research article correction of faulty sensors in phased...

Research ArticleCorrection of Faulty Sensors in Phased ArrayRadars Using Symmetrical Sensor Failure Technique andCultural Algorithm with Differential Evolution

S U Khan1 I M Qureshi23 F Zaman4 B Shoaib4 A Naveed4 and A Basit4

1 School of Engineering amp Applied Sciences ISRA University Islamabad 44000 Pakistan2 Electrical Department Air University Islamabad 44000 Pakistan3 Institute of Signals Systems and Soft Computing (ISSS) Islamabad 44000 Pakistan4 Electronic Engineering Department IIU H-10 Islamabad 44000 Pakistan

Correspondence should be addressed to S U Khan shafqatphyyahoocom

Received 30 August 2013 Accepted 17 November 2013 Published 29 January 2014

Academic Editors Z Cui and X Yang

Copyright copy 2014 S U Khan 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

Three issues regarding sensor failure at any position in the antenna array are discussed We assume that sensor position is knownThe issues include raise in sidelobe levels displacement of nulls from their original positions and diminishing of null depth Therequired null depth is achieved by making the weight of symmetrical complement sensor passive A hybrid method based onmemetic computing algorithm is proposedThe hybridmethod combines the cultural algorithmwith differential evolution (CADE)which is used for the reduction of sidelobe levels and placement of nulls at their original positions Fitness function is used tominimize the error between the desired and estimated beam patterns along with null constraints Simulation results for variousscenarios have been given to exhibit the validity and performance of the proposed algorithm

1 Introduction

In adaptive beamforming null steering and beam steering arehot research areas It has direct application in radar sonarand mobile communication [1ndash3] In the literature variousanalytical and computational methods are available to con-centrate on the issue of null steering [4ndash6] The conditionbecomes more demanding and complicated when a sensorfails in the active antenna array The excitation of thesesensors is to accomplish desired radiation pattern In case ofsensor failure the sidelobe level (SLL) raise and nulls are dis-placedwhich is highly unwanted It is very expensive in termsof time and budget to replace the defective sensor regularlyHence the weights of active sensors in the same array shouldbe recalculated and readjusted to create a new pattern close tothe original one Recently few algorithms have been proposedto correct the damaged pattern of the array [7ndash10]

In the last few decades Radar technology has developedvery rapidly The radar commonly used nowadays is known

as phased array radar In this radar the whole input arraytransmits the same signal with different delay and a beam isformed towards the area of interest [11] The advantages ofbeam include the electronic steering instead of mechanicalsteering and a high processing gain at the transmitter Thephased array radar used the phase shifting in the inputwaveform to steer a beam electronically in the direction of thetarget instead of mechanical steering Array design is one ofthe most active research area in phased array radars in whichthe sensors are arranged together to form an arrayThe phaseshifters adjust the phase in such away that a beam is formed inthe desired direction The width of the beam depends on thenumber of sensors in the array By increasing the number ofsensors in an array the beam becomes sharper and thusmoreefficient in detecting the targets with smaller size Now if oneor more sensors become damaged the radars cannot detectthe target correctly Researchers are still working to recal-culate and adjust the weights of the active array to get thepattern near the original one By recalculating the weights of

Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 852539 10 pageshttpdxdoiorg1011552014852539

2 The Scientific World Journal

the faulty array the radar will improve its capabilities in sucha way that the radar can perform searching tracking andweapon guidance at the same time

The evolutionary computing technique has been doingwell in solving numerous problems in search and optimiza-tion due to the impartial nature of their operations whichcan still be present in situations with no domain knowledgeThe search method used by evolutionary algorithms (EAs) isimpartial having no domain knowledge to guide the searchmethod Domain knowledge serves as a method to reducethe search space by pruning unnecessary parts of the solutionspace and by promoting required parts Cultural algorithm isbased on the principle to bias the search method with priorknowledge about the domain aswell as the knowledge duringthe evolution method

Among EC techniques differential evolution (DE) isconsidered to be one of the powerful and reliable tools to opti-mize the problems in any engineering field [12ndash14]The DE isa technique based on stochastic searches in which functionparameters are programmed as floating-point variables TheDE algorithmhas a simple structure convergence speed flex-ibility and robustness with only some parameters required tobe put by a user The application of CAs in DE is a differentstrategy to get the performance and local search better Theprevious work on null steering in failed antenna arrays ispresented in [15] The technique tries to restore the previousnulls pattern by using particle swarm optimization (PSO)All the above EC based techniques have discussed the SLLreduction and null steering in failed array but no one is solv-ing the issue of null depth and null steering at their originalpositions using CADE for the correction of faulty arraysAuthors in their previous work [16] have used the symmet-rical element failure technique to achieve the required nulldepth level and first null beamwidth and [17] for fault findingin failed array antenna Memetic computing algorithmsare stochastic population based methods that have beenestablished to be dominant and forceful to solve optimizationproblems The advantages of cultural algorithm (CA) withevolutionary algorithms (EAs) include global search capabil-ity and consistent performance in any field of engineering andtechnology [18ndash20]

In this paper the proposed algorithm developed threeissues in case of sensor failure These are raised in sidelobelevels displacement of nulls from their original positions anddiminishing of null depth We propose a symmetrical sensorfailure (SSF) method that provides better results in terms ofnull depth Moreover the SSF method has deeper first nullwhich is another big improvement over single sensor failureThe first null depth in beamforming is of great importanceTo address the other two issues we have used a culturalalgorithm with differential evolution (CADE) to reducethe sidelobe levels and positions of nulls reverse to theiroriginal positions by adjusting the current weights of activesensors A hybrid method based on the memetic computingalgorithm is proposed which combines the cultural algo-rithmwith differential evolution (CADE) for the reduction ofsidelobes and placement of nulls Different simulation resultsare provided to confirm the performance of the proposedapproach The rest of the paper is organized as follows

The problem formulation is discussed in Section 2 whilein Section 3 the proposed solution is provided Section 4describes the simulations and the results while Section 5concludes the paper and proposes some future work

2 Problem Formulation

Consider a linear array of 17 sensors in which all the sensorsare placed symmetrically about the origin The total numberof sensors is119873 = 2119872 + 1 The array factor in this healthy setup with uniformly spaced sensors nonuniform weight andprogressive phase excitation will be [21]

AF (120579119894) =

119872

sum

119899=minus119872

119908119899exp (119895119899 (119896119889 cos 120579

119894+ 120573)) (1)

where 119908119899is the nonuniform weight of 119899th sensor whereas

119899 = 0 plusmn1 plusmn2 plusmn119872 The spacing between the adjacentsensors is 119889 while 120579 is the angle from broadside 119896 = 2120587120582

is the wave number with 120582 as wavelength The progressivephase shift120573 = minus119896119889 cos 120579

119904and 120579119904is steering angle for themain

beam The damage array factor for 7th sensor failure is givenby the expression below

AF (120579119894) =

119872

sum

119899=minus119872119899 = 7

119908119899exp (119895119899 (119896119889 cos 120579

119894+ 120573)) (2)

The nonuniform weights of 17 sensors with 7th symmetrysensor failures are as follows

[119908minus8 119908minus7 119908minus6 119908minus5 119908minus4 119908

minus1 1199080

1199081 119908

4 1199085 1199086 1199087 1199088]

(3)

It is assumed that the 1199087sensor fails in the antenna array

given in (3) One can clearly monitor from Figure 1 that dueto single sensor failure the radiation pattern is damaged interms of sidelobe levels null depth and displacement of thenulls from their original position So the goal of this job isto recover the null depth sidelobe levels and null steeringat their original positions Different methods are available inthe literature to correct the damage pattern of sensor failureshowever none of them is able to achieve the required nulldepth level

3 Proposed Solution

In this section we develop the proposed solution based onSSF As we had assumed the damage of 119908

7sensor we lost the

null depth as given in Figure 1 For SSF method we also forcethe sensor 119908

minus7to be zero as shown in (3) From Figure 2 it is

clear that symmetric sensor failure maintains the null depthalmost as close to that of the original arrayThe damage arrayfactor for 7th symmetrical sensor failure is given by

AF (120579119894) =

119872

sum

119899=minus119872119899 = plusmn7

119908119899exp (119895119899 (119896119889 cos 120579

119894+ 120573)) (4)

The Scientific World Journal 3

0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0

Nor

mal

ized

AF

(dB)

Original7th element failure

120579 (deg)

Figure 1 The original Chebyshev array and the 1199087sensor damage

pattern

0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0

Nor

mal

ized

AF

(dB)

Original7th symmetric sensor failure

120579 (deg)

Figure 2 The original Chebyshev array and the 1199087symmetric

sensor failure

Although we have achieved better null depth level due toSSF but the sidelobe levels and positioning of nulls are still aproblem to be taken into account for which we will use thecultural algorithmwith differential evolution (CADE) for thereduction of sidelobes and placement of nulls in the requiredpositions

31 Differential Evolution (DE) DE is an EA and was devel-oped by Storn and Price which is used to solve real valuedoptimization problems [22] The DE is a method basedon stochastic searches The DE algorithm presents easystructure convergence speed flexibility and robustness withonly some parameters required to be set by a user Howeverthis faster convergence of DE results in a higher possibility ofsearching near a local optimum or getting early convergenceDifferential evolution is based on a mutation operator whichadds an amount obtained by the difference of two randomlychosen individuals of the present populationThe problem to

Adjust

Belief space

Influence function

Fitness evaluationPopulation space

Variate population

Selection

Acceptance function

Figure 3 The flow diagram of the cultural algorithm

Table 1 Parameter used in CADE

CADEParameters SettingPopulation size 300Number of generation 500Value of 119865 05Value of CR 05 le CR le 1

be solved has119873 decision variables 119865 and 119862119877 are parametersgiven by the user and given in Table 1 Computing the differ-ence between two individuals which are selected randomlyfrom the population in fact the algorithm estimating the gra-dient in that region and this technique is a proficient way toself-adapt the mutation operatorThe pseudocode for the DEis given in Pseudocode 1

32 The Cultural Algorithm with Differential Evolution(CADE) Differential evolution is used as a population spacein the cultural algorithm CAs have been developed to modelthe evolution of the cultural component of an evolutionarycomputational system over time as it accumulates experienceAs a result CAs can provide a clear method for globalknowledge and a framework within which to model self-adaptation in an evolutionary system

Cultural algorithms consist of three components a pop-ulation space belief space and a communication protocol asshown in Figure 3 First one is population space that containsthe population to be evolved and the mechanisms for its esti-mate The population space consists of a set of possible solu-tions to the problem in our problem the population space isDE Second one is a belief space that represents the bias thathas been acquired by the population during its problem solv-ing process In CAs the information acquired by a memberof the population can be shared with the entire populationThe third one is the communication protocol that is usedto determine the interface between the population and thebeliefs

CAs model has two levels of evolution One is the popu-lation level and the other is belief space level In addition to apopulation space CA has a belief space in which the beliefsacquired from the populationrsquos evolution can be stored and


Generate the initial population of individualsDo

For each individual 119895 in the populationChoose three numbers 119899

1 1198992 and 119899

3that is 1 le 119899

1 1198992 1198993le 119873 with 119899

1= 1198992= 1198993= 119895

Generate a random integer 119894rand isin (1119873)For each parameter 119894

119910119894119892= 1199091198991 119892 + 119865(119909

1198992 119892 minus 1199091198993 119892)

119911119894119892

119895=

119910119894119892

119895119894119891 rand() le CR 119900119903 119895 = 119895rand

119909119894119892

119895otherwise

End ForReplace 119909119894119892 with the child 119911119894119892 if 119911119894119892 is better

End ForUntil the termination condition is achieved

Pseudocode 1 The pseudocode of the differential evolution algorithm

integrated An acceptance function is used to generate beliefsby gleaning the experience of individuals from the populationspace In return this belief space can bias the evolution of thepopulation component by means of the influence functionThe belief space itself also evolves by the adjust function[23] In the present work the belief space is divided into twoknowledge components

321 Situational Knowledge Situational knowledge storesindividuals from population space which provides the direc-tion for other individuals Situational knowledge consists ofthe best example 119875 found along the evolutionary process Itrepresents a leader for the other individuals to follow Thevariation operators of differential evolution are influenced inthe following way

119910119894119892= 119875119894+ 119865 (119909

1198992119892minus 1199091198993119892) (5)

where 119875119894is the 119894th component of the individual stored in the

situational knowledge This way we use the leader instead ofa randomly chosen individual for the recombination gettingthe children closer to the best point found The update of thesituational knowledge is done by replacing the stored individ-ual 119875 by the best individual found in the current population119909best only if 119909best is better than 119875

322 Normative Knowledge The normative knowledge con-tains the intervals for the decision variables where goodsolutions have been found in order to move new solutionstowards those intervalsThe normative knowledge includes ascaling factor 119889119904

1to influence themutation operator adopted

in differential evolution The following expression shows theinfluence of the normative knowledge of the variation opera-tors

119911119894119892

119895=

1199091198991119892+ 119865 (119909

1198992119892minus 1199091198993119892) if 1199091198991 119892 lt 119897

119894

1199091198991119892minus 119865 (119909

1198992119892minus 1199091198993119892) if 1199091198991 119892 gt 119906

119894

1199091198991119892+ (

119906119894minus 119897119894

119889119904119894

) lowast 119865 (1199091198992119892minus 1199091198993119892) otherwise

(6)

where 119897119894and 119906

119894are the lower and upper bounds respectively

for the 119894th decision variable The update of the normativeknowledge can reduce or expand the intervals stored on itAn extension takes place when the accepted individuals donot fit in the current interval while a reduction occurs whenall the accepted individuals lie in the current interval andthe extreme values have an improved fit and are feasible Thevalues 119889119904

119894are updated with the difference (1199091198992119892minus1199091198993119892) found

of the variation operators of the prior generation Normativeknowledge leads individuals to jump into the good rangeif they are not already there The normative knowledge isupdated as follows let us consider 119909

1198861 1199091198862 1199091198863 119909

119886119899accepted

be the accepted individuals in the current generation and119909min

119894

and 119909max119894

belong to (1198861 1198862 1198863 119899accepted) and theaccepted individualswithminimumandmaximumvalues forthe parameter 119894

119906119894=

119909119894max119894

if 119909119894max119894

gt 119906119894or 119891 (119909max

119894

) lt 119880119894

119906119894

otherwise

119897119894=

119909119894min119894

if 119909119894min119894

lt 119897119894or 119891 (119909min

119894

) lt 119871119894

119897119894

otherwise

(7)

If 119897119894and 119906

119894are updated the values of 119871

119894and 119880

119894will be

done in the same way The 119889119904119894are updated with the greatest

difference of |1199091198941199031minus1199091198941199032| found during the variation operators

at the prior generationThe flow chart and pseudocode for CADE is shown in

Pseudocode 2 and Figure 3

33 Null Constraint (NC) A jamming signal located at a par-ticular angle wants to be eliminated in case of satellite radarand mobile communication applications For a uninformedarray to put a null at a particular angle 120579

119894 we want [24]

AF (120579119894) = w119867s (120579

119894) = 0 (8)


Generate the initial populationCalculate the initial populationInitialize the belief spaceDo

For each individual in the populationApply the variation operator influenced by a randomly knowledge componentCalculate the child generatedReplace the individual with child if the child is better

End forUpdate the belief space with the accepted individuals

Until the termination condition is achieved

Pseudocode 2 The pseudocode of the cultural algorithm

where

s (120579119894) =

[[[[[[[[[[[[[[

[

exp (minus119895 (119873 minus 1

2) 119896119889 cos 120579

119894)


2) 119896119889 cos 120579

119894)

exp(119895 (119873 minus 3

2) 119896119889 cos 120579

119894)

exp (119895 (119873 minus 1

2) 119896119889 cos 120579

119894)

]]]]]]]]]]]]]]

]119873times1

(9)

and w119867 is119873 times 1 vector which is defined as

w = [119908minus119872 119908minus119872+1

1199080 119908

119872minus1 119908119872]119879 (10)

The null constraint is given as

w119867s (120579119894) = 0 119894 = 1 2 119872

0 (11)

We may define an119873 times1198720constraint matrix C as

C = [s (1205791) s (120579

2) s (120579

1198720

)] (12)

where 120579119894for 119894 = 1 2 119872

0is the direction of null Our

goal is to optimize the squared weighting error subject to thecondition that

w119867C = 0 (13)Our constraint is that the columns of C should be

orthogonal to theweight vectorw Accordinglywemaydefine119866119894 119894 = 1 2 and 119866 as follows

1198661=

119875

sum

119894=1

[1003816100381610038161003816AF119889(120579119894) minus AFCADE(120579119894)

1003816100381610038161003816]2 (14)

1198662=10038171003817100381710038171003817w119867C10038171003817100381710038171003817

2

(15)

119866 = 1198661+ 1198662 (16)

Hence 119866 is the fitness function for the problem givenabove which are to be minimized Best chromosome will givethe minimal value of 119866 The first term in (16) is used forSLL reduction where AF

119889(120579119894) represents the desired pattern

and AFCADE(120579119894) is the pattern obtained by using CADE Thesecond term in (16) is used for jammer suppression and place-ment of nulls at their original positions after sensor failure

4 Simulation Results

In the simulation a classical Dolph-Chebyshev linear arrayof 17 sensors with 1205822 intersensor spacing is used as the testantenna The array factor in this case represents a minus35 dBconstant SLL with the nulls at specific angles Analyticaltechniques are used to find out the nonuniform weights forclassical Dolph-Chebyshev array In case of sensor failurecultural algorithmwith differential evolution (CADE) is usedfor the reduction of sidelobes and placement of nulls in therequired positions

Case a At the first instant the 1199087sensor is assumed to fail

After sensor failure the radiation pattern is destroyed whichresults in an increase of the SLL and displacement of nullpositions In order to regain the symmetry its mirror sensorweight 119908

minus7is forced to zero We achieve the desired null

depth level (NDL) and deeper first null depth level (FNDL)as compared to that of nonsymmetric case The SLL rises tominus3232 dB due to the 119908

7sensor failure while due to SSF of

the 1199087sensor the SLL is minus2653 dB The advantage of SSF is

deeper nulls especially the first null The SLL and FNDL fora damage array of single sensor failure and SSF are shownin Table 2 It is clear from Figure 2 that SSF maintains betterFNDL as compared to that of single sensor failure

After optimization by a cultural algorithm with differen-tial evolution (CADE) the SLL of the 119908

7sensor failure are

reduced to minus3299 dB while due to SSF the SLL is reduced tominus2835 dB The recovery of one null due to 7th sensor failureand SSF to its original position 120579

1= 1993

∘ is shown inFigures 4 and 5The comparison of recovered patternwith 7thsensor and SSF for one null imposed is given in Table 3 Therecovered NDL of SSF is seven dB deeper than that of 7thsensor failure

Figures 6 and 7 show the recovery of two nulls at anglesof 1205791= 1993

∘ and 1205792= 3488

∘ respectively for 7th sensorfailure and SSF The SLL and NDL for the correspondingnulls are given in Table 4 The NDL of the SSF is deeper ascompared to the 7th sensor failure

Now we check the recovery of three nulls at the requiredpositions Figures 8 and 9 show the recovery of three nullsoriginally at angles of 120579

1= 1993

∘ 1205792= 3488

∘ and 1205793=

4544∘ for 7th sensor failure and SSF The SLL and NDL for


Table 2 Comparison of FNDL and SLL of the damaged pattern

Comparison of FNDL and SLL of damage pattern of 7th sensor and SSF7th sensor failure SSF

FNDL (dB) SLL (dB) FNDL (dB) SLL (dB)minus3232 minus2945 minus8898 minus2653

Table 3 Recovery of one null

Comparison of NDL and SLL of one sensor failure and SSFCorrection of 7th sensor failure Correction of SSF Recovery of nulls

NDL (dB) SLL (dB) NDL (dB) SLL (dB)minus1047 minus3299 minus1115 minus2835 1st null recovered

0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0

Nor

mal

ized

AF

(dB)

Original7th element failure1 null recovered

120579 (deg)

Figure 4The original radiation pattern the1199087sensor damage and

recovery of one null

0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0

Nor

mal

ized

AF

(dB)

Original7th SSF1 nulls recovered

120579 (deg)

Figure 5The original radiation pattern the1199087SSF and recovery of

one null

0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0

Nor

mal

ized

AF

(dB)

Original7th element failure2 nulls recovered

120579 (deg)


recovery of two nulls

0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0

Nor

mal

ized

AF

(dB)


120579 (deg)


two nulls


Table 4 Recovery of two nulls

Comparison of NDL and SLL of 7th sensor failure and SSFCorrection of 7th sensor failure Correction of SSF Recovery of nulls

NDL (dB) SLL (dB) NDL (dB) SLL (dB)minus953 minus2987 minus1151 minus3112 1st null recoveredminus9365 minus3075 minus1014 2707 2nd null recovered

0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0

Nor

mal

ized

AF

(dB)


120579 (deg)


recovery of three nulls

0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0

Nor

mal

ized

AF

(dB)


120579 (deg)

Figure 9 The original radiation pattern the 1199087SSF and recovery

of three nulls

the corresponding nulls are given in Table 5 The NDL of allnulls in SSF is deeper than that of 7th sensor failure

Now the recovery of five nulls for 7th sensor failure andSSF originally at positions 1993∘ 3488∘ 4544∘6202∘ and6894∘ is carried out and shown in Figures 10 and 11 A

comparison of SLL and NDL for the recovery of five nullsis given in Table 6 In each case SSF produces deeper nulls

0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0

Nor

mal

ized

AF

(dB)


120579 (deg)


recovery of five nulls

compared to the 7th sensor failure From simulation it isobserved that we have received deeper depth of nulls in SSFscenarios as compared to 7th sensor failure discussed above

Case b In this case we discuss the failure of 1199084sensor If the

sensor1199084fails due to any reason the whole radiation pattern

became damage After optimization the SLL reduces andnulls are steered back to their original positions as shown inFigure 12Then to create symmetry we also force119908

minus4equal to

zero to achieve the requirednull depth levelThe advantage ofsymmetry sensor failure is to get deeper null depth level Thenumber of nulls achieved in 7th symmetry sensor failure issix and in case of 4th symmetry sensor failure the number ofnulls received are three From the simulation results it is clearthat the number of nulls reduces by one as the sensors getdamage near the centre sensor After optimization by CADEthe SLL reduces and nulls are steered back to their previouspositions at angles 120579

1= 3489

∘ 1205792= 5431

∘ and 1205793= 689

∘ asshown in Figure 13 The null depth level for single and SEF isgiven in Table 7

Case c In this section we discuss the possibility of gettingfailure near the centre sensor if the sensor 119908

1fails due to

unforeseen reason From Figure 14 it is clear that its SLLincreases and nulls are damaged and also lose null depth Tocreate the symmetry we also force its mirror sensor weight119908minus1

equal to zero The one advantage of symmetry sensorfailure is to get the deeper null depth level and on the otherhand due to 119908

1SSF its beamwidth also decreases In case of


Table 5 Recovery of three nulls


NDL (dB) SLL (dB) NDL (dB) SLL (dB)minus1116 minus3632 minus1185 minus3459 1st null recoveredminus9757 minus3492 minus1056 minus3268 2nd null recoveredminus9501 minus281 minus1033 minus2419 3rd null recovered

Table 6 Recovery of five nulls


NDL (dB) SLL (dB) NDL (dB) SLL (dB)minus1127 minus3734 minus120 minus378 1st null recoveredminus9816 minus3531 minus1084 minus3493 2nd null recoveredminus9533 minus249 minus1053 minus2291 3rd null recoveredminus946 minus3642 minus1025 minus3227 4th null recoveredminus9739 minus2269 minus9968 minus25832 5th null recovered

0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0

Nor

mal

ized

AF

(dB)


120579 (deg)


of five nulls

7th symmetry sensor failure the nulls are six and in 4th SSFthe achievable nulls are three but we received only one null incase of 119908

1SSF as shown in Figure 15 The null depth level for

single and SEF is given in Table 8

Case d The main beam can be steered at any desired angleIf the user changes their position than the main beam can besteered in the desired direction Figure 16 shows the correctedpattern with recovered nulls at main beam pointing at 120579

119904=

110∘ The main beam can be steered in the direction of the

desired user at any particular angleThe array factor for 2119872+

1 sensors in terms of main beam direction 120579119904is given by

AF (120579119894) =

119872

sum

119899=minus119872

119908119899exp 119895119899119896119889 (cos 120579

119894minus cos 120579

119904) (17)

0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0N

orm

aliz

ed A

F (d

B)


120579 (deg)

Figure 12 The original radiation pattern the 1199084sensor failure and


where 120579119904is themain beam direction to which it can be steered

to the desired angles

5 Conclusion and Future Work

We have proposed symmetric sensor failure (SSF) techniquealong cultural algorithm with differential evolution for thecorrection of faulty arrays The null depth of all nullsespecially the first one has been achieved with the help ofSSF technique Null steering at their original positions andsidelobe reduction has been achieved by a cultural algorithmwith differential evolution and using a proper fitness functiondemanding the sidelobe reduction and null constraints Dueto 7th SSF the number of nulls is six and in 4th SSF theachievable nulls are three but in case of 119908

1SSF we received





Table 8 Recovery of one nulls

Comparison of NDL and SLL of 1199081sensor failure and SSF

Correction of 1199081sensor failure Correction of SSF Recovery of nulls


0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0

Nor

mal

ized

AF

(dB)


120579 (deg)


of three nulls

0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0

Nor

mal

ized

AF

(dB)

Original

1 null recovered

120579 (deg)

w1 element failure



0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0

Nor

mal

ized

AF

(dB)

Original

1 null recovered

120579 (deg)

w1 SSF


of one null

0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0

Nor

mal

ized

AF

(dB)

Original7th SSFRecovered nulls

120579 (deg)

Figure 16 The corrected pattern with main beam pointing at 110∘with recovered nulls


only one null with reduced beamwidthThe simulation resultshows that as the faulty sensor gets near the central sensorthe number of nulls reduces by oneThe reduction in the cor-rected sidelobe level comes at the cost of broader main beamThe corrected pattern has beamwidth broader than that of theoriginal one Using the approach of symmetric sensor failurewith the reduction of the SLL we can steer single doubleand multiple nulls in the direction of known interferencesThis method can be extended to planar arrays

Conflict of Interests

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

References

[1] S Applebaum ldquoAdaptive arraysrdquo IEEE Transactions on Anten-nas and Propagation vol 24 no 5 pp 585ndash598 1976

[2] J A Hejres ldquoNull steering in phased arrays by controlling thepositions of selected elementsrdquo IEEE Transactions on Antennasand Propagation vol 52 no 11 pp 2891ndash2895 2004

[3] F Zaman I M Qureshi J A Khan and Z U Khan ldquoAnapplication of artificial intelligence for the joint estimation ofamplitude and two-dimensional direction of arrival of far fieldsources using 2-L-shape arrayrdquo International Journal of Anten-nas and Propagation vol 2013 Article ID 593247 10 pages 2013

[4] F Zaman ldquoAmplitude and directional of arrival estimationcomparison between different techniquesrdquo Progress in Electro-magnetics Research B vol 39 pp 319ndash335 2012

[5] J A Hejres A Peng and J Hijres ldquoFast method for sidelobenulling in a partially adaptive linear array using the elementspositionsrdquo IEEE Antennas andWireless Propagation Letters vol6 pp 332ndash335 2007


[7] T J Peters ldquoA conjugate gradient-based algorithm to minimizethe sidelobe level of planar arrays with element failuresrdquo IEEETransactions on Antennas and Propagation vol 39 no 10 pp1497ndash1504 1991

[8] R J Mailloux ldquoArray failure correction with a digitally beam-formed arrayrdquo IEEE Transactions on Antennas and Propagationvol 44 no 12 pp 1543ndash1550 1996

[9] K Guney and S Basbug ldquoInterference suppression of linearantenna arrays by amplitude-only control using a bacterialforaging algorithmrdquo Progress in Electromagnetics Research vol79 pp 475ndash497 2008

[10] K Guney and M Onay ldquoAmplitude-only pattern nulling oflinear antenna arrays with the use of bees algorithmrdquo Progress inElectromagnetics Research vol 70 pp 21ndash36 2007

[11] H Li and B Himed ldquoTransmit subaperturing forMIMO radarswith co-located antennasrdquo IEEE Journal on Selected Topics inSignal Processing vol 4 no 1 pp 55ndash65 2010

[12] Y W Zhong L J Wang C Y Wang and H Zhang ldquoMulti-agent simulated annealing algorithm on differential evolutionalgorithmrdquo International Journal of Bio-Inspired Computationvol 4 no 4 pp 217ndash228 2012

[13] T J Choi C W Ahn and J An ldquoAn adaptive Cauchy differ-ential evolution algorithm for global numerical optimizationrdquo

The Scientific World Journal vol 2013 Article ID 969734 12pages 2013

[14] Y Wang Z Cai and Q Zhang ldquoDifferential evolution withcomposite trial vector generation strategies and control param-etersrdquo IEEE Transactions on Evolutionary Computation vol 15no 1 pp 55ndash66 2011

[15] O P Acharya A Patnaik and S N Sinha ldquoNull steering infailed antenna arraysrdquo Applied Computational Intelligence andSoft Computing vol 2011 Article ID 692197 9 pages 2011

[16] S U Khan I M Qureshi F Zaman and A Naveed ldquoNullplacement and sidelobe suppression in failed array usingsymmetrical element failure technique and hybrid heuristiccomputationrdquo Progress in Electromagnetics Research B vol 52pp 165ndash184 2013

[17] S U Khan I M Qureshi F Zaman A Basit and W KhanldquoApplication of firefly algorithm to fault finding in linear arraysantennardquoWorld Applied Sciences Journal vol 26 no 2 pp 232ndash238 2013

[18] R G Reynolds ldquoAn introduction to cultural algorithmsrdquo inEvolutionary Programming Proceedings of the Third AnnualConference A V Sebald and L J Fogel Eds pp 131ndash139 WorldScientific Publishing River Edge NJ USA 1994

[19] R G Reynolds and C Chung ldquoThe use of cultural algorithmsto evolve multi agent cooperationrdquo in Proceedings of the Micro-RobotWorld Cup Soccer Tournament pp 53ndash56 Taejon Repub-lic of Korea 1996

[20] X Jin and R G Reynolds ldquoUsing knowledge-based evolu-tionary computation to solve nonlinear constraint optimizationproblems a cultural algorithm approachrdquo in Proceedings ofthe Congress on Evolutionary Computation pp 1672ndash1678Washington DC USA 1999

[21] I Wolf ldquoDetermination of the radiating system which willproduce a specified directional characteristicrdquoProceedings of theInstitute of Radio Engineers vol 25 pp 630ndash643 1937

[22] R Storn and K Price ldquoDifferential evolution a simple and effi-cient adaptive scheme for global optimization over continuousspacesrdquo Tech Rep TR-95-012 International Computer ScienceInstitute Berkeley Calif USA 1995

[23] R L Becerra and C A C Coello ldquoCultured differentialevolution for constrained optimizationrdquo Computer Methods inApplied Mechanics and Engineering vol 195 no 33ndash36 pp4303ndash4322 2006

[24] H Steyskal R A Shore and R L Haupt ldquoMethods for nullcontrol and their effects on the radiation patternrdquo IEEETransac-tions on Antennas and Propagation vol 34 no 3 pp 404ndash4091986

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Distributed Sensor Networks


Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014


ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014


Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications


Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia


Biomedical Imaging


ArtificialNeural Systems

Advances in


RoboticsJournal of



Computational Intelligence and Neuroscience

Industrial EngineeringJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in



the faulty array the radar will improve its capabilities in sucha way that the radar can perform searching tracking andweapon guidance at the same time

The evolutionary computing technique has been doingwell in solving numerous problems in search and optimiza-tion due to the impartial nature of their operations whichcan still be present in situations with no domain knowledgeThe search method used by evolutionary algorithms (EAs) isimpartial having no domain knowledge to guide the searchmethod Domain knowledge serves as a method to reducethe search space by pruning unnecessary parts of the solutionspace and by promoting required parts Cultural algorithm isbased on the principle to bias the search method with priorknowledge about the domain aswell as the knowledge duringthe evolution method

Among EC techniques differential evolution (DE) isconsidered to be one of the powerful and reliable tools to opti-mize the problems in any engineering field [12ndash14]The DE isa technique based on stochastic searches in which functionparameters are programmed as floating-point variables TheDE algorithmhas a simple structure convergence speed flex-ibility and robustness with only some parameters required tobe put by a user The application of CAs in DE is a differentstrategy to get the performance and local search better Theprevious work on null steering in failed antenna arrays ispresented in [15] The technique tries to restore the previousnulls pattern by using particle swarm optimization (PSO)All the above EC based techniques have discussed the SLLreduction and null steering in failed array but no one is solv-ing the issue of null depth and null steering at their originalpositions using CADE for the correction of faulty arraysAuthors in their previous work [16] have used the symmet-rical element failure technique to achieve the required nulldepth level and first null beamwidth and [17] for fault findingin failed array antenna Memetic computing algorithmsare stochastic population based methods that have beenestablished to be dominant and forceful to solve optimizationproblems The advantages of cultural algorithm (CA) withevolutionary algorithms (EAs) include global search capabil-ity and consistent performance in any field of engineering andtechnology [18ndash20]

In this paper the proposed algorithm developed threeissues in case of sensor failure These are raised in sidelobelevels displacement of nulls from their original positions anddiminishing of null depth We propose a symmetrical sensorfailure (SSF) method that provides better results in terms ofnull depth Moreover the SSF method has deeper first nullwhich is another big improvement over single sensor failureThe first null depth in beamforming is of great importanceTo address the other two issues we have used a culturalalgorithm with differential evolution (CADE) to reducethe sidelobe levels and positions of nulls reverse to theiroriginal positions by adjusting the current weights of activesensors A hybrid method based on the memetic computingalgorithm is proposed which combines the cultural algo-rithmwith differential evolution (CADE) for the reduction ofsidelobes and placement of nulls Different simulation resultsare provided to confirm the performance of the proposedapproach The rest of the paper is organized as follows

The problem formulation is discussed in Section 2 whilein Section 3 the proposed solution is provided Section 4describes the simulations and the results while Section 5concludes the paper and proposes some future work

2 Problem Formulation

Consider a linear array of 17 sensors in which all the sensorsare placed symmetrically about the origin The total numberof sensors is119873 = 2119872 + 1 The array factor in this healthy setup with uniformly spaced sensors nonuniform weight andprogressive phase excitation will be [21]

AF (120579119894) =

119872

sum

119899=minus119872

119908119899exp (119895119899 (119896119889 cos 120579

119894+ 120573)) (1)

where 119908119899is the nonuniform weight of 119899th sensor whereas

119899 = 0 plusmn1 plusmn2 plusmn119872 The spacing between the adjacentsensors is 119889 while 120579 is the angle from broadside 119896 = 2120587120582

is the wave number with 120582 as wavelength The progressivephase shift120573 = minus119896119889 cos 120579

119904and 120579119904is steering angle for themain

beam The damage array factor for 7th sensor failure is givenby the expression below

AF (120579119894) =

119872

sum

119899=minus119872119899 = 7

119908119899exp (119895119899 (119896119889 cos 120579

119894+ 120573)) (2)

The nonuniform weights of 17 sensors with 7th symmetrysensor failures are as follows

[119908minus8 119908minus7 119908minus6 119908minus5 119908minus4 119908

minus1 1199080

1199081 119908

4 1199085 1199086 1199087 1199088]

(3)

It is assumed that the 1199087sensor fails in the antenna array

given in (3) One can clearly monitor from Figure 1 that dueto single sensor failure the radiation pattern is damaged interms of sidelobe levels null depth and displacement of thenulls from their original position So the goal of this job isto recover the null depth sidelobe levels and null steeringat their original positions Different methods are available inthe literature to correct the damage pattern of sensor failureshowever none of them is able to achieve the required nulldepth level

3 Proposed Solution

In this section we develop the proposed solution based onSSF As we had assumed the damage of 119908

7sensor we lost the

null depth as given in Figure 1 For SSF method we also forcethe sensor 119908

minus7to be zero as shown in (3) From Figure 2 it is

clear that symmetric sensor failure maintains the null depthalmost as close to that of the original arrayThe damage arrayfactor for 7th symmetrical sensor failure is given by

AF (120579119894) =

119872

sum

119899=minus119872119899 = plusmn7

119908119899exp (119895119899 (119896119889 cos 120579

119894+ 120573)) (4)


0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0

Nor

mal

ized

AF

(dB)


120579 (deg)


pattern

0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0

Nor

mal

ized

AF

(dB)


120579 (deg)


sensor failure



Adjust

Belief space

Influence function


Variate population

Selection

Acceptance function











1 1198992 and 119899


1 1198992 1198993le 119873 with 119899

1= 1198992= 1198993= 119895


119910119894119892= 1199091198991 119892 + 119865(119909

1198992 119892 minus 1199091198993 119892)

119911119894119892

119895=

119910119894119892

119895119894119891 rand() le CR 119900119903 119895 = 119895rand

119909119894119892

119895otherwise






119910119894119892= 119875119894+ 119865 (119909

1198992119892minus 1199091198993119892) (5)






119911119894119892

119895=

1199091198991119892+ 119865 (119909

1198992119892minus 1199091198993119892) if 1199091198991 119892 lt 119897

119894

1199091198991119892minus 119865 (119909

1198992119892minus 1199091198993119892) if 1199091198991 119892 gt 119906

119894

1199091198991119892+ (

119906119894minus 119897119894

119889119904119894


(6)

where 119897119894and 119906





1198861 1199091198862 1199091198863 119909



119894

and 119909max119894


119906119894=

119909119894max119894

if 119909119894max119894

gt 119906119894or 119891 (119909max

119894

) lt 119880119894

119906119894

otherwise

119897119894=

119909119894min119894

if 119909119894min119894

lt 119897119894or 119891 (119909min

119894

) lt 119871119894

119897119894

otherwise

(7)

If 119897119894and 119906


119894and 119880

119894will be






119894 we want [24]

AF (120579119894) = w119867s (120579

119894) = 0 (8)







where

s (120579119894) =

[[[[[[[[[[[[[[

[


2) 119896119889 cos 120579

119894)


2) 119896119889 cos 120579

119894)

exp(119895 (119873 minus 3

2) 119896119889 cos 120579

119894)

exp (119895 (119873 minus 1

2) 119896119889 cos 120579

119894)

]]]]]]]]]]]]]]

]119873times1

(9)


w = [119908minus119872 119908minus119872+1

1199080 119908

119872minus1 119908119872]119879 (10)


w119867s (120579119894) = 0 119894 = 1 2 119872

0 (11)


C = [s (1205791) s (120579

2) s (120579

1198720

)] (12)

where 120579119894for 119894 = 1 2 119872





1198661=

119875

sum

119894=1

[1003816100381610038161003816AF119889(120579119894) minus AFCADE(120579119894)

1003816100381610038161003816]2 (14)

1198662=10038171003817100381710038171003817w119867C10038171003817100381710038171003817

2

(15)

119866 = 1198661+ 1198662 (16)














7sensor failure are


1= 1993



∘ and 1205792= 3488



1= 1993

∘ 1205792= 3488

∘ and 1205793=









0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0

Nor

mal

ized

AF

(dB)


120579 (deg)



0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0

Nor

mal

ized

AF

(dB)


120579 (deg)


one null

0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0

Nor

mal

ized

AF

(dB)


120579 (deg)



0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0

Nor

mal

ized

AF

(dB)


120579 (deg)


two nulls





0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0

Nor

mal

ized

AF

(dB)


120579 (deg)



0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0

Nor

mal

ized

AF

(dB)


120579 (deg)


of three nulls




0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0

Nor

mal

ized

AF

(dB)


120579 (deg)







minus4equal to


1= 3489

∘ 1205792= 5431

∘ and 1205793= 689



1fails due to











0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0

Nor

mal

ized

AF

(dB)


120579 (deg)


of five nulls





119904=




AF (120579119894) =

119872

sum

119899=minus119872

119908119899exp 119895119899119896119889 (cos 120579

119894minus cos 120579

119904) (17)

0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0N

orm

aliz

ed A

F (d

B)


120579 (deg)







1SSF we received









0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0

Nor

mal

ized

AF

(dB)


120579 (deg)


of three nulls

0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0

Nor

mal

ized

AF

(dB)

Original

1 null recovered

120579 (deg)

w1 element failure



0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0

Nor

mal

ized

AF

(dB)

Original

1 null recovered

120579 (deg)

w1 SSF


of one null

0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0

Nor

mal

ized

AF

(dB)


120579 (deg)






References

































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0

Nor

mal

ized

AF

(dB)


120579 (deg)


pattern

0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0

Nor

mal

ized

AF

(dB)


120579 (deg)


sensor failure



Adjust

Belief space

Influence function


Variate population

Selection

Acceptance function











1 1198992 and 119899


1 1198992 1198993le 119873 with 119899

1= 1198992= 1198993= 119895


119910119894119892= 1199091198991 119892 + 119865(119909

1198992 119892 minus 1199091198993 119892)

119911119894119892

119895=

119910119894119892

119895119894119891 rand() le CR 119900119903 119895 = 119895rand

119909119894119892

119895otherwise






119910119894119892= 119875119894+ 119865 (119909

1198992119892minus 1199091198993119892) (5)






119911119894119892

119895=

1199091198991119892+ 119865 (119909

1198992119892minus 1199091198993119892) if 1199091198991 119892 lt 119897

119894

1199091198991119892minus 119865 (119909

1198992119892minus 1199091198993119892) if 1199091198991 119892 gt 119906

119894

1199091198991119892+ (

119906119894minus 119897119894

119889119904119894


(6)

where 119897119894and 119906





1198861 1199091198862 1199091198863 119909



119894

and 119909max119894


119906119894=

119909119894max119894

if 119909119894max119894

gt 119906119894or 119891 (119909max

119894

) lt 119880119894

119906119894

otherwise

119897119894=

119909119894min119894

if 119909119894min119894

lt 119897119894or 119891 (119909min

119894

) lt 119871119894

119897119894

otherwise

(7)

If 119897119894and 119906


119894and 119880

119894will be






119894 we want [24]

AF (120579119894) = w119867s (120579

119894) = 0 (8)







where

s (120579119894) =

[[[[[[[[[[[[[[

[


2) 119896119889 cos 120579

119894)


2) 119896119889 cos 120579

119894)

exp(119895 (119873 minus 3

2) 119896119889 cos 120579

119894)

exp (119895 (119873 minus 1

2) 119896119889 cos 120579

119894)

]]]]]]]]]]]]]]

]119873times1

(9)


w = [119908minus119872 119908minus119872+1

1199080 119908

119872minus1 119908119872]119879 (10)


w119867s (120579119894) = 0 119894 = 1 2 119872

0 (11)


C = [s (1205791) s (120579

2) s (120579

1198720

)] (12)

where 120579119894for 119894 = 1 2 119872





1198661=

119875

sum

119894=1

[1003816100381610038161003816AF119889(120579119894) minus AFCADE(120579119894)

1003816100381610038161003816]2 (14)

1198662=10038171003817100381710038171003817w119867C10038171003817100381710038171003817

2

(15)

119866 = 1198661+ 1198662 (16)














7sensor failure are


1= 1993



∘ and 1205792= 3488



1= 1993

∘ 1205792= 3488

∘ and 1205793=









0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0

Nor

mal

ized

AF

(dB)


120579 (deg)



0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0

Nor

mal

ized

AF

(dB)


120579 (deg)


one null

0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0

Nor

mal

ized

AF

(dB)


120579 (deg)



0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0

Nor

mal

ized

AF

(dB)


120579 (deg)


two nulls





0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0

Nor

mal

ized

AF

(dB)


120579 (deg)



0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0

Nor

mal

ized

AF

(dB)


120579 (deg)


of three nulls




0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0

Nor

mal

ized

AF

(dB)


120579 (deg)







minus4equal to


1= 3489

∘ 1205792= 5431

∘ and 1205793= 689



1fails due to











0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0

Nor

mal

ized

AF

(dB)


120579 (deg)


of five nulls





119904=




AF (120579119894) =

119872

sum

119899=minus119872

119908119899exp 119895119899119896119889 (cos 120579

119894minus cos 120579

119904) (17)

0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0N

orm

aliz

ed A

F (d

B)


120579 (deg)







1SSF we received









0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0

Nor

mal

ized

AF

(dB)


120579 (deg)


of three nulls

0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0

Nor

mal

ized

AF

(dB)

Original

1 null recovered

120579 (deg)

w1 element failure



0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0

Nor

mal

ized

AF

(dB)

Original

1 null recovered

120579 (deg)

w1 SSF


of one null

0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0

Nor

mal

ized

AF

(dB)


120579 (deg)






References

































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in






1 1198992 and 119899


1 1198992 1198993le 119873 with 119899

1= 1198992= 1198993= 119895


119910119894119892= 1199091198991 119892 + 119865(119909

1198992 119892 minus 1199091198993 119892)

119911119894119892

119895=

119910119894119892

119895119894119891 rand() le CR 119900119903 119895 = 119895rand

119909119894119892

119895otherwise






119910119894119892= 119875119894+ 119865 (119909

1198992119892minus 1199091198993119892) (5)






119911119894119892

119895=

1199091198991119892+ 119865 (119909

1198992119892minus 1199091198993119892) if 1199091198991 119892 lt 119897

119894

1199091198991119892minus 119865 (119909

1198992119892minus 1199091198993119892) if 1199091198991 119892 gt 119906

119894

1199091198991119892+ (

119906119894minus 119897119894

119889119904119894


(6)

where 119897119894and 119906





1198861 1199091198862 1199091198863 119909



119894

and 119909max119894


119906119894=

119909119894max119894

if 119909119894max119894

gt 119906119894or 119891 (119909max

119894

) lt 119880119894

119906119894

otherwise

119897119894=

119909119894min119894

if 119909119894min119894

lt 119897119894or 119891 (119909min

119894

) lt 119871119894

119897119894

otherwise

(7)

If 119897119894and 119906


119894and 119880

119894will be






119894 we want [24]

AF (120579119894) = w119867s (120579

119894) = 0 (8)







where

s (120579119894) =

[[[[[[[[[[[[[[

[


2) 119896119889 cos 120579

119894)


2) 119896119889 cos 120579

119894)

exp(119895 (119873 minus 3

2) 119896119889 cos 120579

119894)

exp (119895 (119873 minus 1

2) 119896119889 cos 120579

119894)

]]]]]]]]]]]]]]

]119873times1

(9)


w = [119908minus119872 119908minus119872+1

1199080 119908

119872minus1 119908119872]119879 (10)


w119867s (120579119894) = 0 119894 = 1 2 119872

0 (11)


C = [s (1205791) s (120579

2) s (120579

1198720

)] (12)

where 120579119894for 119894 = 1 2 119872





1198661=

119875

sum

119894=1

[1003816100381610038161003816AF119889(120579119894) minus AFCADE(120579119894)

1003816100381610038161003816]2 (14)

1198662=10038171003817100381710038171003817w119867C10038171003817100381710038171003817

2

(15)

119866 = 1198661+ 1198662 (16)














7sensor failure are


1= 1993



∘ and 1205792= 3488



1= 1993

∘ 1205792= 3488

∘ and 1205793=









0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0

Nor

mal

ized

AF

(dB)


120579 (deg)



0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0

Nor

mal

ized

AF

(dB)


120579 (deg)


one null

0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0

Nor

mal

ized

AF

(dB)


120579 (deg)



0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0

Nor

mal

ized

AF

(dB)


120579 (deg)


two nulls





0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0

Nor

mal

ized

AF

(dB)


120579 (deg)



0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0

Nor

mal

ized

AF

(dB)


120579 (deg)


of three nulls




0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0

Nor

mal

ized

AF

(dB)


120579 (deg)







minus4equal to


1= 3489

∘ 1205792= 5431

∘ and 1205793= 689



1fails due to











0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0

Nor

mal

ized

AF

(dB)


120579 (deg)


of five nulls





119904=




AF (120579119894) =

119872

sum

119899=minus119872

119908119899exp 119895119899119896119889 (cos 120579

119894minus cos 120579

119904) (17)

0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0N

orm

aliz

ed A

F (d

B)


120579 (deg)







1SSF we received









0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0

Nor

mal

ized

AF

(dB)


120579 (deg)


of three nulls

0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0

Nor

mal

ized

AF

(dB)

Original

1 null recovered

120579 (deg)

w1 element failure



0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0

Nor

mal

ized

AF

(dB)

Original

1 null recovered

120579 (deg)

w1 SSF


of one null

0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0

Nor

mal

ized

AF

(dB)


120579 (deg)






References

































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in









where

s (120579119894) =

[[[[[[[[[[[[[[

[


2) 119896119889 cos 120579

119894)


2) 119896119889 cos 120579

119894)

exp(119895 (119873 minus 3

2) 119896119889 cos 120579

119894)

exp (119895 (119873 minus 1

2) 119896119889 cos 120579

119894)

]]]]]]]]]]]]]]

]119873times1

(9)


w = [119908minus119872 119908minus119872+1

1199080 119908

119872minus1 119908119872]119879 (10)


w119867s (120579119894) = 0 119894 = 1 2 119872

0 (11)


C = [s (1205791) s (120579

2) s (120579

1198720

)] (12)

where 120579119894for 119894 = 1 2 119872





1198661=

119875

sum

119894=1

[1003816100381610038161003816AF119889(120579119894) minus AFCADE(120579119894)

1003816100381610038161003816]2 (14)

1198662=10038171003817100381710038171003817w119867C10038171003817100381710038171003817

2

(15)

119866 = 1198661+ 1198662 (16)














7sensor failure are


1= 1993



∘ and 1205792= 3488



1= 1993

∘ 1205792= 3488

∘ and 1205793=









0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0

Nor

mal

ized

AF

(dB)


120579 (deg)



0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0

Nor

mal

ized

AF

(dB)


120579 (deg)


one null

0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0

Nor

mal

ized

AF

(dB)


120579 (deg)



0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0

Nor

mal

ized

AF

(dB)


120579 (deg)


two nulls





0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0

Nor

mal

ized

AF

(dB)


120579 (deg)



0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0

Nor

mal

ized

AF

(dB)


120579 (deg)


of three nulls




0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0

Nor

mal

ized

AF

(dB)


120579 (deg)







minus4equal to


1= 3489

∘ 1205792= 5431

∘ and 1205793= 689



1fails due to











0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0

Nor

mal

ized

AF

(dB)


120579 (deg)


of five nulls





119904=




AF (120579119894) =

119872

sum

119899=minus119872

119908119899exp 119895119899119896119889 (cos 120579

119894minus cos 120579

119904) (17)

0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0N

orm

aliz

ed A

F (d

B)


120579 (deg)







1SSF we received









0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0

Nor

mal

ized

AF

(dB)


120579 (deg)


of three nulls

0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0

Nor

mal

ized

AF

(dB)

Original

1 null recovered

120579 (deg)

w1 element failure



0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0

Nor

mal

ized

AF

(dB)

Original

1 null recovered

120579 (deg)

w1 SSF


of one null

0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0

Nor

mal

ized

AF

(dB)


120579 (deg)






References

































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in










0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0

Nor

mal

ized

AF

(dB)


120579 (deg)



0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0

Nor

mal

ized

AF

(dB)


120579 (deg)


one null

0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0

Nor

mal

ized

AF

(dB)


120579 (deg)



0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0

Nor

mal

ized

AF

(dB)


120579 (deg)


two nulls





0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0

Nor

mal

ized

AF

(dB)


120579 (deg)



0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0

Nor

mal

ized

AF

(dB)


120579 (deg)


of three nulls




0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0

Nor

mal

ized

AF

(dB)


120579 (deg)







minus4equal to


1= 3489

∘ 1205792= 5431

∘ and 1205793= 689



1fails due to











0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0

Nor

mal

ized

AF

(dB)


120579 (deg)


of five nulls





119904=




AF (120579119894) =

119872

sum

119899=minus119872

119908119899exp 119895119899119896119889 (cos 120579

119894minus cos 120579

119904) (17)

0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0N

orm

aliz

ed A

F (d

B)


120579 (deg)







1SSF we received









0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0

Nor

mal

ized

AF

(dB)


120579 (deg)


of three nulls

0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0

Nor

mal

ized

AF

(dB)

Original

1 null recovered

120579 (deg)

w1 element failure



0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0

Nor

mal

ized

AF

(dB)

Original

1 null recovered

120579 (deg)

w1 SSF


of one null

0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0

Nor

mal

ized

AF

(dB)


120579 (deg)






References

































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in







0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0

Nor

mal

ized

AF

(dB)


120579 (deg)



0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0

Nor

mal

ized

AF

(dB)


120579 (deg)


of three nulls




0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0

Nor

mal

ized

AF

(dB)


120579 (deg)







minus4equal to


1= 3489

∘ 1205792= 5431

∘ and 1205793= 689



1fails due to











0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0

Nor

mal

ized

AF

(dB)


120579 (deg)


of five nulls





119904=




AF (120579119894) =

119872

sum

119899=minus119872

119908119899exp 119895119899119896119889 (cos 120579

119894minus cos 120579

119904) (17)

0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0N

orm

aliz

ed A

F (d

B)


120579 (deg)







1SSF we received









0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0

Nor

mal

ized

AF

(dB)


120579 (deg)


of three nulls

0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0

Nor

mal

ized

AF

(dB)

Original

1 null recovered

120579 (deg)

w1 element failure



0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0

Nor

mal

ized

AF

(dB)

Original

1 null recovered

120579 (deg)

w1 SSF


of one null

0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0

Nor

mal

ized

AF

(dB)


120579 (deg)






References

































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in










0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0

Nor

mal

ized

AF

(dB)


120579 (deg)


of five nulls





119904=




AF (120579119894) =

119872

sum

119899=minus119872

119908119899exp 119895119899119896119889 (cos 120579

119894minus cos 120579

119904) (17)

0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0N

orm

aliz

ed A

F (d

B)


120579 (deg)







1SSF we received









0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0

Nor

mal

ized

AF

(dB)


120579 (deg)


of three nulls

0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0

Nor

mal

ized

AF

(dB)

Original

1 null recovered

120579 (deg)

w1 element failure



0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0

Nor

mal

ized

AF

(dB)

Original

1 null recovered

120579 (deg)

w1 SSF


of one null

0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0

Nor

mal

ized

AF

(dB)


120579 (deg)






References

































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in











0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0

Nor

mal

ized

AF

(dB)


120579 (deg)


of three nulls

0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0

Nor

mal

ized

AF

(dB)

Original

1 null recovered

120579 (deg)

w1 element failure



0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0

Nor

mal

ized

AF

(dB)

Original

1 null recovered

120579 (deg)

w1 SSF


of one null

0 20 40 60 80 100 120 140 160 180minus120

minus100

minus80

minus60

minus40

minus20

0

Nor

mal

ized

AF

(dB)


120579 (deg)






References

































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in







References

































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in










Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in



research article correction of faulty sensors in phased...

Documents