Research Article Robust Adaptive Beamforming Using a Low...

Research ArticleRobust Adaptive Beamforming Using aLow-Complexity Steering Vector Estimation andCovariance Matrix Reconstruction Algorithm

Pei Chen Yongjun Zhao and Chengcheng Liu

Zhengzhou Institute of Information Science and Technology Zhengzhou Henan 450000 China

Correspondence should be addressed to Pei Chen clevercpei126com

Received 16 April 2016 Accepted 12 June 2016

Academic Editor Michelangelo Villano

Copyright copy 2016 Pei Chen et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A novel low-complexity robust adaptive beamforming (RAB) technique is proposed in order to overcome the major drawbacksfrom which the recent reported RAB algorithms suffer mainly the high computational cost and the requirement for optimizationprogramsThe proposed algorithm estimates the array steering vector (ASV) using a closed-form formula obtained by a subspace-basedmethod and reconstructs the interference-plus-noise (IPN) covariancematrix by utilizing a sampling progress and employingthe covariance matrix taper (CMT) technique Moreover the proposed beamformer only requires knowledge of the antenna arraygeometry and prior information of the probable angular sector in which the actual ASV lies Simulation results demonstrate theeffectiveness and robustness of the proposed algorithm and prove that this algorithm can achieve superior performance over theexisting RAB methods

1 Introduction

Aiming at receiving a signal from a certain direction and sup-pressing interferences and noise adaptive beamforming hasfound widespread application in many fields ranging fromradar sonar wireless communication and radio astronomyto medical imaging speech processing and so forth [1ndash4]Beamformers can be regarded as spatial filters which haveproper performance under ideal circumstancesThe standardCapon beamformer (SCB) is an optimal spatial filter thatmaximizes the array output signal-to-interference-plus-noiseratio (SINR) [5] However it is sensitive to array steeringvector (ASV) uncertainty and direction of arrival (DOA)estimation error for the desired signal (DS)The performancealso degrades due to the presence of the DS component inthe training data and small sample size In order to improvethe robustness of SCB various robust adaptive beamforming(RAB) techniques have been proposed in the past decades [6]

The existing RAB techniques mainly consist of diag-onal loading (DL) technique [7] eigenspace-based (ESB)technique [8] uncertainty set-based technique [9 10] andthe reconstruction-based technique [11] They have effective

performance in the presence of different mismatches intypical situations However there exist some drawbacks thatlimit their application range which include the difficulty tochoose the optimalDL factor the high probability of subspaceswap for ESBbeamformers at low signal-to-noise ratio (SNR)the cost of reducing output SINR due to the expansion of theuncertainty set when the SV mismatch is large the ad hocnature and the high computational cost

Recently a new approach to the design of RAB based onthe interference-plus-noise (IPN) covariance matrix recon-struction has been introduced in [11] This method utilizesthe Capon spectrum to integrate over an angular sectorseparated from the direction of DS The reconstruction-based beamformer is effective at removing the DS fromthe sample covariance matrix but suffers from its highcomputational complexity The main computational costis due to the large number of samples involved in bothspectrum estimation and summation On the other handthe recent beamformers in [12 13] enhance the robustnessagainst ASV errors However the utilization of complicatedoptimization software restricts their applications in prac-tical situations To improve the reconstruction-based RAB

Hindawi Publishing CorporationInternational Journal of Antennas and PropagationVolume 2016 Article ID 2438183 9 pageshttpdxdoiorg10115520162438183

2 International Journal of Antennas and Propagation

technique a Low-Complexity Shrinkage-Based MismatchEstimation (LOSCME) algorithm is presented in [14] But itcan perform effectively only when the interfering sources areweak

To reduce the computational complexity of the RABmethod in [11] and eliminate its dependence on the opti-mization software in this paper a novel reconstruction-basedRAB algorithm is proposed This algorithm is characterizedby lower complexity and a closed-form formula estimation ofthe actual ASVMoreover it requires very little prior informa-tion and has a superior performance to previously reportedRAB algorithms The only prior information required is theknowledge of the antenna array geometry and the coarseangular sector in which the actual ASV lies Three steps areneeded to achieve this algorithm Firstly a subspace-basedmethod is introduced to obtain a closed-form formula forASV estimation avoiding using the optimization softwareThen the IPN covariance matrix is reconstructed basedon a spatial power spectrum sampling (SPSS) method [15]which is realized by a proposed sample equation Finally thecovariance matrix taper (CMT) technique [16] is utilized toadopt the relatively small size of array systems in practiceSimulation results will be provided to prove the effectivenessand robustness of the proposed beamformer

The rest of this paper is organized as follows The datamodel of array output and some necessary backgroundsabout adaptive beamformer are described in Section 2 InSection 3 a novel RAB algorithm is proposed by employingASV estimation and IPN covariance matrix reconstructionSimulation results are presented in Section 4 Finally conclu-sion is drawn in Section 5

2 Signal Model

Consider a uniform linear array (ULA) of119872 omnidirectionalantenna elements impinged by 119875 narrowband uncorrelatedfar-field signals The 119872 times 1 vector representing the receivedsignal at the 119896th snapshot can be modeled as

x (119896) = s (119896) + i (119896) + n (119896) (1)

where s(119896) i(119896) and n(119896) denote the statistically independent119872times1 vectors of the DS interference and noise respectivelyThe beamformer output is given by y(119896) = w119867x(119896) wherew = [119908

1 1199082 119908

119872]119879 is the complex beamformer weighting

vector and (sdot)119867 and (sdot)

119879 denote the Hermitian transposeand transpose respectively The beamformer output SINR isdefined as

SINR =

1205902

119904

10038161003816100381610038161003816w119867a (120579

119904)10038161003816100381610038161003816

2

w119867Ri+nw (2)

where 120579119904is the assumed DOA of the DS and a(120579

119904)

is the ASV which has the general form a(120579119904) =

[1 119890119895120587 sin 120579

119904 sdot sdot sdot 119890119895120587(119872minus1) sin 120579

119904]119879

Meanwhile 1205902

119904is the

power of the DS and Ri+n = 119864(i(119896) + n(119896))(i(119896) + n(119896))119867 isthe IPN covariance matrix

By maximizing the output SINR of the beamformer theoptimal weight vector can be obtained by

minw w119867Ri+nw

subject to w119867a (120579119904) = 1

(3)

The solution is known as the Capon beamformer [13]

w =Rminus1i+na (120579119904)

a119867 (120579119904)Rminus1i+na (120579119904)

(4)

In practice theoretical covariance matrix Ri+n is usuallyunavailable and sample covariance matrix (5) is used as anapproximation

R119909=

1

119870

119870

sum

119896=1

x (119896) x (119896) (5)

where119870 is the number of data snapshots

3 Proposed Approach

Theperformance of standard beamformers degrades dramat-icallywhen theASVerrors exist and theDSwith a high SNR ispresent in the training snapshots To remove the DS from thesample covariancematrix recently an IPN covariancematrixreconstruction method was proposed [11]

In [11] the IPN covariance matrix can be reconstructedby using the spatial spectrum of the array as

Ri+n = int120579isinΘ

1205902(120579) a (120579) a119867 (120579) 119889120579 (6)

where 1205902(120579) is the Capon spectrum 1205902(120579) = 1a119867(120579)Rminus1

119909a(120579)

Θ is the complement sector ofΘ andΘ stands for the assumeddirection range of theDS that isΘcapΘ = andΘcupΘ coverswhole spatial domain

Thismethod hasmainly two aspects of drawbacks Firstlyconcerning the mismatch between the actual ASV and thenominal ASV the IPNmatrixmay not be reconstructed accu-rately Secondly its high computational complexity restrictsits practical performance [12] In the following part moreprecise estimation of the actual ASV can be achieved andthe computational cost of IPN matrix reconstruction can bereduced

31 Desired Signal Array Steering Vector Estimation Similarto the reconstruction of IPN covariance matrix as (6) a newmatrix Casv can be constructed by

Casv = intΘ

119888 (120579) a (120579) a119867 (120579) 119889120579 (7)

where 119888(120579) denotes a probability density function but thechoice of 119888(120579) is more flexible which means that 119888(120579) can bechosen to be independent of 120579 for example 119888(120579) = 1 forall120579 isin

International Journal of Antennas and Propagation 3

Θ or adjust values depending on the prior informationThenCasv can be eigendecomposed as

Casv =119872

sum

119894=1

120590119894V119894V119867119894= VΣV119867 (8)

where 120590119894 119894 = 1 119872 are the eigenvalues of Casv

in descending order Σ = diag1205901 120590

119872 are diagonal

matrices k119894 119894 = 1 119872 are the eigenvectors associated

with 120590119894 andV = [k

1 k

119872] Following the approach in [17]

let 119878 be the smallest integer such that sum119878119894=1

|120590119894|2sum119872

119894=1|120590119894|2gt

120585 where 0 lt 120585 lt 1 is a predetermined threshold Thenthe eigenvectors associated with 119878 largest eigenvalues of Casvcan be chosen to form the column orthogonal matrix V

119878=

[k1 k

119878] As derivation in [18] the actual ASV a(120579

119904) lies in

the subspace spanned by the columns of V119878

Similar to (8) in order to obtain the eigenvector thesample covariance matrix R

119909can be decomposed as

R119909=

119872

sum

119895=1

120582119895e119895e119867119895= E119904Λ119904E119867119904+ E119899Λ119899E119867119899 (9)

where 120582119895 119895 = 1 119872 are the eigenvalues of R

119909

in descending order Λ119904

= diag1205821 120582

119875 and Λ

119899=

diag120582119875+1

120582119872 are diagonal matrices e

119895 119895 = 1

119872 are the eigenvectors associated with 120582119895 and E

119904=

[e1 e2 e

119875] and E

119899= [e119875+1

e119872] denote the signal-

plus-interference (SPI) subspace and noise subspace respec-tively It is well known that the actual ASV ofDS a(120579

119904) belongs

to the subspace spanned by the columns of E119904[19]

As mentioned above two constraints can be imposed ona

1198621= a a = V

119878120572119881

1198622= a a = E

119904120572119864

(10)

where 120572119881and 120572

119864are the coefficient vectors Then the actual

ASV should be a vector lying within the intersection of 1198620=

1198621cap 1198622 This intersection can be obtained by employing the

theorem of sequential vector space projections [19] and theASV of DS is proved to be estimated by

aes = radic119872119875P1198621

P1198622

(11)

where P1198621

= V119878V119867119878 P1198622

= E119904E119867119904 and 119875sdot denotes the

eigenvector associated with the maximal eigenvalue of amatrix This estimation method can be more efficient whenthe look direction error is large and the method avoidsthe need of optimization software by using a closed-formformula

32 Spatial Power Spectrum Sampling Based IPN Covari-ance Matrix Reconstruction The main computational costof the method in [11] is the integration approximation bysummation where 119878 (the number of sampled values) timesspectrum estimation and vector multiplication operationshave to be performed To reduce the cost the complexspectrum estimation process should be eliminated

Consider the inner product of two steering vector whichis written as

119891 (120579 1205790) =

1

119872a119867 (1205790) a (120579) (12)

where 1205790is a specified reference direction and 120579 120579

0isin

[minus1205872 1205872] Substituting a(120579) into (12) gives

119891 (120579 1205790) =

1

119872

119872minus1

sum

119896=0

119890119895119896120587[sin(120579)minussin(120579

0)] (13)

Let 119909 = 1198722[sin(120579) minus sin(1205790)] isin [(minus1 minus sin(120579

0))1198722 (1 minus

sin(1205790))1198722) (13) can be written as

119891 (119909) =1

119872

119872minus1

sum

119896=0

119890119895(2120587119872)119896119909

=1

119872sdot

sin (120587119909)sin ((120587119872) 119909)

119890119895((119872minus1)119872)120587119909

(14)

From the derivation above 119891(119909) can be regarded as aninverse discrete Fourier transform (IDFT) of an 119872-pointrectangular function in the frequency domain When 119872 islarge enough 119891(119909) can be approximated as a sinc functionthat is 119891(119909) = sinc(120587119909) = sin(120587119909)120587119909 119891(120579 120579

0) will

approximate a Kronecker delta function that is

119891 (120579 1205790) asymp 1205751205791205790

=

1 120579 = 1205790

0 else(15)

The function 119891(120579 1205790) is called the selecting property of the

steering vector in [15]Denote the zeros of 119891(120579 120579

0) by 120579

119896 and consider (14)

119891(119909) = 0 is obtained when 119909 isin 119885 = 119911 | 119911 isin [(minus1 minus

sin(1205790))1198722 (1minussin(120579

0))1198722) 119911 isin Z z = 0 then there are

119872minus1 such values in the set119885denoted by119909119896 119896 = 1 2 119872minus

1 Then 120579119896can be easily obtained by 120579

119896= arcsin(2119909

119896119872 +

sin(1205790))

Define a matrix using 120579119896119872minus1

119896=0

D =1

119872sum

120579119896isinΩ

a (120579119896) a119867 (120579

119896) (16)

whereΩ is a specified angular sector Consider that R can beformed by integrating the spatial spectrum 120590

2(120579) though the

whole region as

R = int120579isin[minus12058721205872)

1205902(120579) a (120579) a119867 (120579) 119889120579 (17)


Then letΩ = Θ (18) can be obtained as

D sdot R sdotD =1

119872sum

120579119896isinΘ

a (120579119896) a119867 (120579

119896)

sdot (int120579isin[minus12058721205872)

1205902(120579) a (120579) a119867 (120579) 119889120579) sdot

1

119872

sdot sum

120579119894isinΘ

a (120579119894) a119867 (120579

119894) = sum

120579119896isinΘ

sum

120579119894isinΘ

a (120579119894)

sdot int120579isin[minus12058721205872)

1205902(120579) [

1

119872a119867 (120579119894) a (120579)] sdot [ 1

119872

sdot a119867 (120579) a (120579119896)] 119889120579 sdot a119867(120579119896) = sum

120579119896isinΘ

sum

120579119894isinΘ

a (120579119894)


1205902(120579) 119891 (120579 120579119894) 119891 (120579119896 120579) 119889120579 a119867 (120579

119896)

(18)

As discussed above when 119872 is large enough D sdot R sdot D is aHermitian matrix and can be approximated as

D sdot R sdotD

asymp sum

120579119896isinΘ

sum

120579119894isinΘ

a (120579119894) int120579isin[minus12058721205872)

120575120579120579119894

120575120579119896120579119889120579 a119867 (120579

119896)

= sum

120579119896isinΘ

1205902(120579119896) a (120579119896) a119867 (120579

119896)

(19)

In this way the IPN covariance matrix is estimated bythe sample matrix D without calculating the spatial powerspectrum In practice R can be replaced by R

119909 which yields

Ri+n asymp D sdot R119909sdotD

However when119872 ≪ infin there will be a large estimationerror because the sampling spacing is not dense enough Forthe purpose of improving the performance the covariancematrix tapering technique introduced in [16] is employedThe tapered matrix is given by R

119879= R ∘T where ldquo∘rdquo denotes

the Hadamard product and T is the taper matrix Here theMailloux-Zatman (MZ) taper is used which is defined as

TMZ = [119886119898119899]119872times119872

= [sinc((119898 minus 119899) Δ

120587)] Δ gt 0 (20)

where Δ corresponds to the width of the dithering areaHenceTMZ should be adopted to taper the sample covariancematrix R

119909and estimated IPN covariancematrix respectively

that is the reconstruction of IPN covariance matrix can beobtain by

Ri+n = (D sdot (R119909∘ TMZ) sdotD) ∘ TMZ (21)

33 The Proposed Beamforming Algorithm Based on thediscussions above the proposed beamforming algorithm canbe summarized by the following steps

Step 1 Construct the two subspaces V119878and E

119904by eigende-

composing Casv and R119909 respectively Then estimate the ASV

of DS aes using (11)

Step 2 Specify theMZ taperTMZ using (20) and constructDby (16) Hence the reconstruction of IPN covariance matrixcan be obtained by using (21)

Step 3 Substituting the reconstructed IPN covariancematrixRi+n and estimated ASV of DS aes into the Capon beam-former the weight vector of the proposed approach iscomputed as

w =Rminus1i+naes (120579119904)

a119867es (120579119904) Rminus1

i+naes (120579119904) (22)

In the proposed algorithm the main computationalcomplexity lies in the eigendecomposition operation and thematrix inversion operation both of which have complexityof 119874(1198723) That means the overall computational complexityis 119874(1198723) Considering that the number of sampling pointsrequired in (7) and (16) is reasonably much less than thatrequired in the beamformer in [11] the proposed beamformerhas a lower cost than that of [11] Additionally compared tothe existing RAB methods in [12 13] which have complexityequal or higher than 119874(119872

35) the proposed algorithm also

avoids the need of optimization software and thus is easierto apply in practice

4 Simulations

In the simulations without loss of generality a ULA with119872 = 30 omnidirectional sensors is considered It is assumedthat there is oneDS from the presumed direction 120579

119904= 5∘ Two

uncorrelated interferences with random waveforms arrivefrom DOA angles of minus53∘ and 32

∘ respectively The noiseis modeled as zero-mean and unity variance spatially andtemporally white Gaussian noise The interference-to-noiseratio (INR) at each sensor is set to be fixed at 30 dB For eachscenario 500 Monte Carlo trials are performed

The proposed beamformer is compared to the diagonalloading sample matrix inversion (LSMI) beamformer [20]the ESB beamformer [8] the worst-case-based beamformer[9] the sequential quadratic programming (SQP) basedbeamformer [18] the reconstruction-based beamformer [11]and the beamformer in [13] The diagonal loading factor ofthe LSMI beamformer is chosen as twice the noise powerThe dimension of the signal-plus-interference subspace isassumed to be always estimated correctly for the eigenspace-based beamformer The value 120576 = 03119872 is selected for theworst-case-based beamformer as it has been recommendedin [9] while the value 120575 = 01 and 20 dominant eigenvectorsof the matrix C are used in the SQP based beamformerFor the proposed beamformer the reconstruction-basedbeamformer and the beamformer in [13] the angular sectorof the DS is presumed to be Θ = [120579

119901minus 5∘ 120579119901+ 5∘] In this

paper 120585 = 095 1205790= 0∘ and Δ = 2 arcsin(2119872) are chosen in

(11) (12) and (20) respectively

Example 1 (mismatch due to signal direction error) The lookdirection mismatch of the DS is assumed to be random anduniformly distributed in [minus4

∘ 4∘] for each simulation run


Proposed beamformerLSMI beamformer

Worst-case beamformerESB beamformer

60 800 20 40minus20minus80 minus60 minus40

DOA (∘)

minus50

minus40

minus30

minus20

minus10

0

10

Beam

pat

tern

(dB)

Figure 1 Comparison of the normalized beampatterns in Example 1with SNR = 0 dB and 119870 = 60

which means the mismatch changes from run to run butkeeps fixed from snapshot to snapshotThe array normalizedbeam patterns for the beamformers in a single simulation runwhere SNR = 0 dB and119870 = 60 are displayed in Figure 1 It canbe seen that in the presence of pointing error the proposedbeamformer and ESB beamformer are able to point the mainlobe to the actual direction while the LSMI beamformerand worst-case-based beamformer cannot Furthermore theproposed beamformer has lower side lobes and deeper nullsat the directions of interferences which make the proposedbeamformer suppress the noise and interferences effectively

Considering the influence of random look direction erroron array output SINR the performance curves versus theinput SNR and the number of snapshots are drawn in Figures2(a) and 2(b) respectively The number of snapshots is fixedto be 119870 = 60 and the INR is kept at 30 dB in Figure 2(a)while a fixed SNR for the DS at 10 dB and a fixed INR at30 dB are considered in Figure 2(b) From the results shownin Figure 2 it can be found that the output SINR of theproposed algorithm is closer to the optimal SINR in a largerange of SNR from minus10 to 50 dB and for all values of numberof snapshots from 10 to 100 which illustrates its high dynamicrange That means the proposed algorithm outperforms theother tested beamformers in the scenario where only signaldirection error exists

The SINR performance versus the pointing error is alsoinvestigated and the results are shown in Figure 3 with fixedSNR and INR at 10 dB and 30 dB respectively It can be seethat the proposed beamformer effectively deals with a widerange of pointing errors and achieves the best performanceamong the tested beamformers even when the pointing erroris as large as plusmn4∘

Table 1 Deviations from the optimal SINR for training data size of119870 = 60 and INR = 30 dB

Beamformers Deviations (dB)at SNR = 0 dB

Deviations (dB)at SNR = 30 dB

Proposed 1330 1347LSMI 12277 57511Worst-case-based 3139 21768ESB 2949 45948SQP 3755 43308Reconstruction-based 1081 1083Beamformer in [13] 3122 39040

Example 2 (mismatch due toASV randomerror) In this sim-ulation example the ASV of DS is assumed to be randomlydistributed in an uncertainty set which can be modeled as

a (120579119904) = a (120579

119904) + e (23)

where a(120579119904) represents the nominal SV corresponding to the

direction 120579119904and e is the random error vector which is drawn

in each simulation run independently from an uncertainty setas follows [12]

e = 120576

radic119872

[1198901198951206010 1198901198951206011 119890

119895120601119872minus1] (24)

where 120576 and 120601119898are uniformly distributed in the intervals

[0 radic3] and [0 2120587] respectively Figures 4(a) and 4(b) cor-respond to the performance of the investigated methodsversus the input SNR and the number of snapshots It canbe seen from the figures that the proposed beamformer andthe reconstruction-based beamformer outperform the othertested beamformers Table 1 shows the deviations from theoptimal SINR for the beamformers at SNR = 0 dB and SNR =

30 dB respectivelyThe performance improvement is a directresult of the ASV estimation and DS elimination especiallyat high SNR values

Furthermore the proposed algorithm performs almost aswell as the reconstruction-based beamformer for the outputSINRbut enjoys a faster convergence rate because of the lowercomputational cost without complex integral computationSince the ASV mismatch is comprehensive and arbitrary-type the proposed beamformer is proved to be effectiveagainst the random error of ASV

Example 3 (mismatch due to incoherent local scattering) Inthis example it is assumed that the desired signal has a time-varying ASV which can be modeled as [11]

a (120579119904 119896) = 119906

0 (119896) a (120579119904) +4

sum

119901=1

1199061015840

119901(119896) a1015840 (1205791015840

119901) (25)

where a1015840(1205791015840119901) without loss of generality 119901 = 1 2 3 4 denotes

the incoherently scattered signal paths Assume that the


Optimal

LSMI beamformerProposed beamformer

Worst-case beamformer

ESB beamformerSQP beamformerReconstruction-basedbeamformerBeamformer in [13]

minus20

minus10

0

10

20

30

40

50

60

70O

utpu

t SIN

R (d

B)

0 10 20 30 40 50minus10

SNR (dB)

(a)

OptimalProposed beamformerLSMI beamformerWorst-case beamformer

ESB beamformerSQP beamformer

Beamformer in [13]

Reconstruction-basedbeamformer

minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

Out

put S

INR

(dB)

8070 90 100504030 602010Number of snapshots

(b)

Figure 2 Performance of the beamformers for the case of mismatch due to signal direction error (a) Output SINR versus SNR for trainingdata size of119870 = 60 and INR = 30 dB (b) Output SINR versus number of snapshots for fixed SNR = 10 dB and INR = 30 dB

minus50

minus40

minus30

minus20

minus10

0

10

20

Out

put S

INR

(dB)

20 3 41minus2minus3 minus1minus4

Pointing error (∘)

Optimal




Figure 3 Output SINR versus pointing error for training data sizeof 119870 = 60 and SNR = 10 dB and INR = 30 dB

directions 1205791015840119901are randomly distributed in a Gaussian distri-

bution with mean 120579119904and standard deviation 2

∘ 1199060(119896) and

1199061015840

119901(119896) are independently and identically distributed complex

Gaussian randomvariables drawn from119873(0 1)which changefrom snapshot to snapshot

In this scenario the signal covariance matrix Rs is nolonger a rank-one matrix thus the output SINR should berewritten in a more general form [9]

SINRopt =w119867Rsww119867Ri+nw

(26)

which is maximized by [9]

wopt = 119875 Rminus1i+nRs (27)

where 119875sdot denotes the same operation as in (11) Figures 5(a)and 5(b) show the performance curves versus the input SNRand the number of snapshots It can be seen from the figuresthat the proposed beamformer presents an effective perfor-mance when handling incoherent local scattering Similar toExample 2 in detail there is about 08 dB performance lossfor the proposed algorithm comparing to the reconstruction-based beamformer The main reason is that the DS may leakinto the complement sector Θ due to the incoherent localscattering which affects the accuracy of the ASV estimation

Example 4 (impacts of some factors on performance) Themain purpose of this example is to study the impacts of somefactors on performance 120585 and Δ are two main factors in theproposed algorithm The former affects the accuracy of theconstruction of subspace in which the actual ASV a(120579

119904) lies

and the latter decides the choice of taper matrix which playsa key role in the reconstruction of IPN covariance matrix

For the purpose of studying the impacts of the two factorsthe model of the mismatch is set to be the same as the firstexample The number of snapshots is fixed to be 119870 = 50


minus20

minus10

0

10

20

30

40

50

60

70

80O

utpu

t SIN

R (d

B)

40 5020 300 10minus10

SNR (dB)

Optimal




(a)

minus20

minus15

minus10

minus5

0

5

10

15

20

25

Out

put S

INR

(dB)

10 3020 50 6040 70 9080 100Number of snapshots



Beamformer in [13]


(b)

Figure 4 Performance of the beamformers for the case of ASV random error (a) Output SINR versus SNR for training data size of 119870 = 60

and INR = 30 dB (b) Output SINR versus number of snapshots for fixed SNR = 10 dB and INR = 30 dB

minus20

minus10

0

10

20

30

40

50

60

70

80

Out

put S

INR

(dB)

40 5020 300 10minus10

SNR (dB)

Optimal




(a)

minus15

minus10

minus5

0

5

10

15

20

25

Out

put S

INR

(dB)

8070 90 10040 5030 602010Number of snapshots



Beamformer in [13]


(b)

Figure 5 Performance of the beamformers for the case of incoherent local scattering (a) Output SINR versus SNR for training data size of119870 = 60 and INR = 30 dB (b) Output SINR versus number of snapshots for fixed SNR = 10 dB and INR = 30 dB

the INR and SNR are kept at 20 dB and 10 dB respectivelyThe performances of the proposed algorithm versus 120585 andΔ are displayed in Figures 6(a) and 6(b) respectively FromFigure 6(a) it can be seen that the output SINR increases as120585 gets larger but the growth range is quite small Considering

that the computational cost will not significantly increase as120585 gets larger setting 120585 = 095 in the proposed algorithm isappropriate Similarly the result of the test of Δ is shown inFigure 6(b) in which the choice of Δ = 2 arcsin(2119872) can befound near the peak of the performance curve To ensure a


Proposed beamformer

2088

209

2092

2094

2096

2098

21

2102

2104

2106

2108O

utpu

t SIN

R (d

B)

08 09 106 0704 0502 03010120585

(a)

OptimalProposed beamformer

05 10 2 2515 35 4 45 53Δ

18

19

20

21

22

23

24

25

Out

put S

INR

(dB)

(b)

Figure 6 Performance of the beamformers for the case of mismatch due to signal direction error (a) Output SINR versus 120585 (b) Output SINRversus Δ for training data size of119870 = 50 and INR = 20 dB and SNR = 10 dB

satisfying output performance Δ can be decided by a test fora typical array before the algorithm operates The procedureis offline and will not dramatically increase the system cost

5 Conclusion

In this paper a novel low-complexity RAB method is pro-posed which is easier to realize in practical situations andmore robust to the look direction mismatch than otherexisting algorithms The ASV is estimated by a closed-form formula so as to avoid utilizing the optimizationsoftware and the IPN covariancematrix is reconstructed by asampling progress The proposed beamformer only requiresprior knowledge of the antenna geometry and the angularsector in which the ASV is located Simulation results havedemonstrated that the proposed beamformer can achievesuperior performance over the existing state of the art RABmethods To simplify the illustration the influence of theelement pattern the polarization and the mutual coupling isnot considered in this paper However these elements will beinvestigated in the future study

Competing Interests

The authors declare that they have no competing interests

References

[1] H L Van TreesOptimumArray Processing JohnWiley amp SonsHoboken NJ USA 2002

[2] W Liu and S Weiss Wideband Beamforming Concepts andTechniques John Wiley amp Sons Chichester UK 2010

[3] H Singh and R M Jha ldquoTrends in adaptive array processingrdquoInternational Journal of Antennas and Propagation vol 2012Article ID 361768 20 pages 2012

[4] U Nickel ldquoArray processing for radar achievements andchallengesrdquo International Journal of Antennas and Propagationvol 2013 Article ID 261230 21 pages 2013

[5] J Capon ldquoHigh-resolution frequency-wavenumber spectrumanalysisrdquo Proceedings of the IEEE vol 57 no 8 pp 1408ndash14181969

[6] S A Vorobyov ldquoPrinciples of minimum variance robust adap-tive beamforming designrdquo Signal Processing vol 93 no 12 pp3264ndash3277 2013

[7] L Du J Li and P Stoica ldquoFully automatic computation ofdiagonal loading levels for robust adaptive beamformingrdquo IEEETransactions on Aerospace and Electronic Systems vol 46 no 1pp 449ndash458 2010

[8] L Chang and C-C Yeh ldquoPerformance of DMI and eigenspace-based beamformersrdquo IEEE Transactions on Antennas and Prop-agation vol 40 no 11 pp 1336ndash1348 1992

[9] S A Vorobyov A B Gershman and Z-Q Luo ldquoRobust adap-tive beamforming using worst-case performance optimizationa solution to the signal mismatch problemrdquo IEEE Transactionson Signal Processing vol 51 no 2 pp 313ndash324 2003

[10] H Li K Wang C Wang Y He and X Zhu ldquoRobust adaptivebeamforming based on worst-case and norm constraintrdquo Inter-national Journal of Antennas amp Propagation vol 2015 ArticleID 765385 7 pages 2015

[11] Y Gu and A Leshem ldquoRobust adaptive beamforming basedon interference covariance matrix reconstruction and steeringvector estimationrdquo IEEE Transactions on Signal Processing vol60 no 7 pp 3881ndash3885 2012

[12] L Huang J Zhang X Xu and Z Ye ldquoRobust adaptivebeamforming with a novel interference-plus-noise covariancematrix reconstruction methodrdquo IEEE Transactions on SignalProcessing vol 63 no 7 pp 1643ndash1650 2015


[13] H Ruan and R C De Lamare ldquoRobust adaptive beamformingusing a low-complexity shrinkage-based mismatch estimationalgorithmrdquo IEEE Signal Processing Letters vol 21 no 1 pp 60ndash64 2014

[14] Z Zhang W Liu W Leng A Wang and H Shi ldquoInterference-plus-noise covariance matrix reconstruction via spatial powerspectrum sampling for robust adaptive beamformingrdquo IEEESignal Processing Letters vol 23 no 1 pp 121ndash125 2016

[15] J R Guerci ldquoTheory and application of covariance matrixtapers for robust adaptive beamformingrdquo IEEE Transactions onSignal Processing vol 47 no 4 pp 977ndash985 1999

[16] A Khabbazibasmenj S A Vorobyov and A HassanienldquoRobust adaptive beamforming based on steering vector esti-mation with as little as possible prior informationrdquo IEEETransactions on Signal Processing vol 60 no 6 pp 2974ndash29872012

[17] F Shen F Chen and J Song ldquoRobust adaptive beamformingbased on steering vector estimation and covariance matrixreconstructionrdquo IEEE Communications Letters vol 19 no 9 pp1636ndash1639 2015

[18] A Hassanien S A Vorobyov and K M Wong ldquoRobust adap-tive beamforming using sequential quadratic programmingan iterative solution to the mismatch problemrdquo IEEE SignalProcessing Letters vol 15 pp 733ndash736 2008

[19] J Zhuang and A Manikas ldquoInterference cancellation beam-forming robust to pointing errorsrdquo IET Signal Processing vol7 no 2 pp 120ndash127 2013

[20] A Elnashar S M Elnoubi and H A El-Mikati ldquoFurtherstudy on robust adaptive beamforming with optimum diagonalloadingrdquo IEEE Transactions on Antennas and Propagation vol54 no 12 pp 3647ndash3658 2006

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



technique a Low-Complexity Shrinkage-Based MismatchEstimation (LOSCME) algorithm is presented in [14] But itcan perform effectively only when the interfering sources areweak

To reduce the computational complexity of the RABmethod in [11] and eliminate its dependence on the opti-mization software in this paper a novel reconstruction-basedRAB algorithm is proposed This algorithm is characterizedby lower complexity and a closed-form formula estimation ofthe actual ASVMoreover it requires very little prior informa-tion and has a superior performance to previously reportedRAB algorithms The only prior information required is theknowledge of the antenna array geometry and the coarseangular sector in which the actual ASV lies Three steps areneeded to achieve this algorithm Firstly a subspace-basedmethod is introduced to obtain a closed-form formula forASV estimation avoiding using the optimization softwareThen the IPN covariance matrix is reconstructed basedon a spatial power spectrum sampling (SPSS) method [15]which is realized by a proposed sample equation Finally thecovariance matrix taper (CMT) technique [16] is utilized toadopt the relatively small size of array systems in practiceSimulation results will be provided to prove the effectivenessand robustness of the proposed beamformer

The rest of this paper is organized as follows The datamodel of array output and some necessary backgroundsabout adaptive beamformer are described in Section 2 InSection 3 a novel RAB algorithm is proposed by employingASV estimation and IPN covariance matrix reconstructionSimulation results are presented in Section 4 Finally conclu-sion is drawn in Section 5

2 Signal Model

Consider a uniform linear array (ULA) of119872 omnidirectionalantenna elements impinged by 119875 narrowband uncorrelatedfar-field signals The 119872 times 1 vector representing the receivedsignal at the 119896th snapshot can be modeled as

x (119896) = s (119896) + i (119896) + n (119896) (1)

where s(119896) i(119896) and n(119896) denote the statistically independent119872times1 vectors of the DS interference and noise respectivelyThe beamformer output is given by y(119896) = w119867x(119896) wherew = [119908

1 1199082 119908

119872]119879 is the complex beamformer weighting

vector and (sdot)119867 and (sdot)

119879 denote the Hermitian transposeand transpose respectively The beamformer output SINR isdefined as

SINR =

1205902

119904

10038161003816100381610038161003816w119867a (120579

119904)10038161003816100381610038161003816

2

w119867Ri+nw (2)

where 120579119904is the assumed DOA of the DS and a(120579

119904)

is the ASV which has the general form a(120579119904) =

[1 119890119895120587 sin 120579

119904 sdot sdot sdot 119890119895120587(119872minus1) sin 120579

119904]119879

Meanwhile 1205902

119904is the

power of the DS and Ri+n = 119864(i(119896) + n(119896))(i(119896) + n(119896))119867 isthe IPN covariance matrix

By maximizing the output SINR of the beamformer theoptimal weight vector can be obtained by

minw w119867Ri+nw

subject to w119867a (120579119904) = 1

(3)

The solution is known as the Capon beamformer [13]

w =Rminus1i+na (120579119904)

a119867 (120579119904)Rminus1i+na (120579119904)

(4)

In practice theoretical covariance matrix Ri+n is usuallyunavailable and sample covariance matrix (5) is used as anapproximation

R119909=

1

119870

119870

sum

119896=1

x (119896) x (119896) (5)

where119870 is the number of data snapshots

3 Proposed Approach

Theperformance of standard beamformers degrades dramat-icallywhen theASVerrors exist and theDSwith a high SNR ispresent in the training snapshots To remove the DS from thesample covariancematrix recently an IPN covariancematrixreconstruction method was proposed [11]

In [11] the IPN covariance matrix can be reconstructedby using the spatial spectrum of the array as

Ri+n = int120579isinΘ

1205902(120579) a (120579) a119867 (120579) 119889120579 (6)

where 1205902(120579) is the Capon spectrum 1205902(120579) = 1a119867(120579)Rminus1

119909a(120579)

Θ is the complement sector ofΘ andΘ stands for the assumeddirection range of theDS that isΘcapΘ = andΘcupΘ coverswhole spatial domain

Thismethod hasmainly two aspects of drawbacks Firstlyconcerning the mismatch between the actual ASV and thenominal ASV the IPNmatrixmay not be reconstructed accu-rately Secondly its high computational complexity restrictsits practical performance [12] In the following part moreprecise estimation of the actual ASV can be achieved andthe computational cost of IPN matrix reconstruction can bereduced

31 Desired Signal Array Steering Vector Estimation Similarto the reconstruction of IPN covariance matrix as (6) a newmatrix Casv can be constructed by

Casv = intΘ

119888 (120579) a (120579) a119867 (120579) 119889120579 (7)

where 119888(120579) denotes a probability density function but thechoice of 119888(120579) is more flexible which means that 119888(120579) can bechosen to be independent of 120579 for example 119888(120579) = 1 forall120579 isin



Casv =119872

sum

119894=1

120590119894V119894V119867119894= VΣV119867 (8)



119872 are diagonal


with 120590119894 andV = [k

1 k



|120590119894|2sum119872

119894=1|120590119894|2gt


119878=

[k1 k


119904) lies in




R119909=

119872

sum

119895=1

120582119895e119895e119867119895= E119904Λ119904E119867119904+ E119899Λ119899E119867119899 (9)


119909


= diag1205821 120582

119875 and Λ

119899=

diag120582119875+1


119895 119895 = 1


119904=

[e1 e2 e

119875] and E

119899= [e119875+1



119904) belongs



1198621= a a = V

119878120572119881

1198622= a a = E

119904120572119864

(10)

where 120572119881and 120572





aes = radic119872119875P1198621

P1198622

(11)

where P1198621

= V119878V119867119878 P1198622





119891 (120579 1205790) =

1

119872a119867 (1205790) a (120579) (12)


0isin


119891 (120579 1205790) =

1

119872

119872minus1

sum

119896=0

119890119895119896120587[sin(120579)minussin(120579

0)] (13)


0))1198722 (1 minus


119891 (119909) =1

119872

119872minus1

sum

119896=0

119890119895(2120587119872)119896119909

=1

119872sdot

sin (120587119909)sin ((120587119872) 119909)

119890119895((119872minus1)119872)120587119909

(14)


0) will


119891 (120579 1205790) asymp 1205751205791205790

=

1 120579 = 1205790

0 else(15)



0) by 120579



sin(1205790))1198722 (1minussin(120579




119896= arcsin(2119909

119896119872 +

sin(1205790))


119896=0

D =1

119872sum

120579119896isinΩ

a (120579119896) a119867 (120579

119896) (16)



whole region as

R = int120579isin[minus12058721205872)

1205902(120579) a (120579) a119867 (120579) 119889120579 (17)



D sdot R sdotD =1

119872sum

120579119896isinΘ

a (120579119896) a119867 (120579

119896)


1205902(120579) a (120579) a119867 (120579) 119889120579) sdot

1

119872

sdot sum

120579119894isinΘ

a (120579119894) a119867 (120579

119894) = sum

120579119896isinΘ

sum

120579119894isinΘ

a (120579119894)


1205902(120579) [

1

119872a119867 (120579119894) a (120579)] sdot [ 1

119872

sdot a119867 (120579) a (120579119896)] 119889120579 sdot a119867(120579119896) = sum

120579119896isinΘ

sum

120579119894isinΘ

a (120579119894)


1205902(120579) 119891 (120579 120579119894) 119891 (120579119896 120579) 119889120579 a119867 (120579

119896)

(18)


D sdot R sdotD

asymp sum

120579119896isinΘ

sum

120579119894isinΘ

a (120579119894) int120579isin[minus12058721205872)

120575120579120579119894

120575120579119896120579119889120579 a119867 (120579

119896)

= sum

120579119896isinΘ

1205902(120579119896) a (120579119896) a119867 (120579

119896)

(19)


119909 which yields





TMZ = [119886119898119899]119872times119872

= [sinc((119898 minus 119899) Δ

120587)] Δ gt 0 (20)







119904by eigende-





w =Rminus1i+naes (120579119904)

a119867es (120579119904) Rminus1

i+naes (120579119904) (22)




4 Simulations


119904= 5∘ Two




119901minus 5∘ 120579119901+ 5∘] In this









DOA (∘)

minus50

minus40

minus30

minus20

minus10

0

10

Beam

pat

tern

(dB)










a (120579119904) = a (120579

119904) + e (23)




e = 120576

radic119872

[1198901198951206010 1198901198951206011 119890

119895120601119872minus1] (24)






a (120579119904 119896) = 119906

0 (119896) a (120579119904) +4

sum

119901=1

1199061015840

119901(119896) a1015840 (1205791015840

119901) (25)




Optimal




minus20

minus10

0

10

20

30

40

50

60

70O

utpu

t SIN

R (d

B)

0 10 20 30 40 50minus10

SNR (dB)

(a)



Beamformer in [13]


minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

Out

put S

INR

(dB)


(b)


minus50

minus40

minus30

minus20

minus10

0

10

20

Out

put S

INR

(dB)



Optimal







∘ 1199060(119896) and

1199061015840





(26)





119904) lies




minus20

minus10

0

10

20

30

40

50

60

70

80O

utpu

t SIN

R (d

B)

40 5020 300 10minus10

SNR (dB)

Optimal




(a)

minus20

minus15

minus10

minus5

0

5

10

15

20

25

Out

put S

INR

(dB)




Beamformer in [13]


(b)



minus20

minus10

0

10

20

30

40

50

60

70

80

Out

put S

INR

(dB)

40 5020 300 10minus10

SNR (dB)

Optimal




(a)

minus15

minus10

minus5

0

5

10

15

20

25

Out

put S

INR

(dB)




Beamformer in [13]


(b)





Proposed beamformer

2088

209

2092

2094

2096

2098

21

2102

2104

2106

2108O

utpu

t SIN

R (d

B)

08 09 106 0704 0502 03010120585

(a)


05 10 2 2515 35 4 45 53Δ

18

19

20

21

22

23

24

25

Out

put S

INR

(dB)

(b)



5 Conclusion


Competing Interests


References
























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











Casv =119872

sum

119894=1

120590119894V119894V119867119894= VΣV119867 (8)



119872 are diagonal


with 120590119894 andV = [k

1 k



|120590119894|2sum119872

119894=1|120590119894|2gt


119878=

[k1 k


119904) lies in




R119909=

119872

sum

119895=1

120582119895e119895e119867119895= E119904Λ119904E119867119904+ E119899Λ119899E119867119899 (9)


119909


= diag1205821 120582

119875 and Λ

119899=

diag120582119875+1


119895 119895 = 1


119904=

[e1 e2 e

119875] and E

119899= [e119875+1



119904) belongs



1198621= a a = V

119878120572119881

1198622= a a = E

119904120572119864

(10)

where 120572119881and 120572





aes = radic119872119875P1198621

P1198622

(11)

where P1198621

= V119878V119867119878 P1198622





119891 (120579 1205790) =

1

119872a119867 (1205790) a (120579) (12)


0isin


119891 (120579 1205790) =

1

119872

119872minus1

sum

119896=0

119890119895119896120587[sin(120579)minussin(120579

0)] (13)


0))1198722 (1 minus


119891 (119909) =1

119872

119872minus1

sum

119896=0

119890119895(2120587119872)119896119909

=1

119872sdot

sin (120587119909)sin ((120587119872) 119909)

119890119895((119872minus1)119872)120587119909

(14)


0) will


119891 (120579 1205790) asymp 1205751205791205790

=

1 120579 = 1205790

0 else(15)



0) by 120579



sin(1205790))1198722 (1minussin(120579




119896= arcsin(2119909

119896119872 +

sin(1205790))


119896=0

D =1

119872sum

120579119896isinΩ

a (120579119896) a119867 (120579

119896) (16)



whole region as

R = int120579isin[minus12058721205872)

1205902(120579) a (120579) a119867 (120579) 119889120579 (17)



D sdot R sdotD =1

119872sum

120579119896isinΘ

a (120579119896) a119867 (120579

119896)


1205902(120579) a (120579) a119867 (120579) 119889120579) sdot

1

119872

sdot sum

120579119894isinΘ

a (120579119894) a119867 (120579

119894) = sum

120579119896isinΘ

sum

120579119894isinΘ

a (120579119894)


1205902(120579) [

1

119872a119867 (120579119894) a (120579)] sdot [ 1

119872

sdot a119867 (120579) a (120579119896)] 119889120579 sdot a119867(120579119896) = sum

120579119896isinΘ

sum

120579119894isinΘ

a (120579119894)


1205902(120579) 119891 (120579 120579119894) 119891 (120579119896 120579) 119889120579 a119867 (120579

119896)

(18)


D sdot R sdotD

asymp sum

120579119896isinΘ

sum

120579119894isinΘ

a (120579119894) int120579isin[minus12058721205872)

120575120579120579119894

120575120579119896120579119889120579 a119867 (120579

119896)

= sum

120579119896isinΘ

1205902(120579119896) a (120579119896) a119867 (120579

119896)

(19)


119909 which yields





TMZ = [119886119898119899]119872times119872

= [sinc((119898 minus 119899) Δ

120587)] Δ gt 0 (20)







119904by eigende-





w =Rminus1i+naes (120579119904)

a119867es (120579119904) Rminus1

i+naes (120579119904) (22)




4 Simulations


119904= 5∘ Two




119901minus 5∘ 120579119901+ 5∘] In this









DOA (∘)

minus50

minus40

minus30

minus20

minus10

0

10

Beam

pat

tern

(dB)










a (120579119904) = a (120579

119904) + e (23)




e = 120576

radic119872

[1198901198951206010 1198901198951206011 119890

119895120601119872minus1] (24)






a (120579119904 119896) = 119906

0 (119896) a (120579119904) +4

sum

119901=1

1199061015840

119901(119896) a1015840 (1205791015840

119901) (25)




Optimal




minus20

minus10

0

10

20

30

40

50

60

70O

utpu

t SIN

R (d

B)

0 10 20 30 40 50minus10

SNR (dB)

(a)



Beamformer in [13]


minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

Out

put S

INR

(dB)


(b)


minus50

minus40

minus30

minus20

minus10

0

10

20

Out

put S

INR

(dB)



Optimal







∘ 1199060(119896) and

1199061015840





(26)





119904) lies




minus20

minus10

0

10

20

30

40

50

60

70

80O

utpu

t SIN

R (d

B)

40 5020 300 10minus10

SNR (dB)

Optimal




(a)

minus20

minus15

minus10

minus5

0

5

10

15

20

25

Out

put S

INR

(dB)




Beamformer in [13]


(b)



minus20

minus10

0

10

20

30

40

50

60

70

80

Out

put S

INR

(dB)

40 5020 300 10minus10

SNR (dB)

Optimal




(a)

minus15

minus10

minus5

0

5

10

15

20

25

Out

put S

INR

(dB)




Beamformer in [13]


(b)





Proposed beamformer

2088

209

2092

2094

2096

2098

21

2102

2104

2106

2108O

utpu

t SIN

R (d

B)

08 09 106 0704 0502 03010120585

(a)


05 10 2 2515 35 4 45 53Δ

18

19

20

21

22

23

24

25

Out

put S

INR

(dB)

(b)



5 Conclusion


Competing Interests


References
























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











D sdot R sdotD =1

119872sum

120579119896isinΘ

a (120579119896) a119867 (120579

119896)


1205902(120579) a (120579) a119867 (120579) 119889120579) sdot

1

119872

sdot sum

120579119894isinΘ

a (120579119894) a119867 (120579

119894) = sum

120579119896isinΘ

sum

120579119894isinΘ

a (120579119894)


1205902(120579) [

1

119872a119867 (120579119894) a (120579)] sdot [ 1

119872

sdot a119867 (120579) a (120579119896)] 119889120579 sdot a119867(120579119896) = sum

120579119896isinΘ

sum

120579119894isinΘ

a (120579119894)


1205902(120579) 119891 (120579 120579119894) 119891 (120579119896 120579) 119889120579 a119867 (120579

119896)

(18)


D sdot R sdotD

asymp sum

120579119896isinΘ

sum

120579119894isinΘ

a (120579119894) int120579isin[minus12058721205872)

120575120579120579119894

120575120579119896120579119889120579 a119867 (120579

119896)

= sum

120579119896isinΘ

1205902(120579119896) a (120579119896) a119867 (120579

119896)

(19)


119909 which yields





TMZ = [119886119898119899]119872times119872

= [sinc((119898 minus 119899) Δ

120587)] Δ gt 0 (20)







119904by eigende-





w =Rminus1i+naes (120579119904)

a119867es (120579119904) Rminus1

i+naes (120579119904) (22)




4 Simulations


119904= 5∘ Two




119901minus 5∘ 120579119901+ 5∘] In this









DOA (∘)

minus50

minus40

minus30

minus20

minus10

0

10

Beam

pat

tern

(dB)










a (120579119904) = a (120579

119904) + e (23)




e = 120576

radic119872

[1198901198951206010 1198901198951206011 119890

119895120601119872minus1] (24)






a (120579119904 119896) = 119906

0 (119896) a (120579119904) +4

sum

119901=1

1199061015840

119901(119896) a1015840 (1205791015840

119901) (25)




Optimal




minus20

minus10

0

10

20

30

40

50

60

70O

utpu

t SIN

R (d

B)

0 10 20 30 40 50minus10

SNR (dB)

(a)



Beamformer in [13]


minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

Out

put S

INR

(dB)


(b)


minus50

minus40

minus30

minus20

minus10

0

10

20

Out

put S

INR

(dB)



Optimal







∘ 1199060(119896) and

1199061015840





(26)





119904) lies




minus20

minus10

0

10

20

30

40

50

60

70

80O

utpu

t SIN

R (d

B)

40 5020 300 10minus10

SNR (dB)

Optimal




(a)

minus20

minus15

minus10

minus5

0

5

10

15

20

25

Out

put S

INR

(dB)




Beamformer in [13]


(b)



minus20

minus10

0

10

20

30

40

50

60

70

80

Out

put S

INR

(dB)

40 5020 300 10minus10

SNR (dB)

Optimal




(a)

minus15

minus10

minus5

0

5

10

15

20

25

Out

put S

INR

(dB)




Beamformer in [13]


(b)





Proposed beamformer

2088

209

2092

2094

2096

2098

21

2102

2104

2106

2108O

utpu

t SIN

R (d

B)

08 09 106 0704 0502 03010120585

(a)


05 10 2 2515 35 4 45 53Δ

18

19

20

21

22

23

24

25

Out

put S

INR

(dB)

(b)



5 Conclusion


Competing Interests


References
























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













DOA (∘)

minus50

minus40

minus30

minus20

minus10

0

10

Beam

pat

tern

(dB)










a (120579119904) = a (120579

119904) + e (23)




e = 120576

radic119872

[1198901198951206010 1198901198951206011 119890

119895120601119872minus1] (24)






a (120579119904 119896) = 119906

0 (119896) a (120579119904) +4

sum

119901=1

1199061015840

119901(119896) a1015840 (1205791015840

119901) (25)




Optimal




minus20

minus10

0

10

20

30

40

50

60

70O

utpu

t SIN

R (d

B)

0 10 20 30 40 50minus10

SNR (dB)

(a)



Beamformer in [13]


minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

Out

put S

INR

(dB)


(b)


minus50

minus40

minus30

minus20

minus10

0

10

20

Out

put S

INR

(dB)



Optimal







∘ 1199060(119896) and

1199061015840





(26)





119904) lies




minus20

minus10

0

10

20

30

40

50

60

70

80O

utpu

t SIN

R (d

B)

40 5020 300 10minus10

SNR (dB)

Optimal




(a)

minus20

minus15

minus10

minus5

0

5

10

15

20

25

Out

put S

INR

(dB)




Beamformer in [13]


(b)



minus20

minus10

0

10

20

30

40

50

60

70

80

Out

put S

INR

(dB)

40 5020 300 10minus10

SNR (dB)

Optimal




(a)

minus15

minus10

minus5

0

5

10

15

20

25

Out

put S

INR

(dB)




Beamformer in [13]


(b)





Proposed beamformer

2088

209

2092

2094

2096

2098

21

2102

2104

2106

2108O

utpu

t SIN

R (d

B)

08 09 106 0704 0502 03010120585

(a)


05 10 2 2515 35 4 45 53Δ

18

19

20

21

22

23

24

25

Out

put S

INR

(dB)

(b)



5 Conclusion


Competing Interests


References
























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Optimal




minus20

minus10

0

10

20

30

40

50

60

70O

utpu

t SIN

R (d

B)

0 10 20 30 40 50minus10

SNR (dB)

(a)



Beamformer in [13]


minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

Out

put S

INR

(dB)


(b)


minus50

minus40

minus30

minus20

minus10

0

10

20

Out

put S

INR

(dB)



Optimal







∘ 1199060(119896) and

1199061015840





(26)





119904) lies




minus20

minus10

0

10

20

30

40

50

60

70

80O

utpu

t SIN

R (d

B)

40 5020 300 10minus10

SNR (dB)

Optimal




(a)

minus20

minus15

minus10

minus5

0

5

10

15

20

25

Out

put S

INR

(dB)




Beamformer in [13]


(b)



minus20

minus10

0

10

20

30

40

50

60

70

80

Out

put S

INR

(dB)

40 5020 300 10minus10

SNR (dB)

Optimal




(a)

minus15

minus10

minus5

0

5

10

15

20

25

Out

put S

INR

(dB)




Beamformer in [13]


(b)





Proposed beamformer

2088

209

2092

2094

2096

2098

21

2102

2104

2106

2108O

utpu

t SIN

R (d

B)

08 09 106 0704 0502 03010120585

(a)


05 10 2 2515 35 4 45 53Δ

18

19

20

21

22

23

24

25

Out

put S

INR

(dB)

(b)



5 Conclusion


Competing Interests


References
























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










minus20

minus10

0

10

20

30

40

50

60

70

80O

utpu

t SIN

R (d

B)

40 5020 300 10minus10

SNR (dB)

Optimal




(a)

minus20

minus15

minus10

minus5

0

5

10

15

20

25

Out

put S

INR

(dB)




Beamformer in [13]


(b)



minus20

minus10

0

10

20

30

40

50

60

70

80

Out

put S

INR

(dB)

40 5020 300 10minus10

SNR (dB)

Optimal




(a)

minus15

minus10

minus5

0

5

10

15

20

25

Out

put S

INR

(dB)




Beamformer in [13]


(b)





Proposed beamformer

2088

209

2092

2094

2096

2098

21

2102

2104

2106

2108O

utpu

t SIN

R (d

B)

08 09 106 0704 0502 03010120585

(a)


05 10 2 2515 35 4 45 53Δ

18

19

20

21

22

23

24

25

Out

put S

INR

(dB)

(b)



5 Conclusion


Competing Interests


References
























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Proposed beamformer

2088

209

2092

2094

2096

2098

21

2102

2104

2106

2108O

utpu

t SIN

R (d

B)

08 09 106 0704 0502 03010120585

(a)


05 10 2 2515 35 4 45 53Δ

18

19

20

21

22

23

24

25

Out

put S

INR

(dB)

(b)



5 Conclusion


Competing Interests


References
























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation




















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Research Article Robust Adaptive Beamforming Using a Low...

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