Research ArticleRobust Recursive Algorithm under Uncertaintiesvia Worst-Case SINR Maximization

Xin Song1 Feng Wang2 Jinkuan Wang1 and Jingguo Ren1

1Engineering Optimization and Smart Antenna Institute Northeastern University at QinhuangdaoQinhuangdao 066004 China2State Grid Ningxia Information amp Communication Company Great Wall East Road No 277 Xingqing DistrictNingxia 750000 China

Correspondence should be addressed to Xin Song sxin78916mailneuqeducn

Received 8 December 2014 Accepted 20 April 2015

Academic Editor Peter Jung

Copyright copy 2015 Xin Song et alThis is an open access article distributed under the Creative CommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The performance of traditional constrained-LMS (CLMS) algorithm is known to degrade seriously in the presence of small trainingdata size and mismatches between the assumed array response and the true array response In this paper we develop a robustconstrained-LMS (RCLMS) algorithm based on worst-case SINR maximization Our algorithm belongs to the class of diagonalloading techniques in which the diagonal loading factor is obtained in a simple form and it decreases the computation costThe updated weight vector is derived by the descent gradient method and Lagrange multiplier method It demonstrates that ourproposed recursive algorithm provides excellent robustness against signal steering vector mismatches and the small training datasize and has fast convergence rate and makes the mean output array signal-to-interference-plus-noise ratio (SINR) consistentlyclose to the optimal one Some simulation results are presented to compare the performance of our robust algorithm with thetraditional CLMS algorithm

1 Introduction

Adaptive beamforming is used for enhancing a desired signalwhile suppressing interference and noise at the output of anarray of sensors It has a long and rich history of practicalapplications to numerous areas such as sonar radar radioastronomy medical imaging and more recently wirelesscommunications [1ndash5]

In the practical applications the adaptive beamformingmethods become very sensitive to any violation of underlyingassumptions on the environment sources or sensor arrayThe performance of the existing adaptive array algorithmsis known to degrade substantially in the presence of evenslight mismatches between the actual and presumed arrayresponses to the desired signal Similar types of degradationcan take place when the signal array response is known pre-cisely but the training sample size is small Therefore robustapproaches to adaptive beamforming appear to be one of theimportant issues There are several efficient approaches to

design robust adaptive beamformers such as the linearly con-strainedminimum variance beamformer [6] the eigenspace-based beamformer [7] and the projection beamformingtechniques [8] For instance additional linear constraintson the array beam pattern have been proposed to betterattenuate the interference and broaden the response aroundthe nominal look direction [9]There is another popular classof robust beamforming techniques called diagonal loading(DL) [10] In these methods the array correlation matrixis loaded with an appropriate multiple called the loadinglevel of the identity matrix in order to satisfy the imposedquadratic constraint However it is somewhat difficult tocalculate the loading level with uncertain bounds of thearray steering vector which may not be available in practicalsituations Based on a spherical or ellipsoidal uncertaintyset of the array steering vectors robust Capon beamformingmaximizes the output power which belongs to the extendedclass of diagonal loading methods but the correspondingvalue of diagonal loading can be calculated precisely [11 12]

Hindawi Publishing CorporationJournal of Electrical and Computer EngineeringVolume 2015 Article ID 458521 8 pageshttpdxdoiorg1011552015458521

2 Journal of Electrical and Computer Engineering

From the above analysis we note that these methods cannotbe expected to provide sufficient robustness improvements

In more recent years some new robust adaptive beam-forming approaches have been proposed [13ndash17] The prob-lem of finding a weight vector maximizes the worst-caseSINR over the uncertainty model With a general convexuncertainty model the worst-case SINR maximization prob-lem can be solved by using convex optimization [13] In [14]a robust downlink beamforming optimization algorithm isproposed for secondary multicast transmission in a multiple-input multiple-output (MIMO) spectrum sharing cognitiveradio (CR) network Recognizing that all channel covariancematrices form a Riemannian manifold Ciochina et al pro-pose worst-case robust downlink beamforming on the Rie-mannian manifold in order to model the set of mismatchedchannel covariance matrices for which robustness shall beguaranteed [15] In [16] a robust beamforming scheme isproposed for the multiantenna nonregenerative cognitiverelay network where the multiantenna relay with imperfectchannel state information (CSI) helps the communication ofsingle-antenna secondary users (SUs) Exploiting imperfectchannel state information (CSI) with its error modeled byaddedGaussian noise robust beamforming in cognitive radiois developed which optimizes the beamforming weights atthe secondary transmitter [17]

Apart from the LMS-type algorithm another well-knowniterative adaptive algorithm is the recursive least square (RLS)algorithm or the complex-valued widely linear RLS algo-rithm which updates the weight vector with small steady-state misadjustment and fast convergence speed Howeverthe RLS-type algorithms have much higher computationcost than the LMS-type algorithms [18 19] In order toreduce the complexity RLS algorithm based on orthonormalpolynomial basis function is proposed which is as simpleas LMS algorithm [20] To yield low complexity cost andkeep fast convergence speed the LMS algorithms based onvariable step size have been presented in [21ndash23] These LMSalgorithms can have faster convergence speed and requireless computational cost per iteration than the RLS-typealgorithms However they cannot enjoy both fast trackingand small misadjustment with simple implementation It isknown that the performance of traditional CLMS algorithmdegrades seriously due to the small training sample sizeand signal steering vector mismatches In this paper inorder to overcome the drawbacks of CLMS algorithm wepropose a robust CLMS algorithm based on worst-case SINRmaximization which provides sufficient robustness againstsome types of mismatches The parameters in our paper canderive in a simple form which decreases computation costThe improved performance of the proposed algorithm isdemonstrated by comparing with traditional linearly CLMSalgorithm via several examples

2 Background

21Mathematical Formulation Weconsider a uniform lineararray (ULA) with119872 omnidirectional sensors spaced by thedistance119889We assume119863narrowband incoherent planewaves

that impinge from directions of arrival 1205790 1205791 120579

119863minus1 The

output of a designed beamformer is expressed as follows

119910 (119896) = w119867x (119896) (1)

where x(119896) = [1199091(119896) 119909

119872(119896)]119879 is the complex vector

of array observation 119872 is the array number and w =

[1199081 119908

119872]119879 is the complex vector of weights here (sdot)119867 and

(sdot)119879 are the Hermitian transpose and transpose respectively

The array observation at time 119896 can be written as

x (119896) = s (119896) + n (119896) + i (119896) = 1199040(119896) a + n (119896) + i (119896) (2)

where s(119896) n(119896) and i(119896) are the desired signal noiseand interference components respectively Here 119904

0(119896) is the

desired signal waveform and a is the signal steering vectorThe weights can be optimized from the following maxi-

mum of the signal-to-interference-plus-noise ratio (SINR)

SINR =1205902

119904

10038161003816100381610038161003816w119867a100381610038161003816100381610038162

w119867R119894+119899

w (3)

where 1205902119904is the signal power and119872 times 119872 interference-plus-

noise correlation matrix R119894+119899

R119894+119899

= 119864 (n (119896) + i (119896)) (n (119896) + i (119896))119867 (4)

22 Linearly Constrained-LMS (CLMS) Algorithm Linearconstrained-LMS algorithm is a real-time constrained algo-rithm for determining the optimal weight vector The prob-lem of finding optimum beamformer weights is as follows

minw w119867R119894+119899

w

subject to w119867a = 1(5)

Using Lagrange multiplier method to solve problem (5)the optimal weight vector can be derived

wopt =Rminus1119894+119899

aaHRminus1119894+119899

a (6)

In practical situations we cannot know completely thesignal characteristics and it is also time-varying circum-stance So we need to update the weights in an iterativemanner The Lagrange function of (5) is written as

119867(w 120582) = w119867R119894+119899

w + 120582 (w119867a minus 1) (7)

Computing the gradient of (7) we can update the weightvector of CLMS algorithm

w (119896 + 1) = w (119896) minus 1205831nabla

= w (119896) minus 1205831[2R119894+119899

w (119896) + 120582a] (8)

where 1205831is the step size and nabla is the gradient vector of

119867(w 120582) Inserting (8) into linear constraint w119867a = 1 we canobtain the Lagrange multiplier

120582 =1

1205831

a119867w (119896) minus 1a119867a

minus 2a119867

a119867aR119894+119899

w (119896) (9)

Journal of Electrical and Computer Engineering 3

According to (8) and (9) the weight vector of traditionalCLMS algorithm can be rewritten as [24]

w (119896 + 1) = P [w (119896) minus 1205831R119894+119899

w (119896)] +Ψ (10)

whereΨ = a[a119867a]minus1 and P = Ι minus a[a119867a]minus1a119867From (10) we note that the performance of the tradi-

tional CLMS algorithm is dependent on exact signal steeringvector and it is sensitive to some types of mismatches Inaddition the interference-plus-noise correlation matrix R

119894+119899

is unknown Hereby the sample covariance matrix

R119873=1

119873

119873

sum

119896=1

x (119896) x119867 (119896) (11)

is used to substitute R119894+119899

in (10) where 119873 is the train-ing sample size Therefore the performance degradation ofCLMS algorithm can occur due to the signal steering vectormismatches and small training sample size

3 Robust CLMS Algorithm Based onWorst-Case SINR Maximization

In order to solve the above-mentioned problems of thelinearly constrained-LMS algorithm we propose a robustrecursive algorithm based onworst-case SINRmaximizationwhich provides robustness against mismatches

We assume that in practical situations the mismatchvector e is norm-bounded by some known constant 120576 gt 0that is

e le 120576 (12)

Then the actual signal steering vector belongs to a ballset

120593 (120576) = c | c = a + e e le 120576 (13)

The weight vector is selected by minimizing the meanoutput power while maintaining a distortionless responsefor the mismatched steering vector So the cost function ofrobust constrained-LMS (RCLMS) algorithm is formulatedas

minw w119867R119873w

subject to mincisin120593(120576)

10038161003816100381610038161003816w119867c10038161003816100381610038161003816 ge 1

(14)

According to [25] the constraint in (14) is equivalent tothe following form

mincisin120593(120576)

10038161003816100381610038161003816w119867c10038161003816100381610038161003816 = 1 (15)

Using (15) problem (14) can be rewritten in the followingway

minw w119867R119873w

subject to 10038161003816100381610038161003816w119867a minus 110038161003816100381610038161003816

2

= 1205762w119867w

(16)

In order to improve the robustness against the mismatchthat may be caused by the small training sample size we canget a further extension of the optimization problem (16) Theactual covariance matrix is

R119889= R119873+ Δ (17)

where Δ is the norm of the error matrix and it is bounded bya certain constant 119903 Δ le 119903

Applying the worst-case performance optimization wecan rewrite

minw maxΔle119903

w119867 (R119873+ Δ)w

subject to 10038161003816100381610038161003816w119867a minus 110038161003816100381610038161003816

2

= 1205762w119867w

(18)

To obtain the optimal weight vector we can first solve thefollowing simpler problem [26]

minΔ

minus w119867 (R119873+ Δ)w

subject to Δ le 119903

(19)

Using Lagrange multiplier method to yield the matrixerror

Δ = 119903ww119867

w (20)

Consequently the minimization problem (18) is con-verted to the following form

minw maxΔle119903

w119867 (R119873+ 119903I)w

subject to 10038161003816100381610038161003816w119867a minus 110038161003816100381610038161003816

2

= 1205762w119867w

(21)

The solution to (21) can be derived by minimizing theLagrange function

119891 (w 120582)

= w119867R119897w

+ 120582 (1205762w119867w minus w119867aa119867w + w119867a + a119867w minus 1)

(22)

where R119897= R119873+ 119903I and 120582 is Lagrange multiplier Computing

the gradient vector of119867(w 120582) we can get the gradient nabla

nabla = (R119897+ 1205821205762Ι minus 120582aa119867)w + 120582a (23)

The gradient of119867(w 120582) is equal to zero andwe can obtainthe optimum weight vector

wopt = 120599 (120582) (R119897 + 1205821205762Ι)minus1

a (24)

where 120599(120582) = 120582(120582a119867(R119897+ 1205821205762Ι)minus1a minus 1)

From (24) we note that the proposed algorithm belongsto the class of diagonal loading techniques but the loadingfactor is calculated in a complicated way

4 Journal of Electrical and Computer Engineering

Using (23) the updated weight vector is obtained by

w (119896 + 1) = w (119896) minus 120583 [Bw (119896) + 120582a] (25)

where B = R119897+ 1205821205762Ι minus 120582aa119867 and 120583 is step size

Next we need to compute the Lagrange multiplier 120582 Thequadratic constraint of the optimization problem (21) is

ℎ (w) = 1 (26)

where

ℎ (w) = w119867 (119896) (1205762I minus aa119867)w (119896) + a119867w (119896)

+ w119867 (119896) a(27)

Inserting (25) into (26) we can obtain the Lagrangemultiplier 120582

120582 =1

21205832 [F119867 (119896) (1205762I minus aa119867) F (119896)](120577 (119896) + 120594 (119896)) (28)

where

F (119896) = (1205762I minus aa119867)w (119896) + a

q (119896) = [I minus 120583 (R119897 + 119903I)]w (119896)

120577 (119896) = 120583 [q119867 (119896) (1205762I minus aa119867) F (119896)

+ F119867 (119896) (1205762I minus aa119867)P (119896) + F119867 (119896) a + a119867F (119896)]

120594 (119896) = P119867 (119896) (1205762I minus aa119867) q (119896) + q119867 (119896) a

+ a119867q (119896)

120588119867(119896) 120588 (119896) = 120577

119867(119896) 120577 (119896) minus 4120583

2F119867 (119896) (1205762I minus aa119867)

sdot F (119896) (120594 (119896) minus 1)

(29)

31TheChoice of Step Size Theweight vector (25) is rewrittenas

w (119896 + 1) = [I minus 120583Β]w (119896) minus 120583120582a (30)

Let

B = ΣΛΣ119867 (31)

where Λ is a diagonal matrix in which the diagonal elementsof matrix 120591

1ge 1205912ge sdot sdot sdot ge 120591

119872are equal to the eigenvalues ofB

and the columns ofΣ contain the corresponding eigenvectorsWe can get the following equation via multiplying (30) byΣ119867

Σ119867w (119896 + 1) = [I minus 120583Λ]Σ119867w (119896) minus 120583120582Σ119867a (32)

As demonstrated in (32) if the proposed algorithmconverges it is required to satisfy the constrained condition

10038161003816100381610038161 minus 1205831205911198941003816100381610038161003816 lt 1 (33)

It follows from (33) that

0 lt 120583 lt2

120591max (34)

where 120591max is the maximum eigenvalue

120591max lt119872

sum

119894=1

120591119894= trace [B] (35)

In recursive algorithm the choice of the step size 120583 is veryimportant and it is varied with each new training snapshot[27 28]

Therefore we can obtain the optimal parameter 120583

120583 =1

5 trace [B] (36)

32TheApproximation of LagrangeMultiplier From (28) and(29) we note that the computation cost of weight vector isvery high Next we obtain the Lagrange multiplier by linearcombination to decrease the computation cost

From (24) the proposed beamformer belongs to theclass of diagonal loading techniques According to [29] weconsider a linear combination of R

119894+119899and R

119897

R119888= 120588R119894+119899+ 120592R119897 (37)

where the parameters 120588 gt 0 and 120592 gt 0 The initial value ofR119894+119899

is assumed to identify matrix I [30] We can rewrite (37)as

R119888= R119897+120588

120592Ι (38)

Contrasting the diagonal loading covariancematrices R119897+

1205821205762Ι in (24) and R

119897+(120588120592)Ι in (38) we note that the parameter

1205821205762 is replaced by 120588120592 In this way we can compute the

Lagrange multiplier 120582 as follows

120582 =120588

120592sdot1

1205762 (39)

From (39) the Lagrange multiplier is calculated sim-ply which decreases the complexity cost of the proposedalgorithm We need first to obtain the parameters 120588 and 120592Minimize the following function

119892 = min119864 1003817100381710038171003817R119888 minus R119909

1003817100381710038171003817

2 (40)

where R119909

= 119864[x(119896)x119867(119896)] is the theoretical covariancematrix By inserting (37) into (40) we have [31]

119892 = 119864 10038171003817100381710038171003817120592 (R119897minus R119909) + 120588Ι minus (1 minus 120592)R

119909

10038171003817100381710038171003817

2

= 120592211986410038171003817100381710038171003817R119897minus R119909

10038171003817100381710038171003817

2

minus 2120588 (1 minus 120592) tr (R119909)

+ (1 minus 120592)2 1003817100381710038171003817R119909

1003817100381710038171003817

2+ 1205882119872

(41)

Journal of Electrical and Computer Engineering 5

Computing the gradient of (41) with respect to 120588 for fixed120592 we can give the optimal value

1205880=(1 minus 120592

0) tr (R

119909)

119872 (42)

Inserting 1205880into (41) and replacing 120592

0by 120592 minimization

problem is written as

min1205921205922119864 10038171003817100381710038171003817R119897minus R119909

10038171003817100381710038171003817

2

+

(1 minus 120592)21003817100381710038171003817R119909

1003817100381710038171003817

2119872minus tr2 (R

119909)

119872

(43)

By minimizing (43) the optimal solution for 120592 is derivedas

1205920=

120572

120572 + 120590 (44)

where 120590 = 119864R119897minus R1199092 and 120572 = R

1199092minus tr2(R

119909)119872

In practical situations R119909is replaced by R

119897to obtain

estimation value

=10038171003817100381710038171003817R119897

10038171003817100381710038171003817

2

minus

tr2 (R119897)

119872 (45)

We can estimate the parameter 120590

=

119872

sum

119898=1

[1

1198732

119873

sum

119899=1

1003817100381710038171003817119903119898 minus x (119899) 119909lowast119898(119899)1003817100381710038171003817

2]

=1

1198732

119873

sum

119899=1

x (119899)4 minus 1

119873

10038171003817100381710038171003817R119897

10038171003817100381710038171003817

2

(46)

where 119909119898(119899) is the119898th element of x(119899)

Substituting (45) and (46) into (44) the estimation valueof 1205920is written as

1205920=

+ (47)

Inserting (45) and (46) into (42) we can obtain estima-tion value of 120588

0

1205880=

(1 minus 1205920) tr (R

119897)

119872 (48)

Consequently we can obtain the expression of Lagrangemultiplier

=1205880

1205920

sdot1

1205762 (49)

Our proposed RCLMS algorithm belongs to the classof diagonal loadings but the diagonal loading factor isderived fully automatically from the observation vectorswithout the need of specifying any user knowledge Theparameter 120576 is determined easily andwe can conclude that theproposed RCLMS algorithm is not sensitive to the choice of

Table 1 The complexity cost of the conventional constrained-LMS

Complexity costR119894+119899

119874(1198722times 119873 + 1)

Ψ 119874(2119872 + 1)

P 119874(21198722+ 1)

Pw(119896) 119874(1198722)

1205831PR119894+119899w(119896) 119874(119872

3+1198722+119872)

Total complexity cost 119874(1198723+ (119873 + 4)119872

2+ 3119872 + 3)

Table 2 The complexity cost of the proposed robust CLMS

Complexity cost 119874(119872

2+119872 times119873 +119873 + 12)

120583a 119874(119872 + 1)

R119897

119874(1198722times 119873 + 1)

1205821205762I 119874(119872

2+ 2)

120582aa119867 119874(21198722)

120583Bw(119896) 119874(1198722+119872)

Total complexity cost 119874((119873 + 5)1198722+ (119873 + 2)119872 +119873 + 16)

parameter 120576 in [28] From the literature [11] it is clear that themajor computational demand of the algorithm comes fromthe eigendecomposition which requires 119874(1198723) flops Thisleads to a high computational cost However the proposedalgorithm which does not need eigendecomposition canreduce the complexity to 119874(1198722) flops In addition robustCapon algorithm and our proposed algorithm belong to theclass of the diagonal loadings in which the loading factorscan be calculated precisely

33 The Analysis of Complexity Cost The complexity cost oftwo algorithms can be shown as in Tables 1 and 2

4 Simulation Results

In this section we present some simulations to justify the per-formance of the proposed robust recursive algorithm basedon worst-case SINR maximization We assume a uniformlinear array with 119872 = 10 omnidirectional sensors spacedhalf a wavelength apart For each scenario 200 simulationruns are used to obtain each simulation point Assume thatboth directions of arrival (DOAs) of the presumed and actualsignal are 0∘ and 3

∘ respectively This corresponds to a3∘ mismatch in the direction of arrival Suppose that twointerfering sources are planewaves impinging from theDOAsminus50∘ and 50∘ respectively We choose the parameter 119903 = 9

Example 1 (output SINR versus the number of snapshots)We assume that signal-to-noise ratio SNR = 10 dB and theparameter 120576 = 3 Figure 1 shows the performance of themethods tested in no mismatch case From Figure 1 we seethat the output SINR of traditional CLMS is about 14 dB Notethat it is sensitive to the small training sample size Howeverour proposed algorithm can provide improved robustness

6 Journal of Electrical and Computer Engineering

0 10 20 30 40 50 60 70 80 90 1000

5

10

15

20

25

Number of snapshots

Out

put S

INR

(dB)

CLMS beamformerOptimal SINRRCLMS beamformer

Figure 1 Output SINR in no-mismatch case

0 10 20 30 40 50 60 70 80 90 100minus15

minus10

minus5

0

5

10

15

20

25

Number of snapshots

Out

put S

INR

(dB)

CLMS beamformerOptimal SINRRCLMS beamformer

Figure 2 Output SINR in a 3∘ mismatch case

Figure 2 shows the array output SINR of the methods testedin a 3∘ mismatch

In Figure 2 we note that the output SINR of CLMS algo-rithm is about minus10 dB which is sensitive to the signal steeringvector mismatches However that of the proposed robustCLMS (RCLMS) algorithm is about 18 dB which is close tothe optimal one In this scenario the proposed recursive algo-rithm outperforms the traditional linear constrained-LMSalgorithm Moreover robust constrained-LMS algorithm hasfaster convergence rate

Example 2 (output SINR versus SNR) In this example thereis a 3∘ mismatch in the signal look direction We assume thatthe fixed training data size119873 is equal to 100 Figure 3 displays

minus10 minus8 minus6 minus4 minus2 0 2 4 6 8 10minus5

0

5

10

15

20

SNR (dB)

Out

put S

INR

(dB)

Optimal SINRCLMS beamformerRCLMS beamformer

Figure 3 Output SINR in no-mismatch case

minus10 minus8 minus6 minus4 minus2 0 2 4 6 8 10minus15

minus10

minus5

0

5

10

15

20

SNR (dB)

Out

put S

INR

(dB)

Optimal SINRCLMS beamformerRCLMS beamformer

Figure 4 Output SINR in a 3∘ mismatch case

the performance of these algorithms versus the SNR in no-mismatch case The performance of these algorithms versusthe SNR in a 3∘ mismatch case is shown in Figure 4

In this example the traditional algorithm is very sensitiveeven to slight mismatches which can easily occur in practicalapplications It is observed from Figure 4 that with theincrease of SNR CLMS algorithm has poor performance atall values of the SNRHowever our proposed recursive robustalgorithm provides improved robustness against signal steer-ing vector mismatches and small training sample size hasfaster convergence rate and yields better output performancethan the CLMS algorithm

Journal of Electrical and Computer Engineering 7

5 Conclusions

In this paper we propose a robust constrained-LMS algo-rithm based on the worst-case SINR maximization TheRCLMS algorithm provides robustness against some types ofmismatches and offers faster convergence rate The updatedweight vector is derived by the gradient descent methodand Lagrange multiplier method in which the diagonalloading factor is obtained in a simple formThis decreases thecomputation cost Some simulation results demonstrate thatthe proposed robust recursive algorithm enjoys better perfor-mance as compared with the traditional CLMS algorithm

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the anonymous reviewersfor their insightful comments that helped improve the qualityof this paper This work is supported by Program for NewCentury Excellent Talents in University (no NCET-12-0103)by the National Natural Science Foundation of China underGrant no 61473066 by the Fundamental Research Funds forthe Central Universities under Grant no N130423005 andby the Natural Science Foundation of Hebei Province underGrant no F2012501044

References

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[2] R T Compton Jr R J Huff W G Swarner and A KsienskildquoAdaptive arrays for communication system an overview of theresearch at the Ohio State Universityrdquo IEEE Transactions onAntennas and Propagation vol AP-24 no 5 pp 599ndash607 1976

[3] A B Gershman E Nemeth and J F Bohme ldquoExperimentalperformance of adaptive beamforming in a sonar environmentwith a towed array and moving interfering sourcesrdquo IEEETransactions on Signal Processing vol 48 no 1 pp 246ndash2502000

[4] Y Kaneda and J Ohga ldquoAdaptive microphone-array system fornoise reductionrdquo IEEE Transactions on Acoustics Speech andSignal Processing vol 34 no 6 pp 1391ndash1400 1986

[5] J Li and P Stoica Robust Adaptive Beamforming John Wiley ampSons New York NY USA 2005

[6] R A Monzingo and T W Miller Introduction to AdaptiveArrays Wiley New York NY USA 1980

[7] 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

[8] D D Feldman and L J Griffiths ldquoProjection approach forrobust adaptive beamformingrdquo IEEE Transactions on SignalProcessing vol 42 no 4 pp 867ndash876 1994

[9] Q Zou Z L Yu and Z Lin ldquoA robust algorithm for linearlyconstrained adaptive beamformingrdquo IEEE Signal ProcessingLetters vol 11 no 1 pp 26ndash29 2004

[10] B D Carlson ldquoCovariance matrix estimation errors and diago-nal loading in adaptive arraysrdquo IEEE Transactions on Aerospaceand Electronic Systems vol 24 no 4 pp 397ndash401 1988

[11] J Li P Stoica and Z Wang ldquoOn robust Capon beamformingand diagonal loadingrdquo IEEE Transactions on Signal Processingvol 51 no 7 pp 1702ndash1715 2003

[12] J Li P Stoica and ZWang ldquoDoubly constrained robust Caponbeamformerrdquo IEEE Transactions on Signal Processing vol 52no 9 pp 2407ndash2423 2004

[13] S-J Kim A Magnani A Mutapcic S P Boyd and Z-QLuo ldquoRobust beamforming viaworst-case SINRmaximizationrdquoIEEE Transactions on Signal Processing vol 56 no 4 pp 1539ndash1547 2008

[14] Y W Huang Q Li W-K Ma and S Z Zhang ldquoRobustmulticast beamforming for spectrum sharing-based cognitiveradiosrdquo IEEE Transactions on Signal Processing vol 60 no 1pp 527ndash533 2012

[15] D Ciochina M Pesavento and K M Wong ldquoWorst caserobust downlink beamforming on the Riemannian manifoldrdquoin Proceedings of the 38th IEEE International Conference onAcoustics Speech and Signal Processing (ICASSP rsquo13) pp 3801ndash3805 May 2013

[16] Q Li Q Zhang and J Qin ldquoRobust beamforming for cognitivemulti-antenna relay networkswith bounded channel uncertain-tiesrdquo IEEE Transactions on Communications vol 62 no 2 pp478ndash487 2014

[17] G Zheng S Ma K-K Wong and T-S Ng ldquoRobust beam-forming in cognitive radiordquo IEEE Transactions on WirelessCommunications vol 9 no 2 pp 570ndash576 2010

[18] S C Douglas ldquoWidely-linear recursive least-squares algorithmfor adaptive beamformingrdquo in Proceedings of the IEEE Inter-national Conference on Acoustics Speech and Signal Processing(ICASSP rsquo09) pp 2041ndash2044 IEEE Taipei Taiwan April 2009

[19] E Eweda ldquoComparison of RLS LMS and sign algorithms fortracking randomly time-varying channelsrdquo IEEE Transactionson Signal Processing vol 42 no 11 pp 2937ndash2944 1994

[20] S J Yao H Qian K Kang and M Y Shen ldquoA recursiveleast squares algorithm with reduced complexity for digitalpredistortion linearizationrdquo in Proceedings of the 38th IEEEInternational Conference on Acoustics Speech and Signal Pro-cessing (ICASSP rsquo13) pp 4736ndash4739 May 2013

[21] J Apolinario Jr M L R Campos and P S R DinizldquoConvergence analysis of the binormalized data-reusing LMSalgorithmrdquo IEEE Transactions on Signal Processing vol 48 no11 pp 3235ndash3242 2000

[22] Y-M Shi L Huang C Qian and H C So ldquoShrinkagelinear and widely linear complex-valued least mean squaresalgorithms for adaptive beamformingrdquo IEEE Transactions onSignal Processing vol 63 no 1 pp 119ndash131 2015

[23] T Aboulnasr and K Mayyas ldquoA robust variable step-size LMS-type algorithm analysis and simulationsrdquo IEEE Transactions onSignal Processing vol 45 no 3 pp 631ndash639 1997

[24] L C Godara ldquoApplication of antenna arrays to mobile com-munications part II beam-forming and direction-of-arrivalconsiderationsrdquo Proceedings of the IEEE vol 85 no 8 pp 1195ndash1245 1997

[25] 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

8 Journal of Electrical and Computer Engineering

[26] S Shahbazpanahi A B Gershman Z-Q Lou and KMWongldquoRobust adaptive beamforming for general-rank signalmodelsrdquoIEEE Transactions on Signal Processing vol 51 no 9 pp 2257ndash2269 2003

[27] A Elnashar ldquoEfficient implementation of robust adaptive beam-forming based on worst-case performance optimisationrdquo IETSignal Processing vol 2 no 4 pp 381ndash393 2008

[28] 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

[29] P Stoica J Li X Zhu and J RGuerci ldquoOnusing a priori knowl-edge in space-time adaptive processingrdquo IEEE Transactions onSignal Processing vol 56 no 6 pp 2598ndash2602 2008

[30] O Ledoit and M Wolf ldquoA well-conditioned estimator forlarge-dimensional covariance matricesrdquo Journal of MultivariateAnalysis vol 88 no 2 pp 365ndash411 2004

[31] J Li L Du and P Stoica ldquoFully automatic computation ofdiagonal loading levels for robust adaptive beamformingrdquo inProceedings of the IEEE International Conference on AcousticsSpeech and Signal Processing (ICASSP 08) pp 2325ndash2328 IEEELas Vegas Nev USA April 2008

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DistributedSensor Networks

International Journal of

2 Journal of Electrical and Computer Engineering

From the above analysis we note that these methods cannotbe expected to provide sufficient robustness improvements

In more recent years some new robust adaptive beam-forming approaches have been proposed [13ndash17] The prob-lem of finding a weight vector maximizes the worst-caseSINR over the uncertainty model With a general convexuncertainty model the worst-case SINR maximization prob-lem can be solved by using convex optimization [13] In [14]a robust downlink beamforming optimization algorithm isproposed for secondary multicast transmission in a multiple-input multiple-output (MIMO) spectrum sharing cognitiveradio (CR) network Recognizing that all channel covariancematrices form a Riemannian manifold Ciochina et al pro-pose worst-case robust downlink beamforming on the Rie-mannian manifold in order to model the set of mismatchedchannel covariance matrices for which robustness shall beguaranteed [15] In [16] a robust beamforming scheme isproposed for the multiantenna nonregenerative cognitiverelay network where the multiantenna relay with imperfectchannel state information (CSI) helps the communication ofsingle-antenna secondary users (SUs) Exploiting imperfectchannel state information (CSI) with its error modeled byaddedGaussian noise robust beamforming in cognitive radiois developed which optimizes the beamforming weights atthe secondary transmitter [17]

Apart from the LMS-type algorithm another well-knowniterative adaptive algorithm is the recursive least square (RLS)algorithm or the complex-valued widely linear RLS algo-rithm which updates the weight vector with small steady-state misadjustment and fast convergence speed Howeverthe RLS-type algorithms have much higher computationcost than the LMS-type algorithms [18 19] In order toreduce the complexity RLS algorithm based on orthonormalpolynomial basis function is proposed which is as simpleas LMS algorithm [20] To yield low complexity cost andkeep fast convergence speed the LMS algorithms based onvariable step size have been presented in [21ndash23] These LMSalgorithms can have faster convergence speed and requireless computational cost per iteration than the RLS-typealgorithms However they cannot enjoy both fast trackingand small misadjustment with simple implementation It isknown that the performance of traditional CLMS algorithmdegrades seriously due to the small training sample sizeand signal steering vector mismatches In this paper inorder to overcome the drawbacks of CLMS algorithm wepropose a robust CLMS algorithm based on worst-case SINRmaximization which provides sufficient robustness againstsome types of mismatches The parameters in our paper canderive in a simple form which decreases computation costThe improved performance of the proposed algorithm isdemonstrated by comparing with traditional linearly CLMSalgorithm via several examples

2 Background

21Mathematical Formulation Weconsider a uniform lineararray (ULA) with119872 omnidirectional sensors spaced by thedistance119889We assume119863narrowband incoherent planewaves

that impinge from directions of arrival 1205790 1205791 120579

119863minus1 The

output of a designed beamformer is expressed as follows

119910 (119896) = w119867x (119896) (1)

where x(119896) = [1199091(119896) 119909

119872(119896)]119879 is the complex vector

of array observation 119872 is the array number and w =

[1199081 119908

119872]119879 is the complex vector of weights here (sdot)119867 and

(sdot)119879 are the Hermitian transpose and transpose respectively

The array observation at time 119896 can be written as

x (119896) = s (119896) + n (119896) + i (119896) = 1199040(119896) a + n (119896) + i (119896) (2)

where s(119896) n(119896) and i(119896) are the desired signal noiseand interference components respectively Here 119904

0(119896) is the

desired signal waveform and a is the signal steering vectorThe weights can be optimized from the following maxi-

mum of the signal-to-interference-plus-noise ratio (SINR)

SINR =1205902

119904

10038161003816100381610038161003816w119867a100381610038161003816100381610038162

w119867R119894+119899

w (3)

where 1205902119904is the signal power and119872 times 119872 interference-plus-

noise correlation matrix R119894+119899

R119894+119899

= 119864 (n (119896) + i (119896)) (n (119896) + i (119896))119867 (4)

22 Linearly Constrained-LMS (CLMS) Algorithm Linearconstrained-LMS algorithm is a real-time constrained algo-rithm for determining the optimal weight vector The prob-lem of finding optimum beamformer weights is as follows

minw w119867R119894+119899

w

subject to w119867a = 1(5)

Using Lagrange multiplier method to solve problem (5)the optimal weight vector can be derived

wopt =Rminus1119894+119899

aaHRminus1119894+119899

a (6)

In practical situations we cannot know completely thesignal characteristics and it is also time-varying circum-stance So we need to update the weights in an iterativemanner The Lagrange function of (5) is written as

119867(w 120582) = w119867R119894+119899

w + 120582 (w119867a minus 1) (7)

Computing the gradient of (7) we can update the weightvector of CLMS algorithm

w (119896 + 1) = w (119896) minus 1205831nabla

= w (119896) minus 1205831[2R119894+119899

w (119896) + 120582a] (8)

where 1205831is the step size and nabla is the gradient vector of

119867(w 120582) Inserting (8) into linear constraint w119867a = 1 we canobtain the Lagrange multiplier

120582 =1

1205831

a119867w (119896) minus 1a119867a

minus 2a119867

a119867aR119894+119899

w (119896) (9)

Journal of Electrical and Computer Engineering 3

According to (8) and (9) the weight vector of traditionalCLMS algorithm can be rewritten as [24]

w (119896 + 1) = P [w (119896) minus 1205831R119894+119899

w (119896)] +Ψ (10)

whereΨ = a[a119867a]minus1 and P = Ι minus a[a119867a]minus1a119867From (10) we note that the performance of the tradi-

tional CLMS algorithm is dependent on exact signal steeringvector and it is sensitive to some types of mismatches Inaddition the interference-plus-noise correlation matrix R

119894+119899

is unknown Hereby the sample covariance matrix

R119873=1

119873

119873

sum

119896=1

x (119896) x119867 (119896) (11)

is used to substitute R119894+119899

in (10) where 119873 is the train-ing sample size Therefore the performance degradation ofCLMS algorithm can occur due to the signal steering vectormismatches and small training sample size

3 Robust CLMS Algorithm Based onWorst-Case SINR Maximization

In order to solve the above-mentioned problems of thelinearly constrained-LMS algorithm we propose a robustrecursive algorithm based onworst-case SINRmaximizationwhich provides robustness against mismatches

We assume that in practical situations the mismatchvector e is norm-bounded by some known constant 120576 gt 0that is

e le 120576 (12)

Then the actual signal steering vector belongs to a ballset

120593 (120576) = c | c = a + e e le 120576 (13)

The weight vector is selected by minimizing the meanoutput power while maintaining a distortionless responsefor the mismatched steering vector So the cost function ofrobust constrained-LMS (RCLMS) algorithm is formulatedas

minw w119867R119873w

subject to mincisin120593(120576)

10038161003816100381610038161003816w119867c10038161003816100381610038161003816 ge 1

(14)

According to [25] the constraint in (14) is equivalent tothe following form

mincisin120593(120576)

10038161003816100381610038161003816w119867c10038161003816100381610038161003816 = 1 (15)

Using (15) problem (14) can be rewritten in the followingway

minw w119867R119873w

subject to 10038161003816100381610038161003816w119867a minus 110038161003816100381610038161003816

2

= 1205762w119867w

(16)

In order to improve the robustness against the mismatchthat may be caused by the small training sample size we canget a further extension of the optimization problem (16) Theactual covariance matrix is

R119889= R119873+ Δ (17)

where Δ is the norm of the error matrix and it is bounded bya certain constant 119903 Δ le 119903

Applying the worst-case performance optimization wecan rewrite

minw maxΔle119903

w119867 (R119873+ Δ)w

subject to 10038161003816100381610038161003816w119867a minus 110038161003816100381610038161003816

2

= 1205762w119867w

(18)

To obtain the optimal weight vector we can first solve thefollowing simpler problem [26]

minΔ

minus w119867 (R119873+ Δ)w

subject to Δ le 119903

(19)

Using Lagrange multiplier method to yield the matrixerror

Δ = 119903ww119867

w (20)

Consequently the minimization problem (18) is con-verted to the following form

minw maxΔle119903

w119867 (R119873+ 119903I)w

subject to 10038161003816100381610038161003816w119867a minus 110038161003816100381610038161003816

2

= 1205762w119867w

(21)

The solution to (21) can be derived by minimizing theLagrange function

119891 (w 120582)

= w119867R119897w

+ 120582 (1205762w119867w minus w119867aa119867w + w119867a + a119867w minus 1)

(22)

where R119897= R119873+ 119903I and 120582 is Lagrange multiplier Computing

the gradient vector of119867(w 120582) we can get the gradient nabla

nabla = (R119897+ 1205821205762Ι minus 120582aa119867)w + 120582a (23)

The gradient of119867(w 120582) is equal to zero andwe can obtainthe optimum weight vector

wopt = 120599 (120582) (R119897 + 1205821205762Ι)minus1

a (24)

where 120599(120582) = 120582(120582a119867(R119897+ 1205821205762Ι)minus1a minus 1)

From (24) we note that the proposed algorithm belongsto the class of diagonal loading techniques but the loadingfactor is calculated in a complicated way

4 Journal of Electrical and Computer Engineering

Using (23) the updated weight vector is obtained by

w (119896 + 1) = w (119896) minus 120583 [Bw (119896) + 120582a] (25)

where B = R119897+ 1205821205762Ι minus 120582aa119867 and 120583 is step size

Next we need to compute the Lagrange multiplier 120582 Thequadratic constraint of the optimization problem (21) is

ℎ (w) = 1 (26)

where

ℎ (w) = w119867 (119896) (1205762I minus aa119867)w (119896) + a119867w (119896)

+ w119867 (119896) a(27)

Inserting (25) into (26) we can obtain the Lagrangemultiplier 120582

120582 =1

21205832 [F119867 (119896) (1205762I minus aa119867) F (119896)](120577 (119896) + 120594 (119896)) (28)

where

F (119896) = (1205762I minus aa119867)w (119896) + a

q (119896) = [I minus 120583 (R119897 + 119903I)]w (119896)

120577 (119896) = 120583 [q119867 (119896) (1205762I minus aa119867) F (119896)

+ F119867 (119896) (1205762I minus aa119867)P (119896) + F119867 (119896) a + a119867F (119896)]

120594 (119896) = P119867 (119896) (1205762I minus aa119867) q (119896) + q119867 (119896) a

+ a119867q (119896)

120588119867(119896) 120588 (119896) = 120577

119867(119896) 120577 (119896) minus 4120583

2F119867 (119896) (1205762I minus aa119867)

sdot F (119896) (120594 (119896) minus 1)

(29)

31TheChoice of Step Size Theweight vector (25) is rewrittenas

w (119896 + 1) = [I minus 120583Β]w (119896) minus 120583120582a (30)

Let

B = ΣΛΣ119867 (31)

where Λ is a diagonal matrix in which the diagonal elementsof matrix 120591

1ge 1205912ge sdot sdot sdot ge 120591

119872are equal to the eigenvalues ofB

and the columns ofΣ contain the corresponding eigenvectorsWe can get the following equation via multiplying (30) byΣ119867

Σ119867w (119896 + 1) = [I minus 120583Λ]Σ119867w (119896) minus 120583120582Σ119867a (32)

As demonstrated in (32) if the proposed algorithmconverges it is required to satisfy the constrained condition

10038161003816100381610038161 minus 1205831205911198941003816100381610038161003816 lt 1 (33)

It follows from (33) that

0 lt 120583 lt2

120591max (34)

where 120591max is the maximum eigenvalue

120591max lt119872

sum

119894=1

120591119894= trace [B] (35)

In recursive algorithm the choice of the step size 120583 is veryimportant and it is varied with each new training snapshot[27 28]

Therefore we can obtain the optimal parameter 120583

120583 =1

5 trace [B] (36)

32TheApproximation of LagrangeMultiplier From (28) and(29) we note that the computation cost of weight vector isvery high Next we obtain the Lagrange multiplier by linearcombination to decrease the computation cost

From (24) the proposed beamformer belongs to theclass of diagonal loading techniques According to [29] weconsider a linear combination of R

119894+119899and R

119897

R119888= 120588R119894+119899+ 120592R119897 (37)

where the parameters 120588 gt 0 and 120592 gt 0 The initial value ofR119894+119899

is assumed to identify matrix I [30] We can rewrite (37)as

R119888= R119897+120588

120592Ι (38)

Contrasting the diagonal loading covariancematrices R119897+

1205821205762Ι in (24) and R

119897+(120588120592)Ι in (38) we note that the parameter

1205821205762 is replaced by 120588120592 In this way we can compute the

Lagrange multiplier 120582 as follows

120582 =120588

120592sdot1

1205762 (39)

From (39) the Lagrange multiplier is calculated sim-ply which decreases the complexity cost of the proposedalgorithm We need first to obtain the parameters 120588 and 120592Minimize the following function

119892 = min119864 1003817100381710038171003817R119888 minus R119909

1003817100381710038171003817

2 (40)

where R119909

= 119864[x(119896)x119867(119896)] is the theoretical covariancematrix By inserting (37) into (40) we have [31]

119892 = 119864 10038171003817100381710038171003817120592 (R119897minus R119909) + 120588Ι minus (1 minus 120592)R

119909

10038171003817100381710038171003817

2

= 120592211986410038171003817100381710038171003817R119897minus R119909

10038171003817100381710038171003817

2

minus 2120588 (1 minus 120592) tr (R119909)

+ (1 minus 120592)2 1003817100381710038171003817R119909

1003817100381710038171003817

2+ 1205882119872

(41)

Journal of Electrical and Computer Engineering 5

Computing the gradient of (41) with respect to 120588 for fixed120592 we can give the optimal value

1205880=(1 minus 120592

0) tr (R

119909)

119872 (42)

Inserting 1205880into (41) and replacing 120592

0by 120592 minimization

problem is written as

min1205921205922119864 10038171003817100381710038171003817R119897minus R119909

10038171003817100381710038171003817

2

+

(1 minus 120592)21003817100381710038171003817R119909

1003817100381710038171003817

2119872minus tr2 (R

119909)

119872

(43)

By minimizing (43) the optimal solution for 120592 is derivedas

1205920=

120572

120572 + 120590 (44)

where 120590 = 119864R119897minus R1199092 and 120572 = R

1199092minus tr2(R

119909)119872

In practical situations R119909is replaced by R

119897to obtain

estimation value

=10038171003817100381710038171003817R119897

10038171003817100381710038171003817

2

minus

tr2 (R119897)

119872 (45)

We can estimate the parameter 120590

=

119872

sum

119898=1

[1

1198732

119873

sum

119899=1

1003817100381710038171003817119903119898 minus x (119899) 119909lowast119898(119899)1003817100381710038171003817

2]

=1

1198732

119873

sum

119899=1

x (119899)4 minus 1

119873

10038171003817100381710038171003817R119897

10038171003817100381710038171003817

2

(46)

where 119909119898(119899) is the119898th element of x(119899)

Substituting (45) and (46) into (44) the estimation valueof 1205920is written as

1205920=

+ (47)

Inserting (45) and (46) into (42) we can obtain estima-tion value of 120588

0

1205880=

(1 minus 1205920) tr (R

119897)

119872 (48)

Consequently we can obtain the expression of Lagrangemultiplier

=1205880

1205920

sdot1

1205762 (49)

Our proposed RCLMS algorithm belongs to the classof diagonal loadings but the diagonal loading factor isderived fully automatically from the observation vectorswithout the need of specifying any user knowledge Theparameter 120576 is determined easily andwe can conclude that theproposed RCLMS algorithm is not sensitive to the choice of

Table 1 The complexity cost of the conventional constrained-LMS

Complexity costR119894+119899

119874(1198722times 119873 + 1)

Ψ 119874(2119872 + 1)

P 119874(21198722+ 1)

Pw(119896) 119874(1198722)

1205831PR119894+119899w(119896) 119874(119872

3+1198722+119872)

Total complexity cost 119874(1198723+ (119873 + 4)119872

2+ 3119872 + 3)

Table 2 The complexity cost of the proposed robust CLMS

Complexity cost 119874(119872

2+119872 times119873 +119873 + 12)

120583a 119874(119872 + 1)

R119897

119874(1198722times 119873 + 1)

1205821205762I 119874(119872

2+ 2)

120582aa119867 119874(21198722)

120583Bw(119896) 119874(1198722+119872)

Total complexity cost 119874((119873 + 5)1198722+ (119873 + 2)119872 +119873 + 16)

parameter 120576 in [28] From the literature [11] it is clear that themajor computational demand of the algorithm comes fromthe eigendecomposition which requires 119874(1198723) flops Thisleads to a high computational cost However the proposedalgorithm which does not need eigendecomposition canreduce the complexity to 119874(1198722) flops In addition robustCapon algorithm and our proposed algorithm belong to theclass of the diagonal loadings in which the loading factorscan be calculated precisely

33 The Analysis of Complexity Cost The complexity cost oftwo algorithms can be shown as in Tables 1 and 2

4 Simulation Results

In this section we present some simulations to justify the per-formance of the proposed robust recursive algorithm basedon worst-case SINR maximization We assume a uniformlinear array with 119872 = 10 omnidirectional sensors spacedhalf a wavelength apart For each scenario 200 simulationruns are used to obtain each simulation point Assume thatboth directions of arrival (DOAs) of the presumed and actualsignal are 0∘ and 3

∘ respectively This corresponds to a3∘ mismatch in the direction of arrival Suppose that twointerfering sources are planewaves impinging from theDOAsminus50∘ and 50∘ respectively We choose the parameter 119903 = 9

Example 1 (output SINR versus the number of snapshots)We assume that signal-to-noise ratio SNR = 10 dB and theparameter 120576 = 3 Figure 1 shows the performance of themethods tested in no mismatch case From Figure 1 we seethat the output SINR of traditional CLMS is about 14 dB Notethat it is sensitive to the small training sample size Howeverour proposed algorithm can provide improved robustness

6 Journal of Electrical and Computer Engineering

0 10 20 30 40 50 60 70 80 90 1000

5

10

15

20

25

Number of snapshots

Out

put S

INR

(dB)

CLMS beamformerOptimal SINRRCLMS beamformer

Figure 1 Output SINR in no-mismatch case

0 10 20 30 40 50 60 70 80 90 100minus15

minus10

minus5

0

5

10

15

20

25

Number of snapshots

Out

put S

INR

(dB)

CLMS beamformerOptimal SINRRCLMS beamformer

Figure 2 Output SINR in a 3∘ mismatch case

Figure 2 shows the array output SINR of the methods testedin a 3∘ mismatch

In Figure 2 we note that the output SINR of CLMS algo-rithm is about minus10 dB which is sensitive to the signal steeringvector mismatches However that of the proposed robustCLMS (RCLMS) algorithm is about 18 dB which is close tothe optimal one In this scenario the proposed recursive algo-rithm outperforms the traditional linear constrained-LMSalgorithm Moreover robust constrained-LMS algorithm hasfaster convergence rate

Example 2 (output SINR versus SNR) In this example thereis a 3∘ mismatch in the signal look direction We assume thatthe fixed training data size119873 is equal to 100 Figure 3 displays

minus10 minus8 minus6 minus4 minus2 0 2 4 6 8 10minus5

0

5

10

15

20

SNR (dB)

Out

put S

INR

(dB)

Optimal SINRCLMS beamformerRCLMS beamformer

Figure 3 Output SINR in no-mismatch case

minus10 minus8 minus6 minus4 minus2 0 2 4 6 8 10minus15

minus10

minus5

0

5

10

15

20

SNR (dB)

Out

put S

INR

(dB)

Optimal SINRCLMS beamformerRCLMS beamformer

Figure 4 Output SINR in a 3∘ mismatch case

the performance of these algorithms versus the SNR in no-mismatch case The performance of these algorithms versusthe SNR in a 3∘ mismatch case is shown in Figure 4

In this example the traditional algorithm is very sensitiveeven to slight mismatches which can easily occur in practicalapplications It is observed from Figure 4 that with theincrease of SNR CLMS algorithm has poor performance atall values of the SNRHowever our proposed recursive robustalgorithm provides improved robustness against signal steer-ing vector mismatches and small training sample size hasfaster convergence rate and yields better output performancethan the CLMS algorithm

Journal of Electrical and Computer Engineering 7

5 Conclusions

In this paper we propose a robust constrained-LMS algo-rithm based on the worst-case SINR maximization TheRCLMS algorithm provides robustness against some types ofmismatches and offers faster convergence rate The updatedweight vector is derived by the gradient descent methodand Lagrange multiplier method in which the diagonalloading factor is obtained in a simple formThis decreases thecomputation cost Some simulation results demonstrate thatthe proposed robust recursive algorithm enjoys better perfor-mance as compared with the traditional CLMS algorithm

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the anonymous reviewersfor their insightful comments that helped improve the qualityof this paper This work is supported by Program for NewCentury Excellent Talents in University (no NCET-12-0103)by the National Natural Science Foundation of China underGrant no 61473066 by the Fundamental Research Funds forthe Central Universities under Grant no N130423005 andby the Natural Science Foundation of Hebei Province underGrant no F2012501044

References

[1] T S Rapapport Ed Smart Antennas Adaptive Arrays Algo-rithms and Wireless Position Location IEEE Piscataway NJUSA 1998

[2] R T Compton Jr R J Huff W G Swarner and A KsienskildquoAdaptive arrays for communication system an overview of theresearch at the Ohio State Universityrdquo IEEE Transactions onAntennas and Propagation vol AP-24 no 5 pp 599ndash607 1976

[3] A B Gershman E Nemeth and J F Bohme ldquoExperimentalperformance of adaptive beamforming in a sonar environmentwith a towed array and moving interfering sourcesrdquo IEEETransactions on Signal Processing vol 48 no 1 pp 246ndash2502000

[4] Y Kaneda and J Ohga ldquoAdaptive microphone-array system fornoise reductionrdquo IEEE Transactions on Acoustics Speech andSignal Processing vol 34 no 6 pp 1391ndash1400 1986

[5] J Li and P Stoica Robust Adaptive Beamforming John Wiley ampSons New York NY USA 2005

[6] R A Monzingo and T W Miller Introduction to AdaptiveArrays Wiley New York NY USA 1980

[7] 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

[8] D D Feldman and L J Griffiths ldquoProjection approach forrobust adaptive beamformingrdquo IEEE Transactions on SignalProcessing vol 42 no 4 pp 867ndash876 1994

[9] Q Zou Z L Yu and Z Lin ldquoA robust algorithm for linearlyconstrained adaptive beamformingrdquo IEEE Signal ProcessingLetters vol 11 no 1 pp 26ndash29 2004

[10] B D Carlson ldquoCovariance matrix estimation errors and diago-nal loading in adaptive arraysrdquo IEEE Transactions on Aerospaceand Electronic Systems vol 24 no 4 pp 397ndash401 1988

[11] J Li P Stoica and Z Wang ldquoOn robust Capon beamformingand diagonal loadingrdquo IEEE Transactions on Signal Processingvol 51 no 7 pp 1702ndash1715 2003

[12] J Li P Stoica and ZWang ldquoDoubly constrained robust Caponbeamformerrdquo IEEE Transactions on Signal Processing vol 52no 9 pp 2407ndash2423 2004

[13] S-J Kim A Magnani A Mutapcic S P Boyd and Z-QLuo ldquoRobust beamforming viaworst-case SINRmaximizationrdquoIEEE Transactions on Signal Processing vol 56 no 4 pp 1539ndash1547 2008

[14] Y W Huang Q Li W-K Ma and S Z Zhang ldquoRobustmulticast beamforming for spectrum sharing-based cognitiveradiosrdquo IEEE Transactions on Signal Processing vol 60 no 1pp 527ndash533 2012

[15] D Ciochina M Pesavento and K M Wong ldquoWorst caserobust downlink beamforming on the Riemannian manifoldrdquoin Proceedings of the 38th IEEE International Conference onAcoustics Speech and Signal Processing (ICASSP rsquo13) pp 3801ndash3805 May 2013

[16] Q Li Q Zhang and J Qin ldquoRobust beamforming for cognitivemulti-antenna relay networkswith bounded channel uncertain-tiesrdquo IEEE Transactions on Communications vol 62 no 2 pp478ndash487 2014

[17] G Zheng S Ma K-K Wong and T-S Ng ldquoRobust beam-forming in cognitive radiordquo IEEE Transactions on WirelessCommunications vol 9 no 2 pp 570ndash576 2010

[18] S C Douglas ldquoWidely-linear recursive least-squares algorithmfor adaptive beamformingrdquo in Proceedings of the IEEE Inter-national Conference on Acoustics Speech and Signal Processing(ICASSP rsquo09) pp 2041ndash2044 IEEE Taipei Taiwan April 2009

[19] E Eweda ldquoComparison of RLS LMS and sign algorithms fortracking randomly time-varying channelsrdquo IEEE Transactionson Signal Processing vol 42 no 11 pp 2937ndash2944 1994

[20] S J Yao H Qian K Kang and M Y Shen ldquoA recursiveleast squares algorithm with reduced complexity for digitalpredistortion linearizationrdquo in Proceedings of the 38th IEEEInternational Conference on Acoustics Speech and Signal Pro-cessing (ICASSP rsquo13) pp 4736ndash4739 May 2013

[21] J Apolinario Jr M L R Campos and P S R DinizldquoConvergence analysis of the binormalized data-reusing LMSalgorithmrdquo IEEE Transactions on Signal Processing vol 48 no11 pp 3235ndash3242 2000

[22] Y-M Shi L Huang C Qian and H C So ldquoShrinkagelinear and widely linear complex-valued least mean squaresalgorithms for adaptive beamformingrdquo IEEE Transactions onSignal Processing vol 63 no 1 pp 119ndash131 2015

[23] T Aboulnasr and K Mayyas ldquoA robust variable step-size LMS-type algorithm analysis and simulationsrdquo IEEE Transactions onSignal Processing vol 45 no 3 pp 631ndash639 1997

[24] L C Godara ldquoApplication of antenna arrays to mobile com-munications part II beam-forming and direction-of-arrivalconsiderationsrdquo Proceedings of the IEEE vol 85 no 8 pp 1195ndash1245 1997

[25] 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

8 Journal of Electrical and Computer Engineering

[26] S Shahbazpanahi A B Gershman Z-Q Lou and KMWongldquoRobust adaptive beamforming for general-rank signalmodelsrdquoIEEE Transactions on Signal Processing vol 51 no 9 pp 2257ndash2269 2003

[27] A Elnashar ldquoEfficient implementation of robust adaptive beam-forming based on worst-case performance optimisationrdquo IETSignal Processing vol 2 no 4 pp 381ndash393 2008

[28] 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

[29] P Stoica J Li X Zhu and J RGuerci ldquoOnusing a priori knowl-edge in space-time adaptive processingrdquo IEEE Transactions onSignal Processing vol 56 no 6 pp 2598ndash2602 2008

[30] O Ledoit and M Wolf ldquoA well-conditioned estimator forlarge-dimensional covariance matricesrdquo Journal of MultivariateAnalysis vol 88 no 2 pp 365ndash411 2004

[31] J Li L Du and P Stoica ldquoFully automatic computation ofdiagonal loading levels for robust adaptive beamformingrdquo inProceedings of the IEEE International Conference on AcousticsSpeech and Signal Processing (ICASSP 08) pp 2325ndash2328 IEEELas Vegas Nev USA April 2008

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Submit your manuscripts athttpwwwhindawicom

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DistributedSensor Networks

International Journal of

Journal of Electrical and Computer Engineering 3

According to (8) and (9) the weight vector of traditionalCLMS algorithm can be rewritten as [24]

w (119896 + 1) = P [w (119896) minus 1205831R119894+119899

w (119896)] +Ψ (10)

whereΨ = a[a119867a]minus1 and P = Ι minus a[a119867a]minus1a119867From (10) we note that the performance of the tradi-

tional CLMS algorithm is dependent on exact signal steeringvector and it is sensitive to some types of mismatches Inaddition the interference-plus-noise correlation matrix R

119894+119899

is unknown Hereby the sample covariance matrix

R119873=1

119873

119873

sum

119896=1

x (119896) x119867 (119896) (11)

is used to substitute R119894+119899

in (10) where 119873 is the train-ing sample size Therefore the performance degradation ofCLMS algorithm can occur due to the signal steering vectormismatches and small training sample size

3 Robust CLMS Algorithm Based onWorst-Case SINR Maximization

In order to solve the above-mentioned problems of thelinearly constrained-LMS algorithm we propose a robustrecursive algorithm based onworst-case SINRmaximizationwhich provides robustness against mismatches

We assume that in practical situations the mismatchvector e is norm-bounded by some known constant 120576 gt 0that is

e le 120576 (12)

Then the actual signal steering vector belongs to a ballset

120593 (120576) = c | c = a + e e le 120576 (13)

The weight vector is selected by minimizing the meanoutput power while maintaining a distortionless responsefor the mismatched steering vector So the cost function ofrobust constrained-LMS (RCLMS) algorithm is formulatedas

minw w119867R119873w

subject to mincisin120593(120576)

10038161003816100381610038161003816w119867c10038161003816100381610038161003816 ge 1

(14)

According to [25] the constraint in (14) is equivalent tothe following form

mincisin120593(120576)

10038161003816100381610038161003816w119867c10038161003816100381610038161003816 = 1 (15)

Using (15) problem (14) can be rewritten in the followingway

minw w119867R119873w

subject to 10038161003816100381610038161003816w119867a minus 110038161003816100381610038161003816

2

= 1205762w119867w

(16)

In order to improve the robustness against the mismatchthat may be caused by the small training sample size we canget a further extension of the optimization problem (16) Theactual covariance matrix is

R119889= R119873+ Δ (17)

where Δ is the norm of the error matrix and it is bounded bya certain constant 119903 Δ le 119903

Applying the worst-case performance optimization wecan rewrite

minw maxΔle119903

w119867 (R119873+ Δ)w

subject to 10038161003816100381610038161003816w119867a minus 110038161003816100381610038161003816

2

= 1205762w119867w

(18)

To obtain the optimal weight vector we can first solve thefollowing simpler problem [26]

minΔ

minus w119867 (R119873+ Δ)w

subject to Δ le 119903

(19)

Using Lagrange multiplier method to yield the matrixerror

Δ = 119903ww119867

w (20)

Consequently the minimization problem (18) is con-verted to the following form

minw maxΔle119903

w119867 (R119873+ 119903I)w

subject to 10038161003816100381610038161003816w119867a minus 110038161003816100381610038161003816

2

= 1205762w119867w

(21)

The solution to (21) can be derived by minimizing theLagrange function

119891 (w 120582)

= w119867R119897w

+ 120582 (1205762w119867w minus w119867aa119867w + w119867a + a119867w minus 1)

(22)

where R119897= R119873+ 119903I and 120582 is Lagrange multiplier Computing

the gradient vector of119867(w 120582) we can get the gradient nabla

nabla = (R119897+ 1205821205762Ι minus 120582aa119867)w + 120582a (23)

The gradient of119867(w 120582) is equal to zero andwe can obtainthe optimum weight vector

wopt = 120599 (120582) (R119897 + 1205821205762Ι)minus1

a (24)

where 120599(120582) = 120582(120582a119867(R119897+ 1205821205762Ι)minus1a minus 1)

From (24) we note that the proposed algorithm belongsto the class of diagonal loading techniques but the loadingfactor is calculated in a complicated way

4 Journal of Electrical and Computer Engineering

Using (23) the updated weight vector is obtained by

w (119896 + 1) = w (119896) minus 120583 [Bw (119896) + 120582a] (25)

where B = R119897+ 1205821205762Ι minus 120582aa119867 and 120583 is step size

Next we need to compute the Lagrange multiplier 120582 Thequadratic constraint of the optimization problem (21) is

ℎ (w) = 1 (26)

where

ℎ (w) = w119867 (119896) (1205762I minus aa119867)w (119896) + a119867w (119896)

+ w119867 (119896) a(27)

Inserting (25) into (26) we can obtain the Lagrangemultiplier 120582

120582 =1

21205832 [F119867 (119896) (1205762I minus aa119867) F (119896)](120577 (119896) + 120594 (119896)) (28)

where

F (119896) = (1205762I minus aa119867)w (119896) + a

q (119896) = [I minus 120583 (R119897 + 119903I)]w (119896)

120577 (119896) = 120583 [q119867 (119896) (1205762I minus aa119867) F (119896)

+ F119867 (119896) (1205762I minus aa119867)P (119896) + F119867 (119896) a + a119867F (119896)]

120594 (119896) = P119867 (119896) (1205762I minus aa119867) q (119896) + q119867 (119896) a

+ a119867q (119896)

120588119867(119896) 120588 (119896) = 120577

119867(119896) 120577 (119896) minus 4120583

2F119867 (119896) (1205762I minus aa119867)

sdot F (119896) (120594 (119896) minus 1)

(29)

31TheChoice of Step Size Theweight vector (25) is rewrittenas

w (119896 + 1) = [I minus 120583Β]w (119896) minus 120583120582a (30)

Let

B = ΣΛΣ119867 (31)

where Λ is a diagonal matrix in which the diagonal elementsof matrix 120591

1ge 1205912ge sdot sdot sdot ge 120591

119872are equal to the eigenvalues ofB

and the columns ofΣ contain the corresponding eigenvectorsWe can get the following equation via multiplying (30) byΣ119867

Σ119867w (119896 + 1) = [I minus 120583Λ]Σ119867w (119896) minus 120583120582Σ119867a (32)

As demonstrated in (32) if the proposed algorithmconverges it is required to satisfy the constrained condition

10038161003816100381610038161 minus 1205831205911198941003816100381610038161003816 lt 1 (33)

It follows from (33) that

0 lt 120583 lt2

120591max (34)

where 120591max is the maximum eigenvalue

120591max lt119872

sum

119894=1

120591119894= trace [B] (35)

In recursive algorithm the choice of the step size 120583 is veryimportant and it is varied with each new training snapshot[27 28]

Therefore we can obtain the optimal parameter 120583

120583 =1

5 trace [B] (36)

32TheApproximation of LagrangeMultiplier From (28) and(29) we note that the computation cost of weight vector isvery high Next we obtain the Lagrange multiplier by linearcombination to decrease the computation cost

From (24) the proposed beamformer belongs to theclass of diagonal loading techniques According to [29] weconsider a linear combination of R

119894+119899and R

119897

R119888= 120588R119894+119899+ 120592R119897 (37)

where the parameters 120588 gt 0 and 120592 gt 0 The initial value ofR119894+119899

is assumed to identify matrix I [30] We can rewrite (37)as

R119888= R119897+120588

120592Ι (38)

Contrasting the diagonal loading covariancematrices R119897+

1205821205762Ι in (24) and R

119897+(120588120592)Ι in (38) we note that the parameter

1205821205762 is replaced by 120588120592 In this way we can compute the

Lagrange multiplier 120582 as follows

120582 =120588

120592sdot1

1205762 (39)

From (39) the Lagrange multiplier is calculated sim-ply which decreases the complexity cost of the proposedalgorithm We need first to obtain the parameters 120588 and 120592Minimize the following function

119892 = min119864 1003817100381710038171003817R119888 minus R119909

1003817100381710038171003817

2 (40)

where R119909

= 119864[x(119896)x119867(119896)] is the theoretical covariancematrix By inserting (37) into (40) we have [31]

119892 = 119864 10038171003817100381710038171003817120592 (R119897minus R119909) + 120588Ι minus (1 minus 120592)R

119909

10038171003817100381710038171003817

2

= 120592211986410038171003817100381710038171003817R119897minus R119909

10038171003817100381710038171003817

2

minus 2120588 (1 minus 120592) tr (R119909)

+ (1 minus 120592)2 1003817100381710038171003817R119909

1003817100381710038171003817

2+ 1205882119872

(41)

Journal of Electrical and Computer Engineering 5

Computing the gradient of (41) with respect to 120588 for fixed120592 we can give the optimal value

1205880=(1 minus 120592

0) tr (R

119909)

119872 (42)

Inserting 1205880into (41) and replacing 120592

0by 120592 minimization

problem is written as

min1205921205922119864 10038171003817100381710038171003817R119897minus R119909

10038171003817100381710038171003817

2

+

(1 minus 120592)21003817100381710038171003817R119909

1003817100381710038171003817

2119872minus tr2 (R

119909)

119872

(43)

By minimizing (43) the optimal solution for 120592 is derivedas

1205920=

120572

120572 + 120590 (44)

where 120590 = 119864R119897minus R1199092 and 120572 = R

1199092minus tr2(R

119909)119872

In practical situations R119909is replaced by R

119897to obtain

estimation value

=10038171003817100381710038171003817R119897

10038171003817100381710038171003817

2

minus

tr2 (R119897)

119872 (45)

We can estimate the parameter 120590

=

119872

sum

119898=1

[1

1198732

119873

sum

119899=1

1003817100381710038171003817119903119898 minus x (119899) 119909lowast119898(119899)1003817100381710038171003817

2]

=1

1198732

119873

sum

119899=1

x (119899)4 minus 1

119873

10038171003817100381710038171003817R119897

10038171003817100381710038171003817

2

(46)

where 119909119898(119899) is the119898th element of x(119899)

Substituting (45) and (46) into (44) the estimation valueof 1205920is written as

1205920=

+ (47)

Inserting (45) and (46) into (42) we can obtain estima-tion value of 120588

0

1205880=

(1 minus 1205920) tr (R

119897)

119872 (48)

Consequently we can obtain the expression of Lagrangemultiplier

=1205880

1205920

sdot1

1205762 (49)

Our proposed RCLMS algorithm belongs to the classof diagonal loadings but the diagonal loading factor isderived fully automatically from the observation vectorswithout the need of specifying any user knowledge Theparameter 120576 is determined easily andwe can conclude that theproposed RCLMS algorithm is not sensitive to the choice of

Table 1 The complexity cost of the conventional constrained-LMS

Complexity costR119894+119899

119874(1198722times 119873 + 1)

Ψ 119874(2119872 + 1)

P 119874(21198722+ 1)

Pw(119896) 119874(1198722)

1205831PR119894+119899w(119896) 119874(119872

3+1198722+119872)

Total complexity cost 119874(1198723+ (119873 + 4)119872

2+ 3119872 + 3)

Table 2 The complexity cost of the proposed robust CLMS

Complexity cost 119874(119872

2+119872 times119873 +119873 + 12)

120583a 119874(119872 + 1)

R119897

119874(1198722times 119873 + 1)

1205821205762I 119874(119872

2+ 2)

120582aa119867 119874(21198722)

120583Bw(119896) 119874(1198722+119872)

Total complexity cost 119874((119873 + 5)1198722+ (119873 + 2)119872 +119873 + 16)

parameter 120576 in [28] From the literature [11] it is clear that themajor computational demand of the algorithm comes fromthe eigendecomposition which requires 119874(1198723) flops Thisleads to a high computational cost However the proposedalgorithm which does not need eigendecomposition canreduce the complexity to 119874(1198722) flops In addition robustCapon algorithm and our proposed algorithm belong to theclass of the diagonal loadings in which the loading factorscan be calculated precisely

33 The Analysis of Complexity Cost The complexity cost oftwo algorithms can be shown as in Tables 1 and 2

4 Simulation Results

In this section we present some simulations to justify the per-formance of the proposed robust recursive algorithm basedon worst-case SINR maximization We assume a uniformlinear array with 119872 = 10 omnidirectional sensors spacedhalf a wavelength apart For each scenario 200 simulationruns are used to obtain each simulation point Assume thatboth directions of arrival (DOAs) of the presumed and actualsignal are 0∘ and 3

∘ respectively This corresponds to a3∘ mismatch in the direction of arrival Suppose that twointerfering sources are planewaves impinging from theDOAsminus50∘ and 50∘ respectively We choose the parameter 119903 = 9

Example 1 (output SINR versus the number of snapshots)We assume that signal-to-noise ratio SNR = 10 dB and theparameter 120576 = 3 Figure 1 shows the performance of themethods tested in no mismatch case From Figure 1 we seethat the output SINR of traditional CLMS is about 14 dB Notethat it is sensitive to the small training sample size Howeverour proposed algorithm can provide improved robustness

6 Journal of Electrical and Computer Engineering

0 10 20 30 40 50 60 70 80 90 1000

5

10

15

20

25

Number of snapshots

Out

put S

INR

(dB)

CLMS beamformerOptimal SINRRCLMS beamformer

Figure 1 Output SINR in no-mismatch case

0 10 20 30 40 50 60 70 80 90 100minus15

minus10

minus5

0

5

10

15

20

25

Number of snapshots

Out

put S

INR

(dB)

CLMS beamformerOptimal SINRRCLMS beamformer

Figure 2 Output SINR in a 3∘ mismatch case

Figure 2 shows the array output SINR of the methods testedin a 3∘ mismatch

In Figure 2 we note that the output SINR of CLMS algo-rithm is about minus10 dB which is sensitive to the signal steeringvector mismatches However that of the proposed robustCLMS (RCLMS) algorithm is about 18 dB which is close tothe optimal one In this scenario the proposed recursive algo-rithm outperforms the traditional linear constrained-LMSalgorithm Moreover robust constrained-LMS algorithm hasfaster convergence rate

Example 2 (output SINR versus SNR) In this example thereis a 3∘ mismatch in the signal look direction We assume thatthe fixed training data size119873 is equal to 100 Figure 3 displays

minus10 minus8 minus6 minus4 minus2 0 2 4 6 8 10minus5

0

5

10

15

20

SNR (dB)

Out

put S

INR

(dB)

Optimal SINRCLMS beamformerRCLMS beamformer

Figure 3 Output SINR in no-mismatch case

minus10 minus8 minus6 minus4 minus2 0 2 4 6 8 10minus15

minus10

minus5

0

5

10

15

20

SNR (dB)

Out

put S

INR

(dB)

Optimal SINRCLMS beamformerRCLMS beamformer

Figure 4 Output SINR in a 3∘ mismatch case

the performance of these algorithms versus the SNR in no-mismatch case The performance of these algorithms versusthe SNR in a 3∘ mismatch case is shown in Figure 4

In this example the traditional algorithm is very sensitiveeven to slight mismatches which can easily occur in practicalapplications It is observed from Figure 4 that with theincrease of SNR CLMS algorithm has poor performance atall values of the SNRHowever our proposed recursive robustalgorithm provides improved robustness against signal steer-ing vector mismatches and small training sample size hasfaster convergence rate and yields better output performancethan the CLMS algorithm

Journal of Electrical and Computer Engineering 7

5 Conclusions

In this paper we propose a robust constrained-LMS algo-rithm based on the worst-case SINR maximization TheRCLMS algorithm provides robustness against some types ofmismatches and offers faster convergence rate The updatedweight vector is derived by the gradient descent methodand Lagrange multiplier method in which the diagonalloading factor is obtained in a simple formThis decreases thecomputation cost Some simulation results demonstrate thatthe proposed robust recursive algorithm enjoys better perfor-mance as compared with the traditional CLMS algorithm

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the anonymous reviewersfor their insightful comments that helped improve the qualityof this paper This work is supported by Program for NewCentury Excellent Talents in University (no NCET-12-0103)by the National Natural Science Foundation of China underGrant no 61473066 by the Fundamental Research Funds forthe Central Universities under Grant no N130423005 andby the Natural Science Foundation of Hebei Province underGrant no F2012501044

References

[1] T S Rapapport Ed Smart Antennas Adaptive Arrays Algo-rithms and Wireless Position Location IEEE Piscataway NJUSA 1998

[2] R T Compton Jr R J Huff W G Swarner and A KsienskildquoAdaptive arrays for communication system an overview of theresearch at the Ohio State Universityrdquo IEEE Transactions onAntennas and Propagation vol AP-24 no 5 pp 599ndash607 1976

[3] A B Gershman E Nemeth and J F Bohme ldquoExperimentalperformance of adaptive beamforming in a sonar environmentwith a towed array and moving interfering sourcesrdquo IEEETransactions on Signal Processing vol 48 no 1 pp 246ndash2502000

[4] Y Kaneda and J Ohga ldquoAdaptive microphone-array system fornoise reductionrdquo IEEE Transactions on Acoustics Speech andSignal Processing vol 34 no 6 pp 1391ndash1400 1986

[5] J Li and P Stoica Robust Adaptive Beamforming John Wiley ampSons New York NY USA 2005

[6] R A Monzingo and T W Miller Introduction to AdaptiveArrays Wiley New York NY USA 1980

[7] 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

[8] D D Feldman and L J Griffiths ldquoProjection approach forrobust adaptive beamformingrdquo IEEE Transactions on SignalProcessing vol 42 no 4 pp 867ndash876 1994

[9] Q Zou Z L Yu and Z Lin ldquoA robust algorithm for linearlyconstrained adaptive beamformingrdquo IEEE Signal ProcessingLetters vol 11 no 1 pp 26ndash29 2004

[10] B D Carlson ldquoCovariance matrix estimation errors and diago-nal loading in adaptive arraysrdquo IEEE Transactions on Aerospaceand Electronic Systems vol 24 no 4 pp 397ndash401 1988

[11] J Li P Stoica and Z Wang ldquoOn robust Capon beamformingand diagonal loadingrdquo IEEE Transactions on Signal Processingvol 51 no 7 pp 1702ndash1715 2003

[12] J Li P Stoica and ZWang ldquoDoubly constrained robust Caponbeamformerrdquo IEEE Transactions on Signal Processing vol 52no 9 pp 2407ndash2423 2004

[13] S-J Kim A Magnani A Mutapcic S P Boyd and Z-QLuo ldquoRobust beamforming viaworst-case SINRmaximizationrdquoIEEE Transactions on Signal Processing vol 56 no 4 pp 1539ndash1547 2008

[14] Y W Huang Q Li W-K Ma and S Z Zhang ldquoRobustmulticast beamforming for spectrum sharing-based cognitiveradiosrdquo IEEE Transactions on Signal Processing vol 60 no 1pp 527ndash533 2012

[15] D Ciochina M Pesavento and K M Wong ldquoWorst caserobust downlink beamforming on the Riemannian manifoldrdquoin Proceedings of the 38th IEEE International Conference onAcoustics Speech and Signal Processing (ICASSP rsquo13) pp 3801ndash3805 May 2013

[16] Q Li Q Zhang and J Qin ldquoRobust beamforming for cognitivemulti-antenna relay networkswith bounded channel uncertain-tiesrdquo IEEE Transactions on Communications vol 62 no 2 pp478ndash487 2014

[17] G Zheng S Ma K-K Wong and T-S Ng ldquoRobust beam-forming in cognitive radiordquo IEEE Transactions on WirelessCommunications vol 9 no 2 pp 570ndash576 2010

[18] S C Douglas ldquoWidely-linear recursive least-squares algorithmfor adaptive beamformingrdquo in Proceedings of the IEEE Inter-national Conference on Acoustics Speech and Signal Processing(ICASSP rsquo09) pp 2041ndash2044 IEEE Taipei Taiwan April 2009

[19] E Eweda ldquoComparison of RLS LMS and sign algorithms fortracking randomly time-varying channelsrdquo IEEE Transactionson Signal Processing vol 42 no 11 pp 2937ndash2944 1994

[20] S J Yao H Qian K Kang and M Y Shen ldquoA recursiveleast squares algorithm with reduced complexity for digitalpredistortion linearizationrdquo in Proceedings of the 38th IEEEInternational Conference on Acoustics Speech and Signal Pro-cessing (ICASSP rsquo13) pp 4736ndash4739 May 2013

[21] J Apolinario Jr M L R Campos and P S R DinizldquoConvergence analysis of the binormalized data-reusing LMSalgorithmrdquo IEEE Transactions on Signal Processing vol 48 no11 pp 3235ndash3242 2000

[22] Y-M Shi L Huang C Qian and H C So ldquoShrinkagelinear and widely linear complex-valued least mean squaresalgorithms for adaptive beamformingrdquo IEEE Transactions onSignal Processing vol 63 no 1 pp 119ndash131 2015

[23] T Aboulnasr and K Mayyas ldquoA robust variable step-size LMS-type algorithm analysis and simulationsrdquo IEEE Transactions onSignal Processing vol 45 no 3 pp 631ndash639 1997

[24] L C Godara ldquoApplication of antenna arrays to mobile com-munications part II beam-forming and direction-of-arrivalconsiderationsrdquo Proceedings of the IEEE vol 85 no 8 pp 1195ndash1245 1997

[25] 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

8 Journal of Electrical and Computer Engineering

[26] S Shahbazpanahi A B Gershman Z-Q Lou and KMWongldquoRobust adaptive beamforming for general-rank signalmodelsrdquoIEEE Transactions on Signal Processing vol 51 no 9 pp 2257ndash2269 2003

[27] A Elnashar ldquoEfficient implementation of robust adaptive beam-forming based on worst-case performance optimisationrdquo IETSignal Processing vol 2 no 4 pp 381ndash393 2008

[28] 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

[29] P Stoica J Li X Zhu and J RGuerci ldquoOnusing a priori knowl-edge in space-time adaptive processingrdquo IEEE Transactions onSignal Processing vol 56 no 6 pp 2598ndash2602 2008

[30] O Ledoit and M Wolf ldquoA well-conditioned estimator forlarge-dimensional covariance matricesrdquo Journal of MultivariateAnalysis vol 88 no 2 pp 365ndash411 2004

[31] J Li L Du and P Stoica ldquoFully automatic computation ofdiagonal loading levels for robust adaptive beamformingrdquo inProceedings of the IEEE International Conference on AcousticsSpeech and Signal Processing (ICASSP 08) pp 2325ndash2328 IEEELas Vegas Nev USA April 2008

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

4 Journal of Electrical and Computer Engineering

Using (23) the updated weight vector is obtained by

w (119896 + 1) = w (119896) minus 120583 [Bw (119896) + 120582a] (25)

where B = R119897+ 1205821205762Ι minus 120582aa119867 and 120583 is step size

Next we need to compute the Lagrange multiplier 120582 Thequadratic constraint of the optimization problem (21) is

ℎ (w) = 1 (26)

where

ℎ (w) = w119867 (119896) (1205762I minus aa119867)w (119896) + a119867w (119896)

+ w119867 (119896) a(27)

Inserting (25) into (26) we can obtain the Lagrangemultiplier 120582

120582 =1

21205832 [F119867 (119896) (1205762I minus aa119867) F (119896)](120577 (119896) + 120594 (119896)) (28)

where

F (119896) = (1205762I minus aa119867)w (119896) + a

q (119896) = [I minus 120583 (R119897 + 119903I)]w (119896)

120577 (119896) = 120583 [q119867 (119896) (1205762I minus aa119867) F (119896)

+ F119867 (119896) (1205762I minus aa119867)P (119896) + F119867 (119896) a + a119867F (119896)]

120594 (119896) = P119867 (119896) (1205762I minus aa119867) q (119896) + q119867 (119896) a

+ a119867q (119896)

120588119867(119896) 120588 (119896) = 120577

119867(119896) 120577 (119896) minus 4120583

2F119867 (119896) (1205762I minus aa119867)

sdot F (119896) (120594 (119896) minus 1)

(29)

31TheChoice of Step Size Theweight vector (25) is rewrittenas

w (119896 + 1) = [I minus 120583Β]w (119896) minus 120583120582a (30)

Let

B = ΣΛΣ119867 (31)

where Λ is a diagonal matrix in which the diagonal elementsof matrix 120591

1ge 1205912ge sdot sdot sdot ge 120591

119872are equal to the eigenvalues ofB

and the columns ofΣ contain the corresponding eigenvectorsWe can get the following equation via multiplying (30) byΣ119867

Σ119867w (119896 + 1) = [I minus 120583Λ]Σ119867w (119896) minus 120583120582Σ119867a (32)

As demonstrated in (32) if the proposed algorithmconverges it is required to satisfy the constrained condition

10038161003816100381610038161 minus 1205831205911198941003816100381610038161003816 lt 1 (33)

It follows from (33) that

0 lt 120583 lt2

120591max (34)

where 120591max is the maximum eigenvalue

120591max lt119872

sum

119894=1

120591119894= trace [B] (35)

In recursive algorithm the choice of the step size 120583 is veryimportant and it is varied with each new training snapshot[27 28]

Therefore we can obtain the optimal parameter 120583

120583 =1

5 trace [B] (36)

32TheApproximation of LagrangeMultiplier From (28) and(29) we note that the computation cost of weight vector isvery high Next we obtain the Lagrange multiplier by linearcombination to decrease the computation cost

From (24) the proposed beamformer belongs to theclass of diagonal loading techniques According to [29] weconsider a linear combination of R

119894+119899and R

119897

R119888= 120588R119894+119899+ 120592R119897 (37)

where the parameters 120588 gt 0 and 120592 gt 0 The initial value ofR119894+119899

is assumed to identify matrix I [30] We can rewrite (37)as

R119888= R119897+120588

120592Ι (38)

Contrasting the diagonal loading covariancematrices R119897+

1205821205762Ι in (24) and R

119897+(120588120592)Ι in (38) we note that the parameter

1205821205762 is replaced by 120588120592 In this way we can compute the

Lagrange multiplier 120582 as follows

120582 =120588

120592sdot1

1205762 (39)

From (39) the Lagrange multiplier is calculated sim-ply which decreases the complexity cost of the proposedalgorithm We need first to obtain the parameters 120588 and 120592Minimize the following function

119892 = min119864 1003817100381710038171003817R119888 minus R119909

1003817100381710038171003817

2 (40)

where R119909

= 119864[x(119896)x119867(119896)] is the theoretical covariancematrix By inserting (37) into (40) we have [31]

119892 = 119864 10038171003817100381710038171003817120592 (R119897minus R119909) + 120588Ι minus (1 minus 120592)R

119909

10038171003817100381710038171003817

2

= 120592211986410038171003817100381710038171003817R119897minus R119909

10038171003817100381710038171003817

2

minus 2120588 (1 minus 120592) tr (R119909)

+ (1 minus 120592)2 1003817100381710038171003817R119909

1003817100381710038171003817

2+ 1205882119872

(41)

Journal of Electrical and Computer Engineering 5

Computing the gradient of (41) with respect to 120588 for fixed120592 we can give the optimal value

1205880=(1 minus 120592

0) tr (R

119909)

119872 (42)

Inserting 1205880into (41) and replacing 120592

0by 120592 minimization

problem is written as

min1205921205922119864 10038171003817100381710038171003817R119897minus R119909

10038171003817100381710038171003817

2

+

(1 minus 120592)21003817100381710038171003817R119909

1003817100381710038171003817

2119872minus tr2 (R

119909)

119872

(43)

By minimizing (43) the optimal solution for 120592 is derivedas

1205920=

120572

120572 + 120590 (44)

where 120590 = 119864R119897minus R1199092 and 120572 = R

1199092minus tr2(R

119909)119872

In practical situations R119909is replaced by R

119897to obtain

estimation value

=10038171003817100381710038171003817R119897

10038171003817100381710038171003817

2

minus

tr2 (R119897)

119872 (45)

We can estimate the parameter 120590

=

119872

sum

119898=1

[1

1198732

119873

sum

119899=1

1003817100381710038171003817119903119898 minus x (119899) 119909lowast119898(119899)1003817100381710038171003817

2]

=1

1198732

119873

sum

119899=1

x (119899)4 minus 1

119873

10038171003817100381710038171003817R119897

10038171003817100381710038171003817

2

(46)

where 119909119898(119899) is the119898th element of x(119899)

Substituting (45) and (46) into (44) the estimation valueof 1205920is written as

1205920=

+ (47)

Inserting (45) and (46) into (42) we can obtain estima-tion value of 120588

0

1205880=

(1 minus 1205920) tr (R

119897)

119872 (48)

Consequently we can obtain the expression of Lagrangemultiplier

=1205880

1205920

sdot1

1205762 (49)

Our proposed RCLMS algorithm belongs to the classof diagonal loadings but the diagonal loading factor isderived fully automatically from the observation vectorswithout the need of specifying any user knowledge Theparameter 120576 is determined easily andwe can conclude that theproposed RCLMS algorithm is not sensitive to the choice of

Table 1 The complexity cost of the conventional constrained-LMS

Complexity costR119894+119899

119874(1198722times 119873 + 1)

Ψ 119874(2119872 + 1)

P 119874(21198722+ 1)

Pw(119896) 119874(1198722)

1205831PR119894+119899w(119896) 119874(119872

3+1198722+119872)

Total complexity cost 119874(1198723+ (119873 + 4)119872

2+ 3119872 + 3)

Table 2 The complexity cost of the proposed robust CLMS

Complexity cost 119874(119872

2+119872 times119873 +119873 + 12)

120583a 119874(119872 + 1)

R119897

119874(1198722times 119873 + 1)

1205821205762I 119874(119872

2+ 2)

120582aa119867 119874(21198722)

120583Bw(119896) 119874(1198722+119872)

Total complexity cost 119874((119873 + 5)1198722+ (119873 + 2)119872 +119873 + 16)

parameter 120576 in [28] From the literature [11] it is clear that themajor computational demand of the algorithm comes fromthe eigendecomposition which requires 119874(1198723) flops Thisleads to a high computational cost However the proposedalgorithm which does not need eigendecomposition canreduce the complexity to 119874(1198722) flops In addition robustCapon algorithm and our proposed algorithm belong to theclass of the diagonal loadings in which the loading factorscan be calculated precisely

33 The Analysis of Complexity Cost The complexity cost oftwo algorithms can be shown as in Tables 1 and 2

4 Simulation Results

In this section we present some simulations to justify the per-formance of the proposed robust recursive algorithm basedon worst-case SINR maximization We assume a uniformlinear array with 119872 = 10 omnidirectional sensors spacedhalf a wavelength apart For each scenario 200 simulationruns are used to obtain each simulation point Assume thatboth directions of arrival (DOAs) of the presumed and actualsignal are 0∘ and 3

∘ respectively This corresponds to a3∘ mismatch in the direction of arrival Suppose that twointerfering sources are planewaves impinging from theDOAsminus50∘ and 50∘ respectively We choose the parameter 119903 = 9

Example 1 (output SINR versus the number of snapshots)We assume that signal-to-noise ratio SNR = 10 dB and theparameter 120576 = 3 Figure 1 shows the performance of themethods tested in no mismatch case From Figure 1 we seethat the output SINR of traditional CLMS is about 14 dB Notethat it is sensitive to the small training sample size Howeverour proposed algorithm can provide improved robustness

6 Journal of Electrical and Computer Engineering

0 10 20 30 40 50 60 70 80 90 1000

5

10

15

20

25

Number of snapshots

Out

put S

INR

(dB)

CLMS beamformerOptimal SINRRCLMS beamformer

Figure 1 Output SINR in no-mismatch case

0 10 20 30 40 50 60 70 80 90 100minus15

minus10

minus5

0

5

10

15

20

25

Number of snapshots

Out

put S

INR

(dB)

CLMS beamformerOptimal SINRRCLMS beamformer

Figure 2 Output SINR in a 3∘ mismatch case

Figure 2 shows the array output SINR of the methods testedin a 3∘ mismatch

In Figure 2 we note that the output SINR of CLMS algo-rithm is about minus10 dB which is sensitive to the signal steeringvector mismatches However that of the proposed robustCLMS (RCLMS) algorithm is about 18 dB which is close tothe optimal one In this scenario the proposed recursive algo-rithm outperforms the traditional linear constrained-LMSalgorithm Moreover robust constrained-LMS algorithm hasfaster convergence rate

Example 2 (output SINR versus SNR) In this example thereis a 3∘ mismatch in the signal look direction We assume thatthe fixed training data size119873 is equal to 100 Figure 3 displays

minus10 minus8 minus6 minus4 minus2 0 2 4 6 8 10minus5

0

5

10

15

20

SNR (dB)

Out

put S

INR

(dB)

Optimal SINRCLMS beamformerRCLMS beamformer

Figure 3 Output SINR in no-mismatch case

minus10 minus8 minus6 minus4 minus2 0 2 4 6 8 10minus15

minus10

minus5

0

5

10

15

20

SNR (dB)

Out

put S

INR

(dB)

Optimal SINRCLMS beamformerRCLMS beamformer

Figure 4 Output SINR in a 3∘ mismatch case

the performance of these algorithms versus the SNR in no-mismatch case The performance of these algorithms versusthe SNR in a 3∘ mismatch case is shown in Figure 4

In this example the traditional algorithm is very sensitiveeven to slight mismatches which can easily occur in practicalapplications It is observed from Figure 4 that with theincrease of SNR CLMS algorithm has poor performance atall values of the SNRHowever our proposed recursive robustalgorithm provides improved robustness against signal steer-ing vector mismatches and small training sample size hasfaster convergence rate and yields better output performancethan the CLMS algorithm

Journal of Electrical and Computer Engineering 7

5 Conclusions

In this paper we propose a robust constrained-LMS algo-rithm based on the worst-case SINR maximization TheRCLMS algorithm provides robustness against some types ofmismatches and offers faster convergence rate The updatedweight vector is derived by the gradient descent methodand Lagrange multiplier method in which the diagonalloading factor is obtained in a simple formThis decreases thecomputation cost Some simulation results demonstrate thatthe proposed robust recursive algorithm enjoys better perfor-mance as compared with the traditional CLMS algorithm

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the anonymous reviewersfor their insightful comments that helped improve the qualityof this paper This work is supported by Program for NewCentury Excellent Talents in University (no NCET-12-0103)by the National Natural Science Foundation of China underGrant no 61473066 by the Fundamental Research Funds forthe Central Universities under Grant no N130423005 andby the Natural Science Foundation of Hebei Province underGrant no F2012501044

References

[1] T S Rapapport Ed Smart Antennas Adaptive Arrays Algo-rithms and Wireless Position Location IEEE Piscataway NJUSA 1998

[2] R T Compton Jr R J Huff W G Swarner and A KsienskildquoAdaptive arrays for communication system an overview of theresearch at the Ohio State Universityrdquo IEEE Transactions onAntennas and Propagation vol AP-24 no 5 pp 599ndash607 1976

[3] A B Gershman E Nemeth and J F Bohme ldquoExperimentalperformance of adaptive beamforming in a sonar environmentwith a towed array and moving interfering sourcesrdquo IEEETransactions on Signal Processing vol 48 no 1 pp 246ndash2502000

[4] Y Kaneda and J Ohga ldquoAdaptive microphone-array system fornoise reductionrdquo IEEE Transactions on Acoustics Speech andSignal Processing vol 34 no 6 pp 1391ndash1400 1986

[5] J Li and P Stoica Robust Adaptive Beamforming John Wiley ampSons New York NY USA 2005

[6] R A Monzingo and T W Miller Introduction to AdaptiveArrays Wiley New York NY USA 1980

[7] 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

[8] D D Feldman and L J Griffiths ldquoProjection approach forrobust adaptive beamformingrdquo IEEE Transactions on SignalProcessing vol 42 no 4 pp 867ndash876 1994

[9] Q Zou Z L Yu and Z Lin ldquoA robust algorithm for linearlyconstrained adaptive beamformingrdquo IEEE Signal ProcessingLetters vol 11 no 1 pp 26ndash29 2004

[10] B D Carlson ldquoCovariance matrix estimation errors and diago-nal loading in adaptive arraysrdquo IEEE Transactions on Aerospaceand Electronic Systems vol 24 no 4 pp 397ndash401 1988

[11] J Li P Stoica and Z Wang ldquoOn robust Capon beamformingand diagonal loadingrdquo IEEE Transactions on Signal Processingvol 51 no 7 pp 1702ndash1715 2003

[12] J Li P Stoica and ZWang ldquoDoubly constrained robust Caponbeamformerrdquo IEEE Transactions on Signal Processing vol 52no 9 pp 2407ndash2423 2004

[13] S-J Kim A Magnani A Mutapcic S P Boyd and Z-QLuo ldquoRobust beamforming viaworst-case SINRmaximizationrdquoIEEE Transactions on Signal Processing vol 56 no 4 pp 1539ndash1547 2008

[14] Y W Huang Q Li W-K Ma and S Z Zhang ldquoRobustmulticast beamforming for spectrum sharing-based cognitiveradiosrdquo IEEE Transactions on Signal Processing vol 60 no 1pp 527ndash533 2012

[15] D Ciochina M Pesavento and K M Wong ldquoWorst caserobust downlink beamforming on the Riemannian manifoldrdquoin Proceedings of the 38th IEEE International Conference onAcoustics Speech and Signal Processing (ICASSP rsquo13) pp 3801ndash3805 May 2013

[16] Q Li Q Zhang and J Qin ldquoRobust beamforming for cognitivemulti-antenna relay networkswith bounded channel uncertain-tiesrdquo IEEE Transactions on Communications vol 62 no 2 pp478ndash487 2014

[17] G Zheng S Ma K-K Wong and T-S Ng ldquoRobust beam-forming in cognitive radiordquo IEEE Transactions on WirelessCommunications vol 9 no 2 pp 570ndash576 2010

[18] S C Douglas ldquoWidely-linear recursive least-squares algorithmfor adaptive beamformingrdquo in Proceedings of the IEEE Inter-national Conference on Acoustics Speech and Signal Processing(ICASSP rsquo09) pp 2041ndash2044 IEEE Taipei Taiwan April 2009

[19] E Eweda ldquoComparison of RLS LMS and sign algorithms fortracking randomly time-varying channelsrdquo IEEE Transactionson Signal Processing vol 42 no 11 pp 2937ndash2944 1994

[20] S J Yao H Qian K Kang and M Y Shen ldquoA recursiveleast squares algorithm with reduced complexity for digitalpredistortion linearizationrdquo in Proceedings of the 38th IEEEInternational Conference on Acoustics Speech and Signal Pro-cessing (ICASSP rsquo13) pp 4736ndash4739 May 2013

[21] J Apolinario Jr M L R Campos and P S R DinizldquoConvergence analysis of the binormalized data-reusing LMSalgorithmrdquo IEEE Transactions on Signal Processing vol 48 no11 pp 3235ndash3242 2000

[22] Y-M Shi L Huang C Qian and H C So ldquoShrinkagelinear and widely linear complex-valued least mean squaresalgorithms for adaptive beamformingrdquo IEEE Transactions onSignal Processing vol 63 no 1 pp 119ndash131 2015

[23] T Aboulnasr and K Mayyas ldquoA robust variable step-size LMS-type algorithm analysis and simulationsrdquo IEEE Transactions onSignal Processing vol 45 no 3 pp 631ndash639 1997

[24] L C Godara ldquoApplication of antenna arrays to mobile com-munications part II beam-forming and direction-of-arrivalconsiderationsrdquo Proceedings of the IEEE vol 85 no 8 pp 1195ndash1245 1997

[25] 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

8 Journal of Electrical and Computer Engineering

[26] S Shahbazpanahi A B Gershman Z-Q Lou and KMWongldquoRobust adaptive beamforming for general-rank signalmodelsrdquoIEEE Transactions on Signal Processing vol 51 no 9 pp 2257ndash2269 2003

[27] A Elnashar ldquoEfficient implementation of robust adaptive beam-forming based on worst-case performance optimisationrdquo IETSignal Processing vol 2 no 4 pp 381ndash393 2008

[28] 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

[29] P Stoica J Li X Zhu and J RGuerci ldquoOnusing a priori knowl-edge in space-time adaptive processingrdquo IEEE Transactions onSignal Processing vol 56 no 6 pp 2598ndash2602 2008

[30] O Ledoit and M Wolf ldquoA well-conditioned estimator forlarge-dimensional covariance matricesrdquo Journal of MultivariateAnalysis vol 88 no 2 pp 365ndash411 2004

[31] J Li L Du and P Stoica ldquoFully automatic computation ofdiagonal loading levels for robust adaptive beamformingrdquo inProceedings of the IEEE International Conference on AcousticsSpeech and Signal Processing (ICASSP 08) pp 2325ndash2328 IEEELas Vegas Nev USA April 2008

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Journal of Electrical and Computer Engineering 5

Computing the gradient of (41) with respect to 120588 for fixed120592 we can give the optimal value

1205880=(1 minus 120592

0) tr (R

119909)

119872 (42)

Inserting 1205880into (41) and replacing 120592

0by 120592 minimization

problem is written as

min1205921205922119864 10038171003817100381710038171003817R119897minus R119909

10038171003817100381710038171003817

2

+

(1 minus 120592)21003817100381710038171003817R119909

1003817100381710038171003817

2119872minus tr2 (R

119909)

119872

(43)

By minimizing (43) the optimal solution for 120592 is derivedas

1205920=

120572

120572 + 120590 (44)

where 120590 = 119864R119897minus R1199092 and 120572 = R

1199092minus tr2(R

119909)119872

In practical situations R119909is replaced by R

119897to obtain

estimation value

=10038171003817100381710038171003817R119897

10038171003817100381710038171003817

2

minus

tr2 (R119897)

119872 (45)

We can estimate the parameter 120590

=

119872

sum

119898=1

[1

1198732

119873

sum

119899=1

1003817100381710038171003817119903119898 minus x (119899) 119909lowast119898(119899)1003817100381710038171003817

2]

=1

1198732

119873

sum

119899=1

x (119899)4 minus 1

119873

10038171003817100381710038171003817R119897

10038171003817100381710038171003817

2

(46)

where 119909119898(119899) is the119898th element of x(119899)

Substituting (45) and (46) into (44) the estimation valueof 1205920is written as

1205920=

+ (47)

Inserting (45) and (46) into (42) we can obtain estima-tion value of 120588

0

1205880=

(1 minus 1205920) tr (R

119897)

119872 (48)

Consequently we can obtain the expression of Lagrangemultiplier

=1205880

1205920

sdot1

1205762 (49)

Our proposed RCLMS algorithm belongs to the classof diagonal loadings but the diagonal loading factor isderived fully automatically from the observation vectorswithout the need of specifying any user knowledge Theparameter 120576 is determined easily andwe can conclude that theproposed RCLMS algorithm is not sensitive to the choice of

Table 1 The complexity cost of the conventional constrained-LMS

Complexity costR119894+119899

119874(1198722times 119873 + 1)

Ψ 119874(2119872 + 1)

P 119874(21198722+ 1)

Pw(119896) 119874(1198722)

1205831PR119894+119899w(119896) 119874(119872

3+1198722+119872)

Total complexity cost 119874(1198723+ (119873 + 4)119872

2+ 3119872 + 3)

Table 2 The complexity cost of the proposed robust CLMS

Complexity cost 119874(119872

2+119872 times119873 +119873 + 12)

120583a 119874(119872 + 1)

R119897

119874(1198722times 119873 + 1)

1205821205762I 119874(119872

2+ 2)

120582aa119867 119874(21198722)

120583Bw(119896) 119874(1198722+119872)

Total complexity cost 119874((119873 + 5)1198722+ (119873 + 2)119872 +119873 + 16)

parameter 120576 in [28] From the literature [11] it is clear that themajor computational demand of the algorithm comes fromthe eigendecomposition which requires 119874(1198723) flops Thisleads to a high computational cost However the proposedalgorithm which does not need eigendecomposition canreduce the complexity to 119874(1198722) flops In addition robustCapon algorithm and our proposed algorithm belong to theclass of the diagonal loadings in which the loading factorscan be calculated precisely

33 The Analysis of Complexity Cost The complexity cost oftwo algorithms can be shown as in Tables 1 and 2

4 Simulation Results

In this section we present some simulations to justify the per-formance of the proposed robust recursive algorithm basedon worst-case SINR maximization We assume a uniformlinear array with 119872 = 10 omnidirectional sensors spacedhalf a wavelength apart For each scenario 200 simulationruns are used to obtain each simulation point Assume thatboth directions of arrival (DOAs) of the presumed and actualsignal are 0∘ and 3

∘ respectively This corresponds to a3∘ mismatch in the direction of arrival Suppose that twointerfering sources are planewaves impinging from theDOAsminus50∘ and 50∘ respectively We choose the parameter 119903 = 9

Example 1 (output SINR versus the number of snapshots)We assume that signal-to-noise ratio SNR = 10 dB and theparameter 120576 = 3 Figure 1 shows the performance of themethods tested in no mismatch case From Figure 1 we seethat the output SINR of traditional CLMS is about 14 dB Notethat it is sensitive to the small training sample size Howeverour proposed algorithm can provide improved robustness

6 Journal of Electrical and Computer Engineering

0 10 20 30 40 50 60 70 80 90 1000

5

10

15

20

25

Number of snapshots

Out

put S

INR

(dB)

CLMS beamformerOptimal SINRRCLMS beamformer

Figure 1 Output SINR in no-mismatch case

0 10 20 30 40 50 60 70 80 90 100minus15

minus10

minus5

0

5

10

15

20

25

Number of snapshots

Out

put S

INR

(dB)

CLMS beamformerOptimal SINRRCLMS beamformer

Figure 2 Output SINR in a 3∘ mismatch case

Figure 2 shows the array output SINR of the methods testedin a 3∘ mismatch

In Figure 2 we note that the output SINR of CLMS algo-rithm is about minus10 dB which is sensitive to the signal steeringvector mismatches However that of the proposed robustCLMS (RCLMS) algorithm is about 18 dB which is close tothe optimal one In this scenario the proposed recursive algo-rithm outperforms the traditional linear constrained-LMSalgorithm Moreover robust constrained-LMS algorithm hasfaster convergence rate

Example 2 (output SINR versus SNR) In this example thereis a 3∘ mismatch in the signal look direction We assume thatthe fixed training data size119873 is equal to 100 Figure 3 displays

minus10 minus8 minus6 minus4 minus2 0 2 4 6 8 10minus5

0

5

10

15

20

SNR (dB)

Out

put S

INR

(dB)

Optimal SINRCLMS beamformerRCLMS beamformer

Figure 3 Output SINR in no-mismatch case

minus10 minus8 minus6 minus4 minus2 0 2 4 6 8 10minus15

minus10

minus5

0

5

10

15

20

SNR (dB)

Out

put S

INR

(dB)

Optimal SINRCLMS beamformerRCLMS beamformer

Figure 4 Output SINR in a 3∘ mismatch case

the performance of these algorithms versus the SNR in no-mismatch case The performance of these algorithms versusthe SNR in a 3∘ mismatch case is shown in Figure 4

In this example the traditional algorithm is very sensitiveeven to slight mismatches which can easily occur in practicalapplications It is observed from Figure 4 that with theincrease of SNR CLMS algorithm has poor performance atall values of the SNRHowever our proposed recursive robustalgorithm provides improved robustness against signal steer-ing vector mismatches and small training sample size hasfaster convergence rate and yields better output performancethan the CLMS algorithm

Journal of Electrical and Computer Engineering 7

5 Conclusions

In this paper we propose a robust constrained-LMS algo-rithm based on the worst-case SINR maximization TheRCLMS algorithm provides robustness against some types ofmismatches and offers faster convergence rate The updatedweight vector is derived by the gradient descent methodand Lagrange multiplier method in which the diagonalloading factor is obtained in a simple formThis decreases thecomputation cost Some simulation results demonstrate thatthe proposed robust recursive algorithm enjoys better perfor-mance as compared with the traditional CLMS algorithm

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the anonymous reviewersfor their insightful comments that helped improve the qualityof this paper This work is supported by Program for NewCentury Excellent Talents in University (no NCET-12-0103)by the National Natural Science Foundation of China underGrant no 61473066 by the Fundamental Research Funds forthe Central Universities under Grant no N130423005 andby the Natural Science Foundation of Hebei Province underGrant no F2012501044

References

[1] T S Rapapport Ed Smart Antennas Adaptive Arrays Algo-rithms and Wireless Position Location IEEE Piscataway NJUSA 1998

[2] R T Compton Jr R J Huff W G Swarner and A KsienskildquoAdaptive arrays for communication system an overview of theresearch at the Ohio State Universityrdquo IEEE Transactions onAntennas and Propagation vol AP-24 no 5 pp 599ndash607 1976

[3] A B Gershman E Nemeth and J F Bohme ldquoExperimentalperformance of adaptive beamforming in a sonar environmentwith a towed array and moving interfering sourcesrdquo IEEETransactions on Signal Processing vol 48 no 1 pp 246ndash2502000

[4] Y Kaneda and J Ohga ldquoAdaptive microphone-array system fornoise reductionrdquo IEEE Transactions on Acoustics Speech andSignal Processing vol 34 no 6 pp 1391ndash1400 1986

[5] J Li and P Stoica Robust Adaptive Beamforming John Wiley ampSons New York NY USA 2005

[6] R A Monzingo and T W Miller Introduction to AdaptiveArrays Wiley New York NY USA 1980

[7] 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

[8] D D Feldman and L J Griffiths ldquoProjection approach forrobust adaptive beamformingrdquo IEEE Transactions on SignalProcessing vol 42 no 4 pp 867ndash876 1994

[9] Q Zou Z L Yu and Z Lin ldquoA robust algorithm for linearlyconstrained adaptive beamformingrdquo IEEE Signal ProcessingLetters vol 11 no 1 pp 26ndash29 2004

[10] B D Carlson ldquoCovariance matrix estimation errors and diago-nal loading in adaptive arraysrdquo IEEE Transactions on Aerospaceand Electronic Systems vol 24 no 4 pp 397ndash401 1988

[11] J Li P Stoica and Z Wang ldquoOn robust Capon beamformingand diagonal loadingrdquo IEEE Transactions on Signal Processingvol 51 no 7 pp 1702ndash1715 2003

[12] J Li P Stoica and ZWang ldquoDoubly constrained robust Caponbeamformerrdquo IEEE Transactions on Signal Processing vol 52no 9 pp 2407ndash2423 2004

[13] S-J Kim A Magnani A Mutapcic S P Boyd and Z-QLuo ldquoRobust beamforming viaworst-case SINRmaximizationrdquoIEEE Transactions on Signal Processing vol 56 no 4 pp 1539ndash1547 2008

[14] Y W Huang Q Li W-K Ma and S Z Zhang ldquoRobustmulticast beamforming for spectrum sharing-based cognitiveradiosrdquo IEEE Transactions on Signal Processing vol 60 no 1pp 527ndash533 2012

[15] D Ciochina M Pesavento and K M Wong ldquoWorst caserobust downlink beamforming on the Riemannian manifoldrdquoin Proceedings of the 38th IEEE International Conference onAcoustics Speech and Signal Processing (ICASSP rsquo13) pp 3801ndash3805 May 2013

[16] Q Li Q Zhang and J Qin ldquoRobust beamforming for cognitivemulti-antenna relay networkswith bounded channel uncertain-tiesrdquo IEEE Transactions on Communications vol 62 no 2 pp478ndash487 2014

[17] G Zheng S Ma K-K Wong and T-S Ng ldquoRobust beam-forming in cognitive radiordquo IEEE Transactions on WirelessCommunications vol 9 no 2 pp 570ndash576 2010

[18] S C Douglas ldquoWidely-linear recursive least-squares algorithmfor adaptive beamformingrdquo in Proceedings of the IEEE Inter-national Conference on Acoustics Speech and Signal Processing(ICASSP rsquo09) pp 2041ndash2044 IEEE Taipei Taiwan April 2009

[19] E Eweda ldquoComparison of RLS LMS and sign algorithms fortracking randomly time-varying channelsrdquo IEEE Transactionson Signal Processing vol 42 no 11 pp 2937ndash2944 1994

[20] S J Yao H Qian K Kang and M Y Shen ldquoA recursiveleast squares algorithm with reduced complexity for digitalpredistortion linearizationrdquo in Proceedings of the 38th IEEEInternational Conference on Acoustics Speech and Signal Pro-cessing (ICASSP rsquo13) pp 4736ndash4739 May 2013

[21] J Apolinario Jr M L R Campos and P S R DinizldquoConvergence analysis of the binormalized data-reusing LMSalgorithmrdquo IEEE Transactions on Signal Processing vol 48 no11 pp 3235ndash3242 2000

[22] Y-M Shi L Huang C Qian and H C So ldquoShrinkagelinear and widely linear complex-valued least mean squaresalgorithms for adaptive beamformingrdquo IEEE Transactions onSignal Processing vol 63 no 1 pp 119ndash131 2015

[23] T Aboulnasr and K Mayyas ldquoA robust variable step-size LMS-type algorithm analysis and simulationsrdquo IEEE Transactions onSignal Processing vol 45 no 3 pp 631ndash639 1997

[24] L C Godara ldquoApplication of antenna arrays to mobile com-munications part II beam-forming and direction-of-arrivalconsiderationsrdquo Proceedings of the IEEE vol 85 no 8 pp 1195ndash1245 1997

[25] 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

8 Journal of Electrical and Computer Engineering

[26] S Shahbazpanahi A B Gershman Z-Q Lou and KMWongldquoRobust adaptive beamforming for general-rank signalmodelsrdquoIEEE Transactions on Signal Processing vol 51 no 9 pp 2257ndash2269 2003

[27] A Elnashar ldquoEfficient implementation of robust adaptive beam-forming based on worst-case performance optimisationrdquo IETSignal Processing vol 2 no 4 pp 381ndash393 2008

[28] 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

[29] P Stoica J Li X Zhu and J RGuerci ldquoOnusing a priori knowl-edge in space-time adaptive processingrdquo IEEE Transactions onSignal Processing vol 56 no 6 pp 2598ndash2602 2008

[30] O Ledoit and M Wolf ldquoA well-conditioned estimator forlarge-dimensional covariance matricesrdquo Journal of MultivariateAnalysis vol 88 no 2 pp 365ndash411 2004

[31] J Li L Du and P Stoica ldquoFully automatic computation ofdiagonal loading levels for robust adaptive beamformingrdquo inProceedings of the IEEE International Conference on AcousticsSpeech and Signal Processing (ICASSP 08) pp 2325ndash2328 IEEELas Vegas Nev USA April 2008

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

6 Journal of Electrical and Computer Engineering

0 10 20 30 40 50 60 70 80 90 1000

5

10

15

20

25

Number of snapshots

Out

put S

INR

(dB)

CLMS beamformerOptimal SINRRCLMS beamformer

Figure 1 Output SINR in no-mismatch case

0 10 20 30 40 50 60 70 80 90 100minus15

minus10

minus5

0

5

10

15

20

25

Number of snapshots

Out

put S

INR

(dB)

CLMS beamformerOptimal SINRRCLMS beamformer

Figure 2 Output SINR in a 3∘ mismatch case

Figure 2 shows the array output SINR of the methods testedin a 3∘ mismatch

In Figure 2 we note that the output SINR of CLMS algo-rithm is about minus10 dB which is sensitive to the signal steeringvector mismatches However that of the proposed robustCLMS (RCLMS) algorithm is about 18 dB which is close tothe optimal one In this scenario the proposed recursive algo-rithm outperforms the traditional linear constrained-LMSalgorithm Moreover robust constrained-LMS algorithm hasfaster convergence rate

Example 2 (output SINR versus SNR) In this example thereis a 3∘ mismatch in the signal look direction We assume thatthe fixed training data size119873 is equal to 100 Figure 3 displays

minus10 minus8 minus6 minus4 minus2 0 2 4 6 8 10minus5

0

5

10

15

20

SNR (dB)

Out

put S

INR

(dB)

Optimal SINRCLMS beamformerRCLMS beamformer

Figure 3 Output SINR in no-mismatch case

minus10 minus8 minus6 minus4 minus2 0 2 4 6 8 10minus15

minus10

minus5

0

5

10

15

20

SNR (dB)

Out

put S

INR

(dB)

Optimal SINRCLMS beamformerRCLMS beamformer

Figure 4 Output SINR in a 3∘ mismatch case

the performance of these algorithms versus the SNR in no-mismatch case The performance of these algorithms versusthe SNR in a 3∘ mismatch case is shown in Figure 4

In this example the traditional algorithm is very sensitiveeven to slight mismatches which can easily occur in practicalapplications It is observed from Figure 4 that with theincrease of SNR CLMS algorithm has poor performance atall values of the SNRHowever our proposed recursive robustalgorithm provides improved robustness against signal steer-ing vector mismatches and small training sample size hasfaster convergence rate and yields better output performancethan the CLMS algorithm

Journal of Electrical and Computer Engineering 7

5 Conclusions

In this paper we propose a robust constrained-LMS algo-rithm based on the worst-case SINR maximization TheRCLMS algorithm provides robustness against some types ofmismatches and offers faster convergence rate The updatedweight vector is derived by the gradient descent methodand Lagrange multiplier method in which the diagonalloading factor is obtained in a simple formThis decreases thecomputation cost Some simulation results demonstrate thatthe proposed robust recursive algorithm enjoys better perfor-mance as compared with the traditional CLMS algorithm

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the anonymous reviewersfor their insightful comments that helped improve the qualityof this paper This work is supported by Program for NewCentury Excellent Talents in University (no NCET-12-0103)by the National Natural Science Foundation of China underGrant no 61473066 by the Fundamental Research Funds forthe Central Universities under Grant no N130423005 andby the Natural Science Foundation of Hebei Province underGrant no F2012501044

References

[1] T S Rapapport Ed Smart Antennas Adaptive Arrays Algo-rithms and Wireless Position Location IEEE Piscataway NJUSA 1998

[2] R T Compton Jr R J Huff W G Swarner and A KsienskildquoAdaptive arrays for communication system an overview of theresearch at the Ohio State Universityrdquo IEEE Transactions onAntennas and Propagation vol AP-24 no 5 pp 599ndash607 1976

[3] A B Gershman E Nemeth and J F Bohme ldquoExperimentalperformance of adaptive beamforming in a sonar environmentwith a towed array and moving interfering sourcesrdquo IEEETransactions on Signal Processing vol 48 no 1 pp 246ndash2502000

[4] Y Kaneda and J Ohga ldquoAdaptive microphone-array system fornoise reductionrdquo IEEE Transactions on Acoustics Speech andSignal Processing vol 34 no 6 pp 1391ndash1400 1986

[5] J Li and P Stoica Robust Adaptive Beamforming John Wiley ampSons New York NY USA 2005

[6] R A Monzingo and T W Miller Introduction to AdaptiveArrays Wiley New York NY USA 1980

[7] 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

[8] D D Feldman and L J Griffiths ldquoProjection approach forrobust adaptive beamformingrdquo IEEE Transactions on SignalProcessing vol 42 no 4 pp 867ndash876 1994

[9] Q Zou Z L Yu and Z Lin ldquoA robust algorithm for linearlyconstrained adaptive beamformingrdquo IEEE Signal ProcessingLetters vol 11 no 1 pp 26ndash29 2004

[10] B D Carlson ldquoCovariance matrix estimation errors and diago-nal loading in adaptive arraysrdquo IEEE Transactions on Aerospaceand Electronic Systems vol 24 no 4 pp 397ndash401 1988

[11] J Li P Stoica and Z Wang ldquoOn robust Capon beamformingand diagonal loadingrdquo IEEE Transactions on Signal Processingvol 51 no 7 pp 1702ndash1715 2003

[12] J Li P Stoica and ZWang ldquoDoubly constrained robust Caponbeamformerrdquo IEEE Transactions on Signal Processing vol 52no 9 pp 2407ndash2423 2004

[13] S-J Kim A Magnani A Mutapcic S P Boyd and Z-QLuo ldquoRobust beamforming viaworst-case SINRmaximizationrdquoIEEE Transactions on Signal Processing vol 56 no 4 pp 1539ndash1547 2008

[14] Y W Huang Q Li W-K Ma and S Z Zhang ldquoRobustmulticast beamforming for spectrum sharing-based cognitiveradiosrdquo IEEE Transactions on Signal Processing vol 60 no 1pp 527ndash533 2012

[15] D Ciochina M Pesavento and K M Wong ldquoWorst caserobust downlink beamforming on the Riemannian manifoldrdquoin Proceedings of the 38th IEEE International Conference onAcoustics Speech and Signal Processing (ICASSP rsquo13) pp 3801ndash3805 May 2013

[16] Q Li Q Zhang and J Qin ldquoRobust beamforming for cognitivemulti-antenna relay networkswith bounded channel uncertain-tiesrdquo IEEE Transactions on Communications vol 62 no 2 pp478ndash487 2014

[17] G Zheng S Ma K-K Wong and T-S Ng ldquoRobust beam-forming in cognitive radiordquo IEEE Transactions on WirelessCommunications vol 9 no 2 pp 570ndash576 2010

[18] S C Douglas ldquoWidely-linear recursive least-squares algorithmfor adaptive beamformingrdquo in Proceedings of the IEEE Inter-national Conference on Acoustics Speech and Signal Processing(ICASSP rsquo09) pp 2041ndash2044 IEEE Taipei Taiwan April 2009

[19] E Eweda ldquoComparison of RLS LMS and sign algorithms fortracking randomly time-varying channelsrdquo IEEE Transactionson Signal Processing vol 42 no 11 pp 2937ndash2944 1994

[20] S J Yao H Qian K Kang and M Y Shen ldquoA recursiveleast squares algorithm with reduced complexity for digitalpredistortion linearizationrdquo in Proceedings of the 38th IEEEInternational Conference on Acoustics Speech and Signal Pro-cessing (ICASSP rsquo13) pp 4736ndash4739 May 2013

[21] J Apolinario Jr M L R Campos and P S R DinizldquoConvergence analysis of the binormalized data-reusing LMSalgorithmrdquo IEEE Transactions on Signal Processing vol 48 no11 pp 3235ndash3242 2000

[22] Y-M Shi L Huang C Qian and H C So ldquoShrinkagelinear and widely linear complex-valued least mean squaresalgorithms for adaptive beamformingrdquo IEEE Transactions onSignal Processing vol 63 no 1 pp 119ndash131 2015

[23] T Aboulnasr and K Mayyas ldquoA robust variable step-size LMS-type algorithm analysis and simulationsrdquo IEEE Transactions onSignal Processing vol 45 no 3 pp 631ndash639 1997

[24] L C Godara ldquoApplication of antenna arrays to mobile com-munications part II beam-forming and direction-of-arrivalconsiderationsrdquo Proceedings of the IEEE vol 85 no 8 pp 1195ndash1245 1997

[25] 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

8 Journal of Electrical and Computer Engineering

[26] S Shahbazpanahi A B Gershman Z-Q Lou and KMWongldquoRobust adaptive beamforming for general-rank signalmodelsrdquoIEEE Transactions on Signal Processing vol 51 no 9 pp 2257ndash2269 2003

[27] A Elnashar ldquoEfficient implementation of robust adaptive beam-forming based on worst-case performance optimisationrdquo IETSignal Processing vol 2 no 4 pp 381ndash393 2008

[28] 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

[29] P Stoica J Li X Zhu and J RGuerci ldquoOnusing a priori knowl-edge in space-time adaptive processingrdquo IEEE Transactions onSignal Processing vol 56 no 6 pp 2598ndash2602 2008

[30] O Ledoit and M Wolf ldquoA well-conditioned estimator forlarge-dimensional covariance matricesrdquo Journal of MultivariateAnalysis vol 88 no 2 pp 365ndash411 2004

[31] J Li L Du and P Stoica ldquoFully automatic computation ofdiagonal loading levels for robust adaptive beamformingrdquo inProceedings of the IEEE International Conference on AcousticsSpeech and Signal Processing (ICASSP 08) pp 2325ndash2328 IEEELas Vegas Nev USA April 2008

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Journal of Electrical and Computer Engineering 7

5 Conclusions

In this paper we propose a robust constrained-LMS algo-rithm based on the worst-case SINR maximization TheRCLMS algorithm provides robustness against some types ofmismatches and offers faster convergence rate The updatedweight vector is derived by the gradient descent methodand Lagrange multiplier method in which the diagonalloading factor is obtained in a simple formThis decreases thecomputation cost Some simulation results demonstrate thatthe proposed robust recursive algorithm enjoys better perfor-mance as compared with the traditional CLMS algorithm

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the anonymous reviewersfor their insightful comments that helped improve the qualityof this paper This work is supported by Program for NewCentury Excellent Talents in University (no NCET-12-0103)by the National Natural Science Foundation of China underGrant no 61473066 by the Fundamental Research Funds forthe Central Universities under Grant no N130423005 andby the Natural Science Foundation of Hebei Province underGrant no F2012501044

References

[1] T S Rapapport Ed Smart Antennas Adaptive Arrays Algo-rithms and Wireless Position Location IEEE Piscataway NJUSA 1998

[2] R T Compton Jr R J Huff W G Swarner and A KsienskildquoAdaptive arrays for communication system an overview of theresearch at the Ohio State Universityrdquo IEEE Transactions onAntennas and Propagation vol AP-24 no 5 pp 599ndash607 1976

[3] A B Gershman E Nemeth and J F Bohme ldquoExperimentalperformance of adaptive beamforming in a sonar environmentwith a towed array and moving interfering sourcesrdquo IEEETransactions on Signal Processing vol 48 no 1 pp 246ndash2502000

[4] Y Kaneda and J Ohga ldquoAdaptive microphone-array system fornoise reductionrdquo IEEE Transactions on Acoustics Speech andSignal Processing vol 34 no 6 pp 1391ndash1400 1986

[5] J Li and P Stoica Robust Adaptive Beamforming John Wiley ampSons New York NY USA 2005

[6] R A Monzingo and T W Miller Introduction to AdaptiveArrays Wiley New York NY USA 1980

[7] 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

[8] D D Feldman and L J Griffiths ldquoProjection approach forrobust adaptive beamformingrdquo IEEE Transactions on SignalProcessing vol 42 no 4 pp 867ndash876 1994

[9] Q Zou Z L Yu and Z Lin ldquoA robust algorithm for linearlyconstrained adaptive beamformingrdquo IEEE Signal ProcessingLetters vol 11 no 1 pp 26ndash29 2004

[10] B D Carlson ldquoCovariance matrix estimation errors and diago-nal loading in adaptive arraysrdquo IEEE Transactions on Aerospaceand Electronic Systems vol 24 no 4 pp 397ndash401 1988

[11] J Li P Stoica and Z Wang ldquoOn robust Capon beamformingand diagonal loadingrdquo IEEE Transactions on Signal Processingvol 51 no 7 pp 1702ndash1715 2003

[12] J Li P Stoica and ZWang ldquoDoubly constrained robust Caponbeamformerrdquo IEEE Transactions on Signal Processing vol 52no 9 pp 2407ndash2423 2004

[13] S-J Kim A Magnani A Mutapcic S P Boyd and Z-QLuo ldquoRobust beamforming viaworst-case SINRmaximizationrdquoIEEE Transactions on Signal Processing vol 56 no 4 pp 1539ndash1547 2008

[14] Y W Huang Q Li W-K Ma and S Z Zhang ldquoRobustmulticast beamforming for spectrum sharing-based cognitiveradiosrdquo IEEE Transactions on Signal Processing vol 60 no 1pp 527ndash533 2012

[15] D Ciochina M Pesavento and K M Wong ldquoWorst caserobust downlink beamforming on the Riemannian manifoldrdquoin Proceedings of the 38th IEEE International Conference onAcoustics Speech and Signal Processing (ICASSP rsquo13) pp 3801ndash3805 May 2013

[16] Q Li Q Zhang and J Qin ldquoRobust beamforming for cognitivemulti-antenna relay networkswith bounded channel uncertain-tiesrdquo IEEE Transactions on Communications vol 62 no 2 pp478ndash487 2014

[17] G Zheng S Ma K-K Wong and T-S Ng ldquoRobust beam-forming in cognitive radiordquo IEEE Transactions on WirelessCommunications vol 9 no 2 pp 570ndash576 2010

[18] S C Douglas ldquoWidely-linear recursive least-squares algorithmfor adaptive beamformingrdquo in Proceedings of the IEEE Inter-national Conference on Acoustics Speech and Signal Processing(ICASSP rsquo09) pp 2041ndash2044 IEEE Taipei Taiwan April 2009

[19] E Eweda ldquoComparison of RLS LMS and sign algorithms fortracking randomly time-varying channelsrdquo IEEE Transactionson Signal Processing vol 42 no 11 pp 2937ndash2944 1994

[20] S J Yao H Qian K Kang and M Y Shen ldquoA recursiveleast squares algorithm with reduced complexity for digitalpredistortion linearizationrdquo in Proceedings of the 38th IEEEInternational Conference on Acoustics Speech and Signal Pro-cessing (ICASSP rsquo13) pp 4736ndash4739 May 2013

[21] J Apolinario Jr M L R Campos and P S R DinizldquoConvergence analysis of the binormalized data-reusing LMSalgorithmrdquo IEEE Transactions on Signal Processing vol 48 no11 pp 3235ndash3242 2000

[22] Y-M Shi L Huang C Qian and H C So ldquoShrinkagelinear and widely linear complex-valued least mean squaresalgorithms for adaptive beamformingrdquo IEEE Transactions onSignal Processing vol 63 no 1 pp 119ndash131 2015

[23] T Aboulnasr and K Mayyas ldquoA robust variable step-size LMS-type algorithm analysis and simulationsrdquo IEEE Transactions onSignal Processing vol 45 no 3 pp 631ndash639 1997

[24] L C Godara ldquoApplication of antenna arrays to mobile com-munications part II beam-forming and direction-of-arrivalconsiderationsrdquo Proceedings of the IEEE vol 85 no 8 pp 1195ndash1245 1997

[25] 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

8 Journal of Electrical and Computer Engineering

[26] S Shahbazpanahi A B Gershman Z-Q Lou and KMWongldquoRobust adaptive beamforming for general-rank signalmodelsrdquoIEEE Transactions on Signal Processing vol 51 no 9 pp 2257ndash2269 2003

[27] A Elnashar ldquoEfficient implementation of robust adaptive beam-forming based on worst-case performance optimisationrdquo IETSignal Processing vol 2 no 4 pp 381ndash393 2008

[28] 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

[29] P Stoica J Li X Zhu and J RGuerci ldquoOnusing a priori knowl-edge in space-time adaptive processingrdquo IEEE Transactions onSignal Processing vol 56 no 6 pp 2598ndash2602 2008

[30] O Ledoit and M Wolf ldquoA well-conditioned estimator forlarge-dimensional covariance matricesrdquo Journal of MultivariateAnalysis vol 88 no 2 pp 365ndash411 2004

[31] J Li L Du and P Stoica ldquoFully automatic computation ofdiagonal loading levels for robust adaptive beamformingrdquo inProceedings of the IEEE International Conference on AcousticsSpeech and Signal Processing (ICASSP 08) pp 2325ndash2328 IEEELas Vegas Nev USA April 2008

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

8 Journal of Electrical and Computer Engineering

[26] S Shahbazpanahi A B Gershman Z-Q Lou and KMWongldquoRobust adaptive beamforming for general-rank signalmodelsrdquoIEEE Transactions on Signal Processing vol 51 no 9 pp 2257ndash2269 2003

[27] A Elnashar ldquoEfficient implementation of robust adaptive beam-forming based on worst-case performance optimisationrdquo IETSignal Processing vol 2 no 4 pp 381ndash393 2008

[28] 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

[29] P Stoica J Li X Zhu and J RGuerci ldquoOnusing a priori knowl-edge in space-time adaptive processingrdquo IEEE Transactions onSignal Processing vol 56 no 6 pp 2598ndash2602 2008

[30] O Ledoit and M Wolf ldquoA well-conditioned estimator forlarge-dimensional covariance matricesrdquo Journal of MultivariateAnalysis vol 88 no 2 pp 365ndash411 2004

[31] J Li L Du and P Stoica ldquoFully automatic computation ofdiagonal loading levels for robust adaptive beamformingrdquo inProceedings of the IEEE International Conference on AcousticsSpeech and Signal Processing (ICASSP 08) pp 2325ndash2328 IEEELas Vegas Nev USA April 2008

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of