Robust Adaptive Beamforming Based on Worst-Case and Norm ...

Research ArticleRobust Adaptive Beamforming Based on Worst-Caseand Norm Constraint

Hongtao Li1 Ke Wang1 Chaoyu Wang12 Yapeng He3 and Xiaohua Zhu1

1Nanjing University of Science and Technology Nanjing Jiangsu 210094 China2China Shipbuilding Industry Corporation Nanjing Jiangsu 210003 China3China Academy of Space Technology Xirsquoan Shanxi 710000 China

Correspondence should be addressed to Ke Wang xywangkegmailcom

Received 24 October 2014 Accepted 9 December 2014

Academic Editor Giampiero Lovat

Copyright copy 2015 Hongtao Li et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A novel robust adaptive beamforming based on worst-case and norm constraint (RAB-WC-NC) is presented The proposedbeamforming possesses superior robustness against array steering vector (ASV) error with finite snapshots by using the normconstraint and worst-case performance optimization (WCPO) techniques Simulation results demonstrate the validity andsuperiority of the proposed algorithm

1 Introduction

As an important branch of array signal processing adap-tive beamforming technique has achieved a wide range ofapplications in the fields of radar sonar wireless communi-cation radio astronomy and so forth [1] However adaptivebeamforming confronts the problem of intense decrease inrobustness in the cases that array steering vector (ASV) haserrors or receipt signal contains desired signal componentor the ideal covariancematrix is replaced by signal covariancematrix with finite snapshots [2] It can be proven that theerrors caused by using signal covariance matrix could betreated equally as the ASV errors at the circumstance offinite snapshots [3] Therefore the research of beamformingrobustness primarily focuses on the ASV errors and thereceipt signal containing desired signal

In order to improve the adaptability of beamformingagainst those above situations plenty of research on beam-forming robustness has been carried out recently [4ndash16] ESBalgorithm possesses excellent robustness against ASV errorswhile its receipt signal must contain comparatively strongdesired signal and prior information or estimation of thedimension of subspace is demanded [4] Diagonal loadingclass-based adaptive beamforming algorithm possesses cer-tain adaptability to the various situations while it is incapableof retaining the maximum gain to the actual desired signal

and thus the signal to interference plus noise ratio (SINR)will to some extent suffer from loss with ASV errors [5ndash13] Magnitude response constraints-based robust adaptivebeamforming algorithm is blessed with favorable robustnessagainst ASV errors by forming flat response in main beamwhile it demands prior information of main beam width andextra interference along with noise can be involved in therange of main beam [14ndash16]

To solve these problems in this paper a robust adap-tive beamforming algorithm based on the worst-case andnorm constraint (RAB-WC-NC) is proposed RAB-WC-NCalgorithm forms the flat response in the main beam widthdetermined by the uncertainty set of ASV and improves theperformance of beamforming by adopting norm constraintunder the circumstance of finite snapshots The proposedalgorithm can improve the robustness of beamforming andsuppress interference with finite snapshots

2 The Signal Model

Consider a uniform linear array (ULA)with119873 array elementsseparated from each other by a distance 119889 119872 far-field nar-rowband incoherent signals are received from the orientationof (1205791 1205792 120579

119872) and then the receiving data of array x(119905)

can be expressed asx (119905) = As (119905) + n (119905) (1)

Hindawi Publishing CorporationInternational Journal of Antennas and PropagationVolume 2015 Article ID 765385 7 pageshttpdxdoiorg1011552015765385

2 International Journal of Antennas and Propagation

where x(119905) = [1199091(119905) 1199092(119905) 119909

119873(119905)]119879 s(119905) = [119904

1(119905) 1199042(119905)

119904119872(119905)]119879 denote the observation vectors of array and signal

at 119905 separatelyn(119905) = [1198991(119905) 1198992(119905) 119899

119873(119905)]119879 is independent

identically distributed gaussian white noise vector A =

[a(1205791) a(1205792) a(120579

119872)] is the array manifold matrix

a (120579119898) =

1

radic119873

sdot [1 1198901198952120587119889 sin(120579

119898)120582 119890

1198952(119873minus1)120587119889 sin(120579119898)120582]119879

(2)

is the steering vector of signal at orientation 120579119898 Assume that

the signal and noise are unrelatedThe output of array is the weight sum of the observation

signals from each array element The weight vector 120596 = [1205961

1205962 120596

119873]119879 where120596

119896denotes the kth weight coefficient and

the output of array is expressed as

119910 (119905) = 120596119867x (119905) (3)

The output power of array is presented as

119864 119910 (119905) 119910119867(119905) = 120596

119867R119909120596 (4)

where 119864sdot denotes the mathematical expectation and R119909=

119864x(119905)x119867(119905) is the covariance matrix of array snapshotIt is well known that MVDR beamforming minimizes

array output power while constraining the desired signalresponse to be unity That is

min 120596119867R119909120596

subject to 120596119867a (1205790) = 1

(5)

where a(1205790) denotes the presumed ASV of the desired signal

The weight vector of MVDR beamforming algorithm canbe derived from Lagrange multiplier method

120596MVDR =Rminus1119909a (1205790)

a (1205790)119867Rminus1119909a (1205790)

(6)

However the presumed steering vector always deviatesfrom the actual one In this case the performance of theMVDR beamformer is severely limited by target signal can-cellation To maintain a fairly stable gain in the region ofinterest the following inequality constraints on the steeringvector are imposed [6]

min 120596119867R119909120596

subject to 10038161003816100381610038161003816120596119867a (1205790)10038161003816100381610038161003816ge 1 forall

1003817100381710038171003817a (1205790) minus a (1205790)10038171003817100381710038172

le 1205760

(7)

where 1205760denotes the uncertainty factors of ASV and a(120579

0)

denotes the actual ASVWCPOalgorithmcan be achieved by analyzing constraint

condition and making the optimal performance of beam-forming on the worst case

min 120596119867R119909120596

subject to 120596119867a (1205790) ge 1 + 120576

0 1205962

(8)

Equation (8) can be transformed to SOCP problem to besolved by internal point method (IPM) algorithm And thenthis algorithm is deduced further to reduce the computationalcomplexity by general iterative method [6]

3 The Proposed Algorithm

In this section we propose a robust beamformer with theworst case performance optimization and the norm con-straint We can formulate the constrained robust problem as

min 120596119867R119909120596


1003817100381710038171003817a (1205790) minus a (1205790)10038171003817100381710038172

le 1205760

1205962 le 1205770

(9)

Meanwhile the physical meaning of the constraint inWCPO algorithm demonstrated in (7) is to ensure theoutput gain of beamforming no less than unity within theerror range of ASV or in other words to form flat responsewithin the main beam Therefore (9) can be transformed to

min 120596119867R119909120596

subject to 10038161003816100381610038161003816120596119867a (120579) minus 120596119867

119872a (120579)10038161003816100381610038161003816 le 120575 120579 isin [120579

119871 120579119880]

1205962 le 1205770

1003817100381710038171003817a(1205790) minus a(1205790)10038171003817100381710038172

le 1205760

(10)

where [120579119871 120579119880] denotes the main beam width 120575 denotes the

ripple of main beam and 120596119872

denotes the weight vector ofexpected flat response beamforming

From the definition of a(120579119897) we know that

10038171003817100381710038171003817a119867(1205790)100381710038171003817100381710038172

= 1 (11)

Using the 3rd constraint of (10) then

1003817100381710038171003817a(1205790)10038171003817100381710038172

=1003817100381710038171003817(a(1205790) minus a(120579

0)) + a(120579

0)10038171003817100381710038172

le1003817100381710038171003817(a(1205790) minus a(120579

0))10038171003817100381710038172+1003817100381710038171003817a(1205790)

10038171003817100381710038172

le 1205760+ 1

(12)

By using the constraint again hence we have

1 =1003817100381710038171003817a (1205790)

10038171003817100381710038172

=1003817100381710038171003817(a(1205790) minus a(120579

0)) + a(120579

0)10038171003817100381710038172

le1003817100381710038171003817(a(1205790) minus a(120579

0))10038171003817100381710038172+1003817100381710038171003817a(1205790)

10038171003817100381710038172

le 1205760+1003817100381710038171003817a (1205790)

10038171003817100381710038172

(13)

Thus according to (12) and (13) we can get

1 minus 1205760le1003817100381710038171003817a (1205790)

10038171003817100381710038172le 1 + 120576

0 (14)

International Journal of Antennas and Propagation 3

Multiplying the constant a119867(1205790)2in both sides of the

3rd constraint of (10)

10038171003817100381710038171003817a119867 (1205790) a (1205790) minus a119867 (120579

0) a (1205790)100381710038171003817100381710038172

le10038171003817100381710038171003817a119867 (1205790)100381710038171003817100381710038172sdot1003817100381710038171003817a (1205790) minus a (120579

0)10038171003817100381710038172

le10038171003817100381710038171003817a119867 (1205790)100381710038171003817100381710038172sdot 1205760

(15)

Since a119867(1205790)2= 1 then

10038171003817100381710038171003817a119867(1205790)a(1205790) minus 1

100381710038171003817100381710038172le 1205760 (16)

According to the property of vector norm

10038161003816100381610038161003816

10038171003817100381710038171003817a119867(1205790)a(1205790)100381710038171003817100381710038172minus 1

10038161003816100381610038161003816le10038171003817100381710038171003817a119867 (1205790) a (1205790) minus 1

100381710038171003817100381710038172le 1205760 (17)

After simplification

1 minus 1205760le10038171003817100381710038171003817a119867 (1205790) a (1205790)100381710038171003817100381710038172

le 1 + 1205760 (18)

According to the symmetry of signal in both space andfrequency domains if a(120579

0) is treated as a ldquosignalrdquo with 119873

point time sequence then the physical meaning of (14) is thatthe energy interval of signal a(120579

0) is [1minus120576

0 1+1205760] Taking a(120579

0)

as the coefficient of 119873-order filter then the filter is a band-pass one with center frequency being 120579

0 Since a119867(120579

0)2=

1 the maximum gain of the filter is unity Combining thephysical meaning of (14) and (18) the energy interval ofoutput signal is [1 minus 120576

0 1 + 120576

0] after the signal a(120579

0) goes

through band-pass filter of which maximum gain is unityThe position of peak value of ldquosignalrdquo a(120579

0) in frequency

domain corresponds to the position of main beam in spacedomain Consider the worst case when the energy of inputldquosignalrdquo a(120579

0) is 1 + 120576

0while the output energy of signal

through the filter is 1 minus 1205760 So the maximum attenuation of

signal through filter in frequency domain is (1 minus 1205760)(1 + 120576

0)

Assume that the attenuation of filter is equal to or less than(1 minus 120576

0)(1 + 120576

0) and the corresponding frequency range is

[119891119871 119891119880] so the main peak of ldquosignalrdquo a(120579

0)must drop in the

range of [119891119871 119891119880]

Assume that the corresponding angle of ASV of real mainbeam in space domain is 120579

0 then the value interval of sin(120579

0)

is [119891119871 119891119880] according to the symmetry of space domain and

frequency domain Therefore the real range of main beam[120579119871 120579119880] can be easily obtained by solving the arc-sin function

So the parameters of main beamwidth 120579119871and 120579119880 can be

achieved by solving

10038171003817100381710038171003817a119867(1205790)100381710038171003817100381710038172

=(1 minus 120576

0)

(1 + 1205760) (19)

At last amending (10) using the main beam widthinformation obtained from (19) RAB-WC-NC algorithm canbe easily drawn

min 120596119867R119909120596


119872a (120579)10038161003816100381610038161003816 le 120575 120579 isin [120579

119871 120579119880]

1205962 le 1205770

10038171003817100381710038171003817a119867 (120579119894)100381710038171003817100381710038172

=(1 minus 120576

0)

(1 + 1205760) 120579119894= 120579119871 120579119880

(20)

Using the Cholesky decomposition covariance matrix ofarray snapshot can be given as

R119909= V119867V (21)

Hence (4) can be transformed to

120596119867R119909120596 = (V120596)119867 (V120596) = V1205962

2 (22)

Thus (20) can be rewritten as

min 120578


119872a (120579)10038161003816100381610038161003816 le 120575 120579 isin [120579

119871 120579119880]

1205962 le 1205770

V1205962 le 120578

10038171003817100381710038171003817a119867 (120579119894)100381710038171003817100381710038172

=(1 minus 120576

0)

(1 + 1205760) 120579119894= 120579119871 120579119880

(23)

As mentioned before (23) can be solved by IPM algorithm aswell

In conclusion the step of RAB-WC-NC algorithm can begeneralized as follows

Worst-Case and Norm Constraint-Based Robust AdaptiveBeamforming Algorithm

Step 1 Use (19) to calculate the main beam width [120579119871 120579119880]

Step 2 Adopt IPM algorithm to solve (24) to obtain the RAB-WC-NV algorithm weight vector

min 120578


119872a (120579)10038161003816100381610038161003816 le 120575 120579 isin [120579

119871 120579119880]

1205962 le 1205770

V1205962 le 120578

(24)

4 Simulation

In our simulations a ULA of119873 = 32 array elements spaced ahalf wavelength apart is used Desired signal and interferenceare both far-field narrow-band signal the additive noise is


Beam

pat

tern

(dB)

0

0

minus10

minus20

minus20 20

minus30

minus40

minus40 40

minus50

minus60

1205760 = 01

1205760 = 02

1205760 = 03

120579 (∘)

(a) Beam patternsBe

am p

atte

rn (d

B)

0 5 10 15

minus2

minus4

minus6

minus5

minus8

minus10

minus10

minus12

minus14

minus15

minus16

minus18

minus20

1205760 = 01

1205760 = 02

1205760 = 03

120579 (∘)

(b) Main beams

Figure 1 Various beamformersrsquo beam patterns with 20 times 119873 snapshots

modeled as a complex circularly symmetric Gaussian zero-mean spatially and temporally white process RAB-WC-NC algorithm is compared with WCPO norm constraintCapon beamforming (NCCB) SMI and LSMI algorithmwithdifferent snapshots input SNR and errors of ASV

Example 1 (RAB-WC-NC algorithm with different errorupper bonds and sufficient snapshots) We assume that theorientation of desired signal is 0∘ SNR of signal is 10 dB anddirections of arrival (DOA) of two interferences are minus20

∘10∘ separately the INR of which is 40 dB while snapshots

are 20 times 119873 = 640 Figure 1(a) shows the beam patternsof RAB-WC-NC algorithm with different error upper bonds1205760 As illustrated in Figure 1(a) RAB-WC-NC algorithm can

still form effective beams when the receipt signal containscomparatively strong desired signal With the increase of 120576

0

the main beam width in the beam patterns of RAB-WC-NC algorithm widens accordingly and forms nulls in severalinterference points the depth of which can be minus55 dB satis-fying the requirement of interference suppression Figure 1(b)shows the main beam of RAB-WC-NC algorithm

Example 2 (RAB-WC-NC algorithm with different errorupper bonds and finite snapshots) As illustrated inFigure 2(a) with the decrease of snapshots nulls producedby RAB-WC-NC algorithm with different main beam widthsbecome slightly shallower while the depth of them still meetsthe requirement of interference suppression Figure 2(b)shows the main beam of RAB-WC-NC algorithm

Example 3 (beam patterns with high SNR) We assume thatthe orientation of desired signal is 0∘ orientation of actualdesired signal is 03∘ SNR of signal is 30 dB DOAs of twointerferences are minus20

∘ 10∘ separately the INR of which is

40 dB the error upper bond 1205760= 03 and LNR of LSMI

algorithm is 10 dB Figure 3(a) demonstrates the comparisonamong the beam patterns of RAB-WC-NC algorithm NCCBalgorithm SMI algorithm and LSMI algorithm at the snap-shots of 2 times 119873 = 64 As shown in Figure 3(b) the array gainof RAB-WC-NC algorithm at 03∘ loses 002 dB compared toit at 0∘ but the gain of desired signal suffers almost no loss Incontrast the array gain of NCCB algorithm in the directionof 03∘ loses more than 15 dB compared to the maximumone and the SMI algorithm and the LMSI algorithm bothproduce nulls in the direction of 03∘ Figure 3(b) showsthe comparison among the beam patterns of RAB-WC-NCalgorithm NCCB algorithm SMI algorithm and LSMIalgorithm at the snapshots 2 times 119873 = 64 From Figure 3(b) itcan be seen that the depth of nulls of RAB-WC-NC algorithmwith finite snapshots tends to be slightly smaller than it withsufficient snapshots which nevertheless can still meet therequirement of interference suppression NCCB algorithmSMI algorithm and LSMI algorithm all share the phenomenathat the depth of nulls becomes smaller and the sidelobetends to be higher leading to the degradation of performance

Example 4 (output SINR with different error upper bonds)We assume that DOAs of two interferences are minus20

∘ 10∘separately the INR of which is 50 dB SNR of signal is 10 dBthe snapshots are 20 times 119873 = 640 and 500 times of MonteCarlo experiments have been done Figure 4 shows the outputSINR versus different error upper bond 120576

0 of RAB-WC-

NC algorithm WCPO algorithm NCCB algorithm SMIalgorithm and LSMI algorithm From Figure 4 it can be seenthat the output SINR of RAB-WC-NC algorithm remainsbasically unchanged with the increase of error upper bond1205760 which means that RAB-WC-NC algorithm possesses

outstanding adaptability against the ASV errors The output


Beam

pat

tern

(dB)

0

0

minus10

minus20

minus20 20

minus30

minus40

minus40 40

minus50

minus60

1205760 = 01

1205760 = 02

1205760 = 03

120579 (∘)

(a) Beam patternsBe

am p

atte

rn (d

B)

minus10

minus15

minus5

minus20

0minus10 10minus5 5

1205760 = 01

1205760 = 02

1205760 = 03

120579 (∘)

(b) Main beams


0

0

10

20

20 40

RAB-WC-NCNCCB

SMILSMI

Beam

pat

tern

(dB)

minus10

minus20

minus20

minus30

minus40

minus40

minus50

minus60

120579 (∘)

(a) 20 times 119873 snapshots

RAB-WC-NCNCCB

SMILSMI

0

0

10

20

Beam

pat

tern

(dB)

minus10

minus20

minus20

minus30

minus40

minus40

minus50

minus60

20 40

120579 (∘)

(b) 2 times 119873 snapshots

Figure 3 Various beamformersrsquo beam patterns with high SNR

SINR of NCCB algorithm and WCPO algorithm slightlydecrease with the increase of error upper bond 120576

0because

they cannot retain the maximum gain to desired signalDue to the appearance of target signal cancellation theoutput SINR of SMI algorithm and LSMI algorithm decreasedrastically with the increase of error upper bond 120576

0 Due to

the diagonal loading the performance of LSMI algorithmslightly exceeds that of SMI

Example 5 (output SINR with different input SNR) Weassume that DOAs of two interferences are minus10∘ 20∘ sepa-rately the INR of which is 50 dB the DOA of desired signalis 1∘ the snapshots are 20 times 119873 = 640 and 500 times ofMonte Carlo experiments have been done Figure 5 showsthe output SINR versus different input SNR of RAB-WC-NC algorithm WCPO algorithm NCCB algorithm SMIalgorithm and LSMI algorithm From Figure 5 it can be


0 002 004 006 008 01

0

10

20

Out

put S

INR

(dB)

OptimalRABminusWCminusNCNCCB

WCPOSMILSMI

minus40

minus30

minus20

minus10

1205760

Figure 4 Output SINR versus 1205760


WCPOSMILSMI

minus20 minus10 0 10 20 30 40

0

20

40

60

SNR (dB)

Out

put S

INR

(dB)

minus60

minus40

minus20

Figure 5 Output SINR versus input SNR

seen that the performances of the proposed beamformers arealways close to the optimal SINR in a large range fromminus20 dBto 40 dB Furthermore the proposed algorithms enjoy muchfaster convergence rates than others

5 Conclusion

RAB-WC-NC algorithm is deduced specifically to handle thesituationswhereASVhas errors and the receipt data containsdesired signal with finite snapshots The uncertain set ofASV is adopted to determine the main beam width withinwhich the flat response is formed increasing the robustness

of beamforming against ASV error and the performance ofthe proposed algorithm in the situations that receipt datacontains desired signal with finite snapshots is improvedby utilizing norm constraint To sum up RAB-WC-NCalgorithm is blessed with certain adaptability to any kind oferrors and effectively increases the output SINR under thecircumstance of various errors consequently proven to be arobust adaptive beamforming algorithm

Conflict of Interests

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

Acknowledgment

This work was supported by the National Natural ScienceFoundation of China under Grant 61401204

References

[1] J Li and P Stoica Robust Adaptive Beamforming Wiley NewYork NY USA 2005

[2] Y Huang Q Li W-K Ma and S Zhang ldquoRobust multicastbeamforming for spectrum sharing-based cognitive radiosrdquoIEEE Transactions on Signal Processing vol 60 no 1 pp 527ndash533 2012

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

[4] 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 1336ndash1347 1992

[5] H Cox R Zeskind and M Owen ldquoRobust adaptive beam-formingrdquo IEEE Transactions on Acoustic Speech and SignalProcessing vol 35 no 10 pp 1365ndash1376 1987

[6] 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

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

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

[9] 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

[10] YWangH-W Liu B Jiu andX-C Yang ldquoRobust transmittingbeamforming for MIMO radarrdquo Journal of Electronics amp Infor-mation Technology vol 34 no 2 pp 318ndash323 2012

[11] C-Y Chen and P P Vaidyanathan ldquoQuadratically constrainedbeamforming robust against direction-of-arrival mismatchrdquoIEEE Transactions on Signal Processing vol 55 no 8 pp 4139ndash4150 2007

[12] M Rubsamen and M Pesavento ldquoMaximally robust caponbeamformerrdquo IEEETransactions on Signal Processing vol 61 no8 pp 2030ndash2041 2013


[13] B-B Xie L Gan and L-P Li ldquoRobust capon beamformingwithorthomodular constraintrdquo Journal of Electronics and Informa-tion Technology vol 33 no 9 pp 2045ndash2049 2011

[14] Z L Yu W Ser M H Er Z Gu and Y Li ldquoRobust adaptivebeamformers based onworst-case optimization and constraintson magnitude responserdquo IEEE Transactions on Signal Process-ing vol 57 no 7 pp 2615ndash2628 2009

[15] D Xu R He and F Shen ldquoRobust beamforming with magni-tude response constraints and conjugate symmetric constraintrdquoIEEE Communications Letters vol 17 no 3 pp 561ndash564 2013

[16] H-T Li Y-P He X-H Zhu and W Hu ldquoSpectral analysisbased robust adaptive beamforming algorithmrdquoChinese Journalof Radio Science vol 27 no 1 pp 147ndash151 2012

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where x(119905) = [1199091(119905) 1199092(119905) 119909

119873(119905)]119879 s(119905) = [119904

1(119905) 1199042(119905)

119904119872(119905)]119879 denote the observation vectors of array and signal

at 119905 separatelyn(119905) = [1198991(119905) 1198992(119905) 119899

119873(119905)]119879 is independent

identically distributed gaussian white noise vector A =

[a(1205791) a(1205792) a(120579

119872)] is the array manifold matrix

a (120579119898) =

1

radic119873

sdot [1 1198901198952120587119889 sin(120579

119898)120582 119890

1198952(119873minus1)120587119889 sin(120579119898)120582]119879

(2)

is the steering vector of signal at orientation 120579119898 Assume that

the signal and noise are unrelatedThe output of array is the weight sum of the observation

signals from each array element The weight vector 120596 = [1205961

1205962 120596

119873]119879 where120596

119896denotes the kth weight coefficient and

the output of array is expressed as

119910 (119905) = 120596119867x (119905) (3)

The output power of array is presented as

119864 119910 (119905) 119910119867(119905) = 120596

119867R119909120596 (4)

where 119864sdot denotes the mathematical expectation and R119909=

119864x(119905)x119867(119905) is the covariance matrix of array snapshotIt is well known that MVDR beamforming minimizes

array output power while constraining the desired signalresponse to be unity That is

min 120596119867R119909120596

subject to 120596119867a (1205790) = 1

(5)

where a(1205790) denotes the presumed ASV of the desired signal

The weight vector of MVDR beamforming algorithm canbe derived from Lagrange multiplier method

120596MVDR =Rminus1119909a (1205790)

a (1205790)119867Rminus1119909a (1205790)

(6)

However the presumed steering vector always deviatesfrom the actual one In this case the performance of theMVDR beamformer is severely limited by target signal can-cellation To maintain a fairly stable gain in the region ofinterest the following inequality constraints on the steeringvector are imposed [6]

min 120596119867R119909120596


1003817100381710038171003817a (1205790) minus a (1205790)10038171003817100381710038172

le 1205760

(7)

where 1205760denotes the uncertainty factors of ASV and a(120579

0)

denotes the actual ASVWCPOalgorithmcan be achieved by analyzing constraint

condition and making the optimal performance of beam-forming on the worst case

min 120596119867R119909120596

subject to 120596119867a (1205790) ge 1 + 120576

0 1205962

(8)

Equation (8) can be transformed to SOCP problem to besolved by internal point method (IPM) algorithm And thenthis algorithm is deduced further to reduce the computationalcomplexity by general iterative method [6]

3 The Proposed Algorithm

In this section we propose a robust beamformer with theworst case performance optimization and the norm con-straint We can formulate the constrained robust problem as

min 120596119867R119909120596


1003817100381710038171003817a (1205790) minus a (1205790)10038171003817100381710038172

le 1205760

1205962 le 1205770

(9)

Meanwhile the physical meaning of the constraint inWCPO algorithm demonstrated in (7) is to ensure theoutput gain of beamforming no less than unity within theerror range of ASV or in other words to form flat responsewithin the main beam Therefore (9) can be transformed to

min 120596119867R119909120596


119872a (120579)10038161003816100381610038161003816 le 120575 120579 isin [120579

119871 120579119880]

1205962 le 1205770

1003817100381710038171003817a(1205790) minus a(1205790)10038171003817100381710038172

le 1205760

(10)

where [120579119871 120579119880] denotes the main beam width 120575 denotes the

ripple of main beam and 120596119872

denotes the weight vector ofexpected flat response beamforming

From the definition of a(120579119897) we know that

10038171003817100381710038171003817a119867(1205790)100381710038171003817100381710038172

= 1 (11)

Using the 3rd constraint of (10) then

1003817100381710038171003817a(1205790)10038171003817100381710038172

=1003817100381710038171003817(a(1205790) minus a(120579

0)) + a(120579

0)10038171003817100381710038172

le1003817100381710038171003817(a(1205790) minus a(120579

0))10038171003817100381710038172+1003817100381710038171003817a(1205790)

10038171003817100381710038172

le 1205760+ 1

(12)

By using the constraint again hence we have

1 =1003817100381710038171003817a (1205790)

10038171003817100381710038172

=1003817100381710038171003817(a(1205790) minus a(120579

0)) + a(120579

0)10038171003817100381710038172

le1003817100381710038171003817(a(1205790) minus a(120579

0))10038171003817100381710038172+1003817100381710038171003817a(1205790)

10038171003817100381710038172

le 1205760+1003817100381710038171003817a (1205790)

10038171003817100381710038172

(13)

Thus according to (12) and (13) we can get

1 minus 1205760le1003817100381710038171003817a (1205790)

10038171003817100381710038172le 1 + 120576

0 (14)




10038171003817100381710038171003817a119867 (1205790) a (1205790) minus a119867 (120579

0) a (1205790)100381710038171003817100381710038172

le10038171003817100381710038171003817a119867 (1205790)100381710038171003817100381710038172sdot1003817100381710038171003817a (1205790) minus a (120579

0)10038171003817100381710038172

le10038171003817100381710038171003817a119867 (1205790)100381710038171003817100381710038172sdot 1205760

(15)

Since a119867(1205790)2= 1 then

10038171003817100381710038171003817a119867(1205790)a(1205790) minus 1

100381710038171003817100381710038172le 1205760 (16)


10038161003816100381610038161003816

10038171003817100381710038171003817a119867(1205790)a(1205790)100381710038171003817100381710038172minus 1

10038161003816100381610038161003816le10038171003817100381710038171003817a119867 (1205790) a (1205790) minus 1

100381710038171003817100381710038172le 1205760 (17)


1 minus 1205760le10038171003817100381710038171003817a119867 (1205790) a (1205790)100381710038171003817100381710038172

le 1 + 1205760 (18)




0) is [1minus120576

0 1+1205760] Taking a(120579

0)


0 Since a119867(120579

0)2=


0 1 + 120576


0) goes


0) in frequency


0) is 1 + 120576




0)


0)(1 + 120576



0)must drop in the

range of [119891119871 119891119880]



0)




achieved by solving

10038171003817100381710038171003817a119867(1205790)100381710038171003817100381710038172

=(1 minus 120576

0)

(1 + 1205760) (19)


min 120596119867R119909120596


119872a (120579)10038161003816100381610038161003816 le 120575 120579 isin [120579

119871 120579119880]

1205962 le 1205770

10038171003817100381710038171003817a119867 (120579119894)100381710038171003817100381710038172

=(1 minus 120576

0)

(1 + 1205760) 120579119894= 120579119871 120579119880

(20)


R119909= V119867V (21)


120596119867R119909120596 = (V120596)119867 (V120596) = V1205962

2 (22)


min 120578


119872a (120579)10038161003816100381610038161003816 le 120575 120579 isin [120579

119871 120579119880]

1205962 le 1205770

V1205962 le 120578

10038171003817100381710038171003817a119867 (120579119894)100381710038171003817100381710038172

=(1 minus 120576

0)

(1 + 1205760) 120579119894= 120579119871 120579119880

(23)






min 120578


119872a (120579)10038161003816100381610038161003816 le 120575 120579 isin [120579

119871 120579119880]

1205962 le 1205770

V1205962 le 120578

(24)

4 Simulation



Beam

pat

tern

(dB)

0

0

minus10

minus20

minus20 20

minus30

minus40

minus40 40

minus50

minus60

1205760 = 01

1205760 = 02

1205760 = 03

120579 (∘)

(a) Beam patternsBe

am p

atte

rn (d

B)

0 5 10 15

minus2

minus4

minus6

minus5

minus8

minus10

minus10

minus12

minus14

minus15

minus16

minus18

minus20

1205760 = 01

1205760 = 02

1205760 = 03

120579 (∘)

(b) Main beams







0









0 of RAB-WC-




Beam

pat

tern

(dB)

0

0

minus10

minus20

minus20 20

minus30

minus40

minus40 40

minus50

minus60

1205760 = 01

1205760 = 02

1205760 = 03

120579 (∘)

(a) Beam patternsBe

am p

atte

rn (d

B)

minus10

minus15

minus5

minus20

0minus10 10minus5 5

1205760 = 01

1205760 = 02

1205760 = 03

120579 (∘)

(b) Main beams


0

0

10

20

20 40

RAB-WC-NCNCCB

SMILSMI

Beam

pat

tern

(dB)

minus10

minus20

minus20

minus30

minus40

minus40

minus50

minus60

120579 (∘)


RAB-WC-NCNCCB

SMILSMI

0

0

10

20

Beam

pat

tern

(dB)

minus10

minus20

minus20

minus30

minus40

minus40

minus50

minus60

20 40

120579 (∘)




0because


0 Due to




0 002 004 006 008 01

0

10

20

Out

put S

INR

(dB)


WCPOSMILSMI

minus40

minus30

minus20

minus10

1205760



WCPOSMILSMI

minus20 minus10 0 10 20 30 40

0

20

40

60

SNR (dB)

Out

put S

INR

(dB)

minus60

minus40

minus20



5 Conclusion





Acknowledgment


References




















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












10038171003817100381710038171003817a119867 (1205790) a (1205790) minus a119867 (120579

0) a (1205790)100381710038171003817100381710038172

le10038171003817100381710038171003817a119867 (1205790)100381710038171003817100381710038172sdot1003817100381710038171003817a (1205790) minus a (120579

0)10038171003817100381710038172

le10038171003817100381710038171003817a119867 (1205790)100381710038171003817100381710038172sdot 1205760

(15)

Since a119867(1205790)2= 1 then

10038171003817100381710038171003817a119867(1205790)a(1205790) minus 1

100381710038171003817100381710038172le 1205760 (16)


10038161003816100381610038161003816

10038171003817100381710038171003817a119867(1205790)a(1205790)100381710038171003817100381710038172minus 1

10038161003816100381610038161003816le10038171003817100381710038171003817a119867 (1205790) a (1205790) minus 1

100381710038171003817100381710038172le 1205760 (17)


1 minus 1205760le10038171003817100381710038171003817a119867 (1205790) a (1205790)100381710038171003817100381710038172

le 1 + 1205760 (18)




0) is [1minus120576

0 1+1205760] Taking a(120579

0)


0 Since a119867(120579

0)2=


0 1 + 120576


0) goes


0) in frequency


0) is 1 + 120576




0)


0)(1 + 120576



0)must drop in the

range of [119891119871 119891119880]



0)




achieved by solving

10038171003817100381710038171003817a119867(1205790)100381710038171003817100381710038172

=(1 minus 120576

0)

(1 + 1205760) (19)


min 120596119867R119909120596


119872a (120579)10038161003816100381610038161003816 le 120575 120579 isin [120579

119871 120579119880]

1205962 le 1205770

10038171003817100381710038171003817a119867 (120579119894)100381710038171003817100381710038172

=(1 minus 120576

0)

(1 + 1205760) 120579119894= 120579119871 120579119880

(20)


R119909= V119867V (21)


120596119867R119909120596 = (V120596)119867 (V120596) = V1205962

2 (22)


min 120578


119872a (120579)10038161003816100381610038161003816 le 120575 120579 isin [120579

119871 120579119880]

1205962 le 1205770

V1205962 le 120578

10038171003817100381710038171003817a119867 (120579119894)100381710038171003817100381710038172

=(1 minus 120576

0)

(1 + 1205760) 120579119894= 120579119871 120579119880

(23)






min 120578


119872a (120579)10038161003816100381610038161003816 le 120575 120579 isin [120579

119871 120579119880]

1205962 le 1205770

V1205962 le 120578

(24)

4 Simulation



Beam

pat

tern

(dB)

0

0

minus10

minus20

minus20 20

minus30

minus40

minus40 40

minus50

minus60

1205760 = 01

1205760 = 02

1205760 = 03

120579 (∘)

(a) Beam patternsBe

am p

atte

rn (d

B)

0 5 10 15

minus2

minus4

minus6

minus5

minus8

minus10

minus10

minus12

minus14

minus15

minus16

minus18

minus20

1205760 = 01

1205760 = 02

1205760 = 03

120579 (∘)

(b) Main beams







0









0 of RAB-WC-




Beam

pat

tern

(dB)

0

0

minus10

minus20

minus20 20

minus30

minus40

minus40 40

minus50

minus60

1205760 = 01

1205760 = 02

1205760 = 03

120579 (∘)

(a) Beam patternsBe

am p

atte

rn (d

B)

minus10

minus15

minus5

minus20

0minus10 10minus5 5

1205760 = 01

1205760 = 02

1205760 = 03

120579 (∘)

(b) Main beams


0

0

10

20

20 40

RAB-WC-NCNCCB

SMILSMI

Beam

pat

tern

(dB)

minus10

minus20

minus20

minus30

minus40

minus40

minus50

minus60

120579 (∘)


RAB-WC-NCNCCB

SMILSMI

0

0

10

20

Beam

pat

tern

(dB)

minus10

minus20

minus20

minus30

minus40

minus40

minus50

minus60

20 40

120579 (∘)




0because


0 Due to




0 002 004 006 008 01

0

10

20

Out

put S

INR

(dB)


WCPOSMILSMI

minus40

minus30

minus20

minus10

1205760



WCPOSMILSMI

minus20 minus10 0 10 20 30 40

0

20

40

60

SNR (dB)

Out

put S

INR

(dB)

minus60

minus40

minus20



5 Conclusion





Acknowledgment


References




















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Beam

pat

tern

(dB)

0

0

minus10

minus20

minus20 20

minus30

minus40

minus40 40

minus50

minus60

1205760 = 01

1205760 = 02

1205760 = 03

120579 (∘)

(a) Beam patternsBe

am p

atte

rn (d

B)

0 5 10 15

minus2

minus4

minus6

minus5

minus8

minus10

minus10

minus12

minus14

minus15

minus16

minus18

minus20

1205760 = 01

1205760 = 02

1205760 = 03

120579 (∘)

(b) Main beams







0









0 of RAB-WC-




Beam

pat

tern

(dB)

0

0

minus10

minus20

minus20 20

minus30

minus40

minus40 40

minus50

minus60

1205760 = 01

1205760 = 02

1205760 = 03

120579 (∘)

(a) Beam patternsBe

am p

atte

rn (d

B)

minus10

minus15

minus5

minus20

0minus10 10minus5 5

1205760 = 01

1205760 = 02

1205760 = 03

120579 (∘)

(b) Main beams


0

0

10

20

20 40

RAB-WC-NCNCCB

SMILSMI

Beam

pat

tern

(dB)

minus10

minus20

minus20

minus30

minus40

minus40

minus50

minus60

120579 (∘)


RAB-WC-NCNCCB

SMILSMI

0

0

10

20

Beam

pat

tern

(dB)

minus10

minus20

minus20

minus30

minus40

minus40

minus50

minus60

20 40

120579 (∘)




0because


0 Due to




0 002 004 006 008 01

0

10

20

Out

put S

INR

(dB)


WCPOSMILSMI

minus40

minus30

minus20

minus10

1205760



WCPOSMILSMI

minus20 minus10 0 10 20 30 40

0

20

40

60

SNR (dB)

Out

put S

INR

(dB)

minus60

minus40

minus20



5 Conclusion





Acknowledgment


References




















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Beam

pat

tern

(dB)

0

0

minus10

minus20

minus20 20

minus30

minus40

minus40 40

minus50

minus60

1205760 = 01

1205760 = 02

1205760 = 03

120579 (∘)

(a) Beam patternsBe

am p

atte

rn (d

B)

minus10

minus15

minus5

minus20

0minus10 10minus5 5

1205760 = 01

1205760 = 02

1205760 = 03

120579 (∘)

(b) Main beams


0

0

10

20

20 40

RAB-WC-NCNCCB

SMILSMI

Beam

pat

tern

(dB)

minus10

minus20

minus20

minus30

minus40

minus40

minus50

minus60

120579 (∘)


RAB-WC-NCNCCB

SMILSMI

0

0

10

20

Beam

pat

tern

(dB)

minus10

minus20

minus20

minus30

minus40

minus40

minus50

minus60

20 40

120579 (∘)




0because


0 Due to




0 002 004 006 008 01

0

10

20

Out

put S

INR

(dB)


WCPOSMILSMI

minus40

minus30

minus20

minus10

1205760



WCPOSMILSMI

minus20 minus10 0 10 20 30 40

0

20

40

60

SNR (dB)

Out

put S

INR

(dB)

minus60

minus40

minus20



5 Conclusion





Acknowledgment


References




















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 002 004 006 008 01

0

10

20

Out

put S

INR

(dB)


WCPOSMILSMI

minus40

minus30

minus20

minus10

1205760



WCPOSMILSMI

minus20 minus10 0 10 20 30 40

0

20

40

60

SNR (dB)

Out

put S

INR

(dB)

minus60

minus40

minus20



5 Conclusion





Acknowledgment


References




















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation
















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Robust Adaptive Beamforming Based on Worst-Case and Norm ...

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Transcript of Robust Adaptive Beamforming Based on Worst-Case and Norm ...