm into parallel operational modules.Each operational module adjusts its own adaptive weightvector for a subsection with the data it receives and asmall amount of data (intermediate results) from othersubsections. Although some steps of the PRRLCMValgorithm have to be processed sequentially, mostcomputation of this algorithm can be executed in parallel.The important property of the proposed PRRLCMV algo-rithm is that some steps of it for different subsections canbe computed in parallel for each iteration, which isdescribed in Section 4. It is suitable to be applied in adistributed-parallel-processing system to reduce thecomputational load and to solve the bottleneck of high-rate data transmission for a large-scale adaptive array. Theimplementation scheme of the proposed PRRLCMV adap-tive beamforming algorithm based on a distributed-parallel-processing system is also proposed in this paper.The simulation results show that the PRRLCMV algorithmcan effectively estimate the actual steering vector andachieve a high signal-to-interference plus noise ratio(SINR). The PRRLCMV algorithm can be used to deal withreal time digital beamforming for large-scale adaptivearrays.

This paper is organized as follows. In Section 2, thePLCMV adaptive algorithm is reviewed. In Section 3, thenew proposed RRLCMV and PRRLCMV adaptive algorithmis discussed in detail. The implementation scheme of theproposed PRRLCMV adaptive beamforming algorithmbased on a distributed-parallel-processing system isintroduced in Section 4. Some numerical studies are

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presented in Section 5 to illustrate the effectiveness of theproposed PRRLCMV algorithm. Finally, a brief conclusionis given in Section 6.

2. The PLCMV adaptive beamforming algorithm

2.1. Signal model

Consider a uniform linear array comprising N isotropicantenna elements with half wavelength spacing. Assumethat K (KoN) uncorrelated narrowband and far fieldsignals impinge on the array. The N�1 dimensionalreceived signal vector present on the antenna elementsat time t is denoted by

xðtÞ ¼XK�1

i¼0

siðtÞaðyiÞ þ nðtÞ (1)

where aðyiÞ ¼ ½1; ejp sin yi ; . . . ; ejpðN�1Þ sin yi �T is the steering

vector corresponding to the direction of arrival yi, siðtÞ isthe complex waveform of the ith signal, and nðtÞ is anadditive white Gaussian noise. In this paper, the sourcesand noise are assumed to be statistically uncorrelated.

The N�N received data covariance matrix R is given by

R ¼ E½xðtÞxHðtÞ� (2)

where the superscript H denotes the complex conjugatetranspose. In practice applications, R is replaced by a finitesample covariance matrix

bR � 1

L

XL

t¼1

xðtÞxHðtÞ (3)

where L is the training sample size.

2.2. The PLCMV adaptive algorithm

In the LCMV adaptive algorithm, the optimal weightvector w is obtained by minimizing the array outputpower subject to J linear or derivative constraints asfollows [10–12]:

Minimize E½jyðtÞj2� ¼ wHRw Subject to CHw 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. Theoptimal solution for (4) is given by

w ¼ R�1CðCHR�1CÞ�1f (5)

If a constraint of unit gain in the desired signal directionis imposed (J ¼ 1, f ¼ 1), then (5) becomes w ¼

R�1CðCHR�1CÞ�1. And the array mean output power ofthe desired signal is

Ps ¼1

CHR�1C(6)

The recursive LCMV algorithm can be described asfollows [11,12]:

wðkþ 1Þ ¼ P½wðkÞ � mxðkÞyHðkÞ� þ F; k ¼ 1;2;3 . . . (7)

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

extension of the work of [11] in two aspects. First, thecomplex weight vectors (as against real weight discussedin [11]) are treated in [12]. Second, the look direction ischosen as any direction (not only perpendicular to the lineof antennas but also other directions). In this paper,beamformers for narrowband signals are considered. Theassumption is that signals impinging on the array arenarrowband signals and there is only one constraint in thedesired signal direction (J ¼ 1, f ¼ 1). The C matrix isdefined as C ¼ ½1; ejp sin y0 ; . . . ; ejpðN�1Þ sin y0 �T, which is aspecial case of the C matrix in [12], and it has the sameform as the C matrix in [20].

Computational load and high-rate data transmissionare two problems for the implementation of the LCMVbeamforming algorithm for large-scale antenna arrays. In[9], the PLCMV algorithm has been proposed to alleviatethese drawbacks. The weight updated equation of therecursive LCMV algorithm is restructured as [9]

wðkþ 1Þ ¼ ½I � SCCH�½wðkÞ � myHðkÞxðkÞ� þ SC

¼ wðkÞ � SCDðkÞ � myHðkÞxðkÞ þ myHðkÞSCQ ðkÞ þ SC

¼ wðkÞ þ ðS½1� DðkÞ þ myðkÞQ ðkÞ�ÞC � myðkÞxðkÞ (8)

where S ¼ (CHC)�1, which is a constant that canbe obtained before updating and will not change dur-ing 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 dimensionalreceived signal vector x(k) is partitioned into M subsec-tions:

xðkÞ ¼ ½XT1ðkÞ;X

T2ðkÞ; . . . ;X

TMðkÞ�

T (9)

where Xi(k) is an Ni�1 dimensional vector fori ¼ 1,2,y,M, Ni is the number of array elements of theith subsection,

PMi¼1Ni ¼ N, and M is the number of

subsections. Also, C and w(k) is correspondingly parti-tioned as

C ¼ ½CT1;C

T2; . . . ;C

TM�

T (10)

wðkÞ ¼ ½wT1ðkÞ;w

T2ðkÞ; . . . ;w

TMðkÞ�

T (11)

Having partitioned the vectors, the PLCMV algorithmcan be summarized as

wiðkþ 1Þ ¼ wiðkÞ þ SDQ ðkÞCi � myðkÞXiðkÞ; i ¼ 1;2; . . . ;M

(12)

SDQ ðkÞ ¼ S½1� DðkÞ þ myðkÞQ ðkÞ� (13)

where

S ¼XMi¼1

Si

!�1

¼XMi¼1

CHi Ci

!�1

(14)

Q ðkÞ ¼XMi¼1

QiðkÞ ¼XMi¼1

CHi XiðkÞ (15)

yðkÞ ¼XMi¼1

yiðkÞ ¼XMi¼1

wHi ðkÞXiðkÞ (16)

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DðkÞ ¼XMi¼1

DiðkÞ ¼XMi¼1

CHi wiðkÞ (17)

myðkÞ ¼ myHðkÞ. (18)

3. Robust adaptive beamforming for large-scale arrays

The PLCMV algorithm can be easily implemented in aparallel DSP or FPGA processing system due to some stepsof this algorithm that can be executed in parallel. It can beused to deal with real time adaptive beamforming forlarge-scale arrays. If there is no mismatch between thepresumed steering vector and the actual direction vector,which means the presumed C is perfect, the arrayprocessor can output the desired signal without distortionand 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 arrayimperfections, some underlying assumptions on adaptivearrays can be invalid, resulting in a mismatch between thepresumed C and the actual C. The mismatch betweenthem will degrade the performance of the PLCMVbeamforming algorithm.

The array imperfections, such as direction-of-arrivalmismatch and array geometry error, can be modeled asgeneralized array phase errors in the steering vector[20,21]. As a consequence of the multiple errors, the actualsteering vector of the desired signal can be expressed as

Cðy0;DFÞ ¼ ½ejw0t1ðy0ÞþjDf1 ; . . . ; ejw0tN ðy0ÞþjDfN �T (19)

where w0 ¼ 2pf 0, f 0 is the frequency of the desired signal,tjðy0Þ ¼ dj sin y0=c, for j ¼ 1,y,N, denotes the ideal steer-ing delay corresponding to the assumed target directiony0 of an ideal array, dj is the distance between the jth andthe first antenna, and c denotes the velocity of the light.Dfj, for j ¼ 1,y,N, denotes the generalized phase error ofthe array processor. DF is an N�1 vector which isexpressed as DF ¼ ½Df1; . . . ; Dfi; . . . ; DfN�

T. The keyissue of this robust beamforming algorithm is to estimateDF by maximizing the array output power of the desiredsignal [20,21], which in turn can be expressed as anoptimization problem:

minDF

P0S ¼ minDF

CHðy0;DFÞR�1Cðy0;DFÞ (20)

Next, the gradient search method is used to iterativelyadjust the phase error vector DF

DFðkþ 1Þ ¼ DFðkÞ � m0@P0sðkÞ

@DF

����DF¼DFðkÞ

(21)

where k is an iteration index, m0 is a small, positive, stepsize parameter, and @P0sðkÞ=@DF is the gradient of P0s withrespect to DF. The gradient vector @P0sðkÞ=@DF can begiven by

@P0sðkÞ

@DF¼ 2RefCH

DFðy0;DFkÞR�1Cðy0;DFkÞg (22)

where Ref�g denotes the operator to get the real part of thecomplex vector, Cðy0;DFkÞ is the steering vector of the

desired signal obtained in the (k�1)th iteration which canbe expressed as

Cðy0;DFkÞ ¼ diagfejDf1ðkÞ; . . . ; ejDfN ðkÞgCðy0;DF0Þ (23)

where Cðy0;DF0Þ is the initial steering vector, and diagf�gis the operator used to expand the vector into a diagonalmatrix. Cðy0;DFkÞ is denoted as CðkÞ hereafter, andCðy0;DF0Þ is denoted as Cð0Þ. CDFðy0;DFkÞ ¼ jdiagfCðkÞgdenotes the gradient of CðkÞ with respect to DF, andCDFðy0;DFkÞ is denoted as CDFðkÞ hereafter.

Eq. (21) can be expressed approximately as

@P0sðkÞ

@DF� 2RefCH

DFðkÞwðkÞg (24)

since the weight vector w can be expressed as

wðkÞ ¼ aR�1CðkÞ (25)

where a ¼ CHðkÞR�1CðkÞ is a constant.

First, based on the restructured recursive LCMValgorithm and the gradient-based optimization methodintroduced above, a new robust recursive LCMV algorithmis proposed. In the RRLCMV adaptive algorithm, thegradient-based optimization method shown in Eqs.(19)–(25) is used to search out the actual steering vectorto improve the robust performance of LCMV, and therestructured recursive LCMV algorithm shown in Eq. (8) isused to obtain the optimum algorithm. The computationalload of the RRLCMV algorithm is on the order of o(N),which is less than that of the traditional robust adaptivealgorithm based on the gradient-based optimizationmethod [20,21]. Then, in order to realize real timeprocessing for large-scale arrays, the parallel robustrecursive LCMV beamforming algorithm is proposed,which effectively partitions the RRLCMV adaptive algo-rithm into a number of operational modules.

In the proposed PRRLCMV algorithm, the vector spaceof DFðkÞ is partitioned into a number of subsections:

DFðkÞ ¼ ½DFT1ðkÞ;DF

T2ðkÞ; . . . ;DF

TMðkÞ� (26)

where M is the number of subsections, DFiðkÞ is an Ni�1data vector and

PMi¼1Ni ¼ N. CðkÞ can be correspondingly

partitioned as CðkÞ ¼ ½CT1ðkÞ;C

T2ðkÞ; . . . ;C

TMðkÞ�

T with

CiðkÞ ¼ diagfejDFiðkÞgCið0Þ for i ¼ 1;2; . . . ;M (27)

And CDFðkÞ is correspondingly partitioned asCDFðkÞ ¼ ½ðCDFÞ

T1ðkÞ; ðCDFÞ

T2ðkÞ; . . . ; ðCDFÞ

TMðkÞ�

T, whereðCDFÞiðkÞ ¼ jdiagfCiðkÞg for i ¼ 1,2,y,M.

Furthermore, a legitimate way to partition @P0sðkÞ=@DFneeds to be determined. As @P0sðkÞ=@DF can be approxi-mately expressed as @P0sðkÞ=@DF � 2RefCH

DFðkÞwðkÞg, it canbe partitioned as

@P0sðkÞ

@DF¼

@P1ðkÞ

@DF;@P2ðkÞ

@DF; . . . ;

@PMðkÞ

@DF

� �(28)

with

@PiðkÞ

@DF¼ 2RefðCDFÞ

Hi ðkÞwiðkÞg ði ¼ 1;2; . . . ;MÞ (29)

where ðCDFÞiðkÞ can be achieved in the (k�1)th iteration,and wiðkÞ can also be obtained in the (k�1)th iteration bythe PLCMV algorithm which was described in Section 2,

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with the except that Eqs. (12)–(14) of the PLCMValgorithm are revised as

wiðkþ 1Þ ¼ wiðkÞ þ SDQ ðkÞCiðkÞ � myðkÞXiðkÞ ði ¼ 1;2; . . . ;MÞ

(30)

SDQ ðkÞ ¼ SðkÞ½1� DðkÞ þ myðkÞQ ðkÞ� (31)

SðkÞ ¼XMi¼1

SiðkÞ

!�1

¼XMi¼1

CHi ðkÞCiðkÞ

!�1

(32)

Consequently, Eq. (21) can be given by

DFiðkþ 1Þ ¼ DFiðkÞ � m0@PiðkÞ

@DF

����DFi¼DFiðkÞ

ði ¼ 1;2; . . . ;MÞ

(33)

where k is an iteration index and M the numberof subsections. The updated steering vector of thedesired signal Ci(k+1) for each subsection can be ob-tained by substituting DFiðkþ 1Þ into Eq. (27) fori ¼ 1,2,y,M.

The proposed PRRLCMV adaptive beamforming algo-rithm 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 @PiðkÞ=@DF by

Eq. (29) for i ¼ 1,2,y,M.Step 5. Update the phase error vectors DFiðkþ 1Þ by

Eq. (33) for i ¼ 1,2,y,M.Step 6. Update the steering vectors Ciðkþ 1Þby equa-

tion (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 throughstep 7 until jjDfðkþ 1Þ � DfðkÞjjo�, where e is the errortolerance. The optimal weight vector can be constructedfrom the weight vectors of subsections by Eq. (11).

Assume the number of subsections is M, and theweighting vector wi of the ith subsection is an Ni � 1vector for i ¼ 1,2,y,M. The number of the complexmultiplications required for finding Si(k), Qi(k), yi(k) andDi(k) in step 1 is about 4Ni for i ¼ 1,2,y,M. Step 2 requires3 complex multiplications to obtain the SDQ(k) value.Moreover, step 3 requires about 2Ni complex multi-plications to obtain each wi(k+1) by Eq. (30) fori ¼ 1,2,y,M. In step 4, to obtain @PiðkÞ=@DF by Eq. (29)approximately 2Ni complex multiplications for i ¼ 1,2,y,M are needed. It requires Ni complex multiplicationsto find DFiðkþ 1Þ by Eq. (33) in step 5 for i ¼ 1, 2,y,M.Moreover, in step 6 it requires about Ni complex multi-plications to obtain the steering vector Ci by using Eq. (27)for i ¼ 1, 2,y,M. Hence, the total number of the complexmultiplications required for computing the optimalweight vector w is given by

CMpu ¼XMi¼1

6Ni þ 4N þ 3 ¼ 10N þ 3 (34)

which is on the order of o(N) whereas the computationalload of the robust algorithm based on the gradientmethod proposed in [20] are about o(N3), and the robustmethod in [21] is on the order of o(N2). There is nodifference in computation load between the PRRLCMVand RRLCMV adaptive beamforming algorithm. Comparedwith the RRLCMV algorithm, the PRRLCMV algorithmcan be applied to a highly efficient distributed-parallel-processing system to achieve high computational effi-ciency.

4. Parallel processing scheme of the proposed PRRLCMVadaptive algorithm

The proposed PRRLCMV adaptive algorithm is a hybridalgorithm. Steps 1 and 3–6 of this algorithm for differentsubsections can be processed in parallel for each iteration.Moreover, steps 4–6 can be processed in parallel withstep1 in each subsection. But steps 4–6 are processedsequentially. Also, to obtain the sub-weight vectors instep 3 for each iteration, the information from othersubsections (SDQ(k)) is required. Although some steps ofthe PRRLCMV algorithm have to be processed sequentially,most computation of this algorithm can be executed inparallel.

Due to steps 1 and 3–6 for different subsections of thePRRLCMV algorithm can be computed in parallel for eachiteration, the computation of different subsections inthese steps can be distributed to different computingnodes. Fig. 1 shows the flowchart of the proposedPRRLCMV algorithm. It can be seen that the PRRLCMValgorithm can be mapped onto a highly efficient dis-tributed-parallel-processing system. In Fig. 1, each blockrepresents a computing node.

In the flowchart, the received data of each subsectionare distributed to a computing node. The input signal ofthe ith subsection block is XiðkÞ in each iteration fori ¼ 1,2,y,M. In the first snapshot, yi(k), SiðkÞ, DiðkÞ, andQiðkÞ for i ¼ 1,2,y,M are obtained in each subsectionblock in parallel. Then, the results that are obtained insubsection blocks are transmitted to Block 2 computingnode. Moreover, in the first snapshot, @PiðkÞ=@DF can becomputed in the ith subsection block for i ¼ 1,2,y,M, inparallel. In the second snapshot, SDQ(k) and myðkÞ areobtained in Block 2 and then are transmitted back to thesubsection blocks. Also, DFiðkþ 1Þ can be achieved byEq. (33) in the ith subsection block for i ¼ 1,2,y,M, in thesecond snapshot. In the third snapshot, Ciðkþ 1Þ can beachieved by Eq. (27) in the ith subsection block fori 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 betransmitted to Block 2 where i ¼ 1,2,y,M. Finally, In Block2, w(k+1) is obtained by Eq. (11) in the fourth snapshot.Moreover, in the fourth snapshot, each section block cango on with the next iteration.

This implementation scheme of the proposedPRRLCMV adaptive beamforming algorithm needs onlythree snapshots for each iteration. Steps 1 and 3–6 insubsection blocks can be performed in parallel for eachiteration, 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.

F. Huang et al. / Signal Processing 90 (2010) 165–172170

step 1 in each subsection block. The PRRLCMV adaptivealgorithm is suitable to be applied in a distributed-parallel-processing system to achieve high computationalefficiency. It can be used for real time digital beamformingfor large-scale antenna arrays.

5. Numerical results

In this section, several simulation studies are carriedout to evaluate the performance of the proposedPRRLCMV adaptive algorithm. The studied array is auniform linear array with the number of array elementsN and half wavelength for interelement spacing. Assumethe number of array elements of each subsection areequal, 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 isthe number of subsections. The initial w(1) is a zerovector. In this study, the non-directional noise is assumedto be a spatially white Gaussian noise with unit covar-iance. It is worth mentioning that the m and m0 have to bechosen properly. The larger the m and m0 values are, thefaster the convergence is. But the larger the m and m0values are, the more severe the jitter of the signal-to-interference plus noise ratio curve is. Both m and m0 areempirically determined to be appropriate for each exam-ple. Two kinds of uncertainty in the array steering vectorare considered. One is the well-studied steering directionerror. The other is the array geometry error. In thissimulation study, all the simulation results are obtainedvia 50 Monte-Carlo runs.

5.1. Performance evaluation: steering direction error

In the first experiment, the effect of the number ofelements for each subsection on array performance isconsidered when the direction-of-arrival mismatch ispresent. For this case the number of elements in the arrayis N ¼ 16. The incident angle of the desired signal is 01,and the presumed steering angle is �51. Signal-to-noiseratio is equal to 0 dB. An interference signal is presentwith incident angle of �201, and interference-to-noiseratio is equal to 10 dB. The empirically found values of mand m0 are 0.004 and 0.013, respectively. To obtain theresults, 2500 iterations steps have been taken.

Fig. 2 shows the array beampatterns formed by thePRRLCM algorithm when steering vector error existedwith different subsection sizes (Ni ¼ 8, Ni ¼ 4, and Ni ¼ 1).The solid curve denotes the beampattern with Ni ¼ 4. Theplus signs denote the beampattern with Ni ¼ 1, and thecircles denote the beampattern with Ni ¼ 8. Fig. 2 showsthat all the plus signs and circles are on the solid curve.This indicates that the PRRLCMV algorithm has the samebeam performance with different subsection sizes. Theoutput SINRs for Ni ¼ 4, Ni ¼ 8, and Ni ¼ 1 are all 11.7 dB,showing the same SINR performance. Simulation resultsshow that the PRRLCMV algorithm can reduce thedegradation effect due to steering angle error, regardlesshow an array is partitioned.

In the second example, the effect of the direction-of-arrival 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. 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. 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. 5. Performance comparison of the PRRLCMV algorithm with

different subsection sizes when array geometry error is present.

F. Huang et al. / Signal Processing 90 (2010) 165–172 171

is N ¼ 12. The incident angle of the desired signal is 01while the steering angle is �51, i.e., the steering angleerror is 51. Signal-to-noise ratio is 0 dB. Two interferencesignals are present with incident angles of 101 and �201.Both interference-to-noise ratios are 10 dB. The values of mand m0 are 0.0029 and 0.0072, respectively, and thesevalues are empirically found to be appropriate. Fig. 3shows that the output SINRs of the ERLS algorithm(Ni ¼ 1) with erroneous steering vector (ESV), thePRRLCMV algorithm (Ni ¼ 1) with erroneous steeringvector (ESV) and the standard LCMV(SLCMV) algorithmwith actual steering vector (ASV) are �4.3, 10.4, and10.8 dB, respectively. The array beampatterns of the threecases are shown in Fig. 4. Then, Figs. 3 and 4 demonstratethat the PRRLCMV algorithm can solve the problem ofsteering angle error in adaptive array beamforming. And itworks well even when Ni ¼ 1, which demonstrates thatthe number of elements of the smallest subsection can beless than the number of signal sources.

5.2. Performance evaluation: array geometry error

In the third example, we investigate how the subsec-tion size affects array performance when the arraygeometry error is present. In this example, the numberof whole array elements is N ¼ 16. The incident angle ofthe desired signal is 101. Signal-to-noise ratio is 0 dB. Twointerference signals are present with incident angles of401 and �201. Both interference-to-noise ratios are 10 dB.The variance of array geometry error is (0.05l)2. Theempirically found values of m and m0 are 0.0016 and 0.006,respectively. The array beampatterns shown in Fig. 5 areobtained by the PRRLCMV algorithm with Ni ¼ 8, Ni ¼ 4,and Ni ¼ 1. The asterisks denote the beampattern withNi ¼ 1, and the circles denote the beampattern withNi ¼ 8. Fig. 5 shows that all the asterisks and circles areon the solid curve which is the beampattern with Ni ¼ 4.

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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.

F. Huang et al. / Signal Processing 90 (2010) 165–172172

It indicates that the beam performance of the PRRLCMValgorithm does not vary with different decompositions ofan array when the array geometry error is present. Theoutput 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 arraygeometry error on array output SINR is considered. Theantenna array parameters in this example are the same asthose used in example three. Fig. 6 shows that the outputSINR of the PLCMV algorithm (Ni ¼ 1) is �3.7 dB and thatof the PRRLCMV algorithm (Ni ¼ 1) is 11.5 dB, whereas theideal output SINR is 11.9 dB. Simulation results indicatethat the PRRLCMV algorithm can also reduce the degrada-tion caused by the array geometry error, and that thenumber of elements of the smallest subsection can be lessthan the number of signal sources.

6. Conclusions

This paper has presented an efficient parallel robustadaptive beamforming method for large-scale arrays inthe presence of array imperfections, including steeringdirection error, array geometry error, etc. First, based on arestructured recursive linear constraint minimum var-iance algorithm and a gradient-based optimization meth-od, a new robust recursive LCMV algorithm is proposed.The computational load of the RRLCMV algorithm is onthe order of o(N), which is less than that of the traditionalrobust adaptive algorithm based on the gradient method[20,21]. Then, in order to realize real time adaptivebeamforming processing, a new efficient parallel RRLCMVadaptive algorithm for implementing the RRLCMV adap-tive algorithm is proposed, which effectively partitions theRRLCMV adaptive algorithm into a number of operationalmodules. The PRRLCMV adaptive algorithm can be easilyimplemented in a distributed-parallel-processing systemdue to its parallel property. The proposed PRRLCMVmethod can significantly reduce computational workload

and alleviate the bottleneck of high-rate data transmis-sion. Simulation results show that the PRRLCMV algo-rithm can also reduce the degradation of the arrayperformance caused by array imperfections. It is expectedthat the proposed algorithm will be used in adaptivebeamforming systems to deal with real time array signalprocessing for large-scale arrays.

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