Research Article Sidelobe Suppression with Null Steering...

Research ArticleSidelobe Suppression with Null Steering byIndependent Weight Control

Zafar-Ullah Khan1 Aqdas Naveed Malik1 Fawad Zaman2

Syed Azmat Hussain3 and Abdul-Rehman Khan2

1Department of Electronic Engineering IIU H-10 Islamabad 44000 Pakistan2Department of Electrical Engineering COMSATS Institute of Information Technology Attock Campus Attock 43600 Pakistan3School of Engineering and Applied Sciences ISRA University Islamabad Campus Islamabad 44000 Pakistan

Correspondence should be addressed to Zafar-Ullah Khan zafarullahphdee13iiuedupk

Received 20 April 2015 Accepted 18 May 2015

Academic Editor Vincenzo Galdi

Copyright copy 2015 Zafar-Ullah Khan et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

A uniform linear array of n antenna elements can steer up to 119899 minus 1 nulls In situations where less than 119899 minus 1 nulls are required tobe steered the existing algorithms have no criterion to utilize the remaining weights for sidelobe suppressionThis work combinessidelobe suppression capability with null steering by independent weight control For this purpose the array factor is transformedas the product of two polynomials One of the polynomials is used for null steering by independent weight control while the secondone is for sidelobe suppression whose coefficients or weights are determined by using convex optimization Finally a new structureis proposed to incorporate the product of two polynomials such that sidelobe suppression weights are decoupled from those of nullsteering weights Simulation results validate the effectiveness of the proposed scheme

1 Introduction

Null steering for adaptive beamforming is an important areaof research due to its military and commercial applications[1ndash6] Null steering techniques can broadly be classifiedas Direction of Arrival (DOA) based beamformers andoptimal adaptive beamformers [1] Existing DOA based nullsteering techniques either control the excitation of elementsrsquoamplitude only [2 3] phase only [4 5] or both [6] Thereare also techniques that achieve null steering by changing theposition andor elevations of the array elements [7ndash10]

Elementsrsquo phase and amplitude excitation techniquesare slow for large arrays because even if a single nullchanges its position the whole set of weights is requiredto be reevaluated which is obviously time consuming andcomplicated On the other hand null steering by controllingthe excitation amplitude only is easy to implement and lesssensitive to quantization error but reduces the number ofsteerable nulls [2 3] The techniques like phase only andposition perturbations of the array elements are nonlinear

problems and cannot be solved directly by analyticalmethods[11] Besides position perturbations methods require servomotors for themovement of the elements and in case of largearrays the complexity to control element position increasesdue to increase in computational time to find new positionperturbations

The solution for this problem is the null steering byindependent weight control where if a single null changesits position only the weight set corresponding to that nullis evaluated and changed [3 6] Unfortunately these DOAbased beamformers do not have sidelobe suppression capa-bility Although this problem has been addressed for optimaladaptive beamforming algorithms [12 13] the method cannotbe used in DOA based beamformers because in these algo-rithms pattern synthesis weights are not determined by usingoptimization techniques

This paper presents a technique where the array factoris transformed as product of two polynomials such that onepolynomial denoted by AF1 provides independent steering ofall available nulls while the other denoted by AF2 suppresses

Hindawi Publishing CorporationInternational Journal of Antennas and PropagationVolume 2015 Article ID 136826 6 pageshttpdxdoiorg1011552015136826

2 International Journal of Antennas and Propagation

sidelobes by utilizing remaining weights The coefficients ofAF1 are determined by forcing the zeros of AF1 to lie onunit circle in the complex plane [6] while the coefficients ofAF2 are evaluated using convex optimization Furthermore astructure is presented to implement this array factor wherethe sidelobe suppression weights are decoupled from theweights meant for independent null steering Therefore thecoefficients of AF2 will not be required to change with thechange in position of any interference

The rest of the work is organized as follows Section 2presents problem formulation while proposed technique andstructure are discussed in Sections 3 and 4 respectivelySimulation results are given in Section 5 Finally conclusionand future work recommendation are given in Section 6

2 Problem Formulation

Consider a uniform linear array (ULA) of119873 omnidirectionalelements Let 119889 be the element spacing 120572 the progressivephase and 120579 the angle of arrival of the plane wave impingingon the array Figure 1 shows the path difference of this waveon the adjacent elements of the ULA that is 119909 = 119889cos120579The output of the array elements is multiplied by the properlyselected coefficients1198600 1198601 119860119873minus1 to steer the nulls in thedirections of interferencesWe take the first element (element0) of the array as reference and 120595 = 120572 + (2120587120582)119889cos120579 where119896 = 2120587120582 is known as wave numberThe signal at element 2 isthe delayed version of the signal at element 1 and is expressedas 119890119895120595 Let 119890119895120595 = 119911 then the array factor can be given as

AF1 = 1198600 +1198601119911 + sdot sdot sdot +119860119873minus1119911(119873minus1)

(1)

This AF1 is the 119873 minus 1-degree polynomial and has 119873 minus 1roots (nulls) dependent on coefficients 1198600 1198601 119860119873minus1 Infactorized form (1) becomes

AF1 = (119911 minus 1199111) (119911 minus 1199112) sdot sdot sdot (119911 minus 119911119873minus1) (2)

where 1199111 1199112 119911119873minus1 are zeros forced to exist on the unitcircle in the complex plane [2 3 6 14] and their position onthe circle depends on coefficients 1198600 1198601 119860119873minus1

To steer the main beam in the desired direction 120579119904 the

progressive phase shift is given by

120572 = minus(

2120587120582

)119889 cos 120579119904 (3)

In order to incorporate expression (1) factors of (2) are tobe multiplied iteratively by combining the coefficients of thesame powers of 119911 at each step This process of multiplicationis complicated and time consuming for large arrays Thealgorithms proposed in [3 6] avoid this multiplication bydecoupling null steering weights while the current workpresents a method to add sidelobe suppression capability tothese algorithms by adjusting and decoupling the weightsof spare elements of the array from those of null steeringweights

d d

x

3

2 1N

Output

120579

A0A2 A1ANminus1

Σ

Figure 1 Uniform linear array of 119873 elements with 119873 minus 1 steerablenulls

3 Proposed Algorithm for Independent NullSteering (INS) with Suppressed Sidelobes

According to Schelkunoff [14] the product of two or morepolynomials results in a polynomial and there exists a lineararray corresponding to this resultant polynomial Using thesame fact consider the desired array factor AF given below

AF = AF1 sdotAF2 (4)

where AF1 is meant for independent steering of1198731 nulls andcan be given as

AF1 =1198731

prod

119894=1(119911 minus 119911

119894) (5)

The purpose of AF2 is to suppress sidelobes and it can beexpressed by

AF2 = 1198870 + 1198871119911 + sdot sdot sdot + 1198871198732minus11199111198732minus1

= s1198732b (6)

where b = [1198870 1198871 1198871198732minus1]119879 represents the complex weight

vector and s1198732=[1 119911 1199111198732minus1] is the steering vector The

condition on the vector b is that it guarantees the sidelobesuppression and steers the main beam in the same directionas that of AF1 To find out b consider a set 119878

119877that is union

of two sets containing angles from the left and right sideloberegions of the main beam with beam-width 120579mb that is

119878119877= 0lt 120579le 120579

119904minus

120579mb2

cup120579119904+

120579mb2

lt 120579le 180∘ (7)

Then 119901 discrete angles are selected from 119878119877and placed in

another set 119860119878119877given by

119860119878119877= 120579119894| 120579119894isin 119878119877and 120579119894= 1205790 + 119894120575120579 sube 119878

119877 (8)

International Journal of Antennas and Propagation 3

0 20 40 60 80 100 120 140 160 180 Angle (deg)

Out

put p

ower

(dB)

t

minus50

minus40

minus30

minus20

minus10

0

y

HSLL = t

(a)

0 20 40 60 80 100 120 140 160 180Angle (deg)

Out

put p

ower

(dB)

t

minus40

minus30

minus20

minus10

0

y

HSLL = t

(b)

Figure 2 (a) Highest sidelobe level (HSLL = 119905) is required to be minimized when 1198732 = 6 120579119904= 95∘ (b) 119910 = AF2 pattern when highest

sidelobe level (HSLL = 119905) has been minimized for1198732 = 6 120579119904= 95∘ and 120579mb = 24∘

where 119894 = 0 1 119901 minus 1 and 1205790 is the starting angle and 120575120579 isthe step size

Now consider another matrix A containing steeringvectors corresponding to the angles contained in119860

119878119877 that is

A = [s1198732

(1205790) s1198732 (1205791) sdot sdot sdot s1198732 (120579119901minus1)]119879

(9)

where s1198732(120579119894) = [1 119911

119894 119911

1198732minus1119894

] with 119911119894

= exp(119895(120572 +

(2120587120582)119889cos120579119894)) Now the requirement is to minimize the

array output power along the angles 120579119894isin 119860119878119877

subject to theunit output along the desired direction 120579

119904

The requirement is fulfilled if we restrict the peak outputpower in the sidelobe region that is the highest sidelobelevel (HSLL) to have minimum value as shown in Figure 2Minimizing this power along the angles 120579

119894isin 119860119878119877is the same

as minimizing the absolute output along these angles that is

minb

max (10038161003816100381610038161003816s1198732

(120579119894) b1003816100381610038161003816

1003816) 120579

119894isin 119860119878119877

subject to s1198732

(120579119904) b = 1

(10)

Since the steering vectors along these angles are contained inmatrix A the absolute output along these angles is the vector|Ab| and the above minimization problem becomes

minb

max |Ab|

subject to s1198732

(120579119904) b = 1

(11)

In (11) the objective function controls the sidelobe level andthe constraint forms the main beam in the desired directionThis problem can be cast as the second-order cone programin the following manner

Minimize 119905

subject to |Ab| le 119905

s1198732

(120579119904) b = 1

(12)

The inequalities in (12) are called second-order cone con-straints and (Ab 119905) isin second-order cone inR119901+1[15] Second-order cone program is the subclass of convex optimizationand hence problem (11) can be solved using [16]

The polynomial AF2 suppresses sidelobe levels and hasoverlapping main beam with that of AF1 Due to patternmultiplication AF = AF1 sdot AF2 will give the independentnull steering capability with suppressed sidelobes The AF2is required to be calculated once and remains unchanged aslong as the position of themain beam is unchanged Howeverif the position of the main beam is changed the coefficientsof both the polynomials that is AF1 and AF2 will have to berecalculated and changed

4 Proposed Structure

In this section a structure is proposed to steer 5 nulls inde-pendently and to use three weights for sidelobe suppressionthat is1198731 = 51198732 = 3 This structure can easily be extendedfor any values of 1198731 and 1198732 The structure starts with ULAof 8 elements that is (1198731 +1198732) It can be seen from Figure 3that for first 1198731 (which is 5 in this case) stages each adderhas 2 inputs In the structure first stage controls the positionof first null where first input of each adder is multiplied by(minus1199111) The 119895th output of this stage is represented by 1199101119895 with119895 = 1 1198731 + 1198732 minus 1 and is given by

1199101119895 = 119911119895minus1

(119911 minus 1199111) for 119895 = 1 1198731 + 1198732 minus 1 (13)

Similarly second stage controls second null by multiplyingfirst input of each of its adders by (minus1199112) These outputs are

1199102119895 = 119911119895minus1

(119911 minus 1199111) (119911 minus 1199112)

for 119895 = 1 1198731 + 1198732 minus 2(14)

It is clear from Figure 3 that we will get 1198732 outputs after 1198731stages and these outputs will be

1199101198731119895

= 119911119895minus11198731

prod

119894=1(119911 minus 119911

119894) = 119911119895minus1AF1

for 119895 = 1 1198732

(15)

It means these outputs are multiples of AF1 as shown below

[1199101198731 1 1199101198731 2 11991011987311198732]

119879

= AF1 [1 119911 1199111198732minus1

]

119879

(16)


Stage 1 Stage 2 Stage 3 Stage 4 Stage 5 Stage 6z7

z6

z5

z4

z3

z2

z

minusz1

minusz2

minusz3

minusz4

minusz5

minusz5

minusz5

minusz4

minusz4

minusz4

minusz3

minusz3

minusz3

minusz3

minusz2

minusz2

minusz2

minusz2

minusz2

minusz1

minusz1

minusz1

minusz1

minusz1

minusz1 +

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+ +

+

+

+

+

+ y17

y16

y15

y14

y13

y12

y11

y26

y25

y24

y23

y22

y21

y35

y34

y33

y32

y31

y44

y43

y42

y41

y53

y52

y51

y

b2

b1

b01

Figure 3 Structure for independent null steering with suppressed sidelobes

Thus first 1198731 stages will incorporate AF1 and steer 1198731 nullsby independent weight control After that the resulting 1198732outputs are weighted to give AF2 for sidelobe suppressionFinally these weighted outputs are summed to result in AF =

AF1 sdot AF2 given by (17) and the corresponding structure isshown in Figure 3 Consider

119910 = AF1 [1 119911 1199111198732minus1

] [1198870 1198871 1198871198732minus1]119879

= AF1AF2(17)

As mentioned earlier convex optimization is used to find outthe1198732 complexweightsThese1198732 weights remain unchangedeven if one or more nulls change their position On the otherhand if the main beam changes its position the whole set ofweights including INS weights for1198731 stages and1198732 weightsfor sidelobe control is changed Even in such situationimplementation of AF1 in Figure 3 is not time consuming ascompared to Figure 1 because it incorporates factors directlywithout multiplication and forming a single polynomial

5 Simulation and Results

In this section several simulation examples are carried outin MATLAB to verify the effectiveness of the proposedalgorithm In these examples element spacing 119889 is taken as1205822

Example 1 This example compares the performance of theproposed technique with two algorithms that is INS bydecoupling real weights (Realwt) and INS by decouplingcomplex weights (Cmplxwt) For this purpose main beam istaken along 95∘ while the four interferences are placed at 20∘70∘ 120∘ and 160∘ In this case INS by decoupling realweights

and INS by decoupling complexweights requireULAs having9 and 5 elements respectively while the proposed algorithmis utilized using a ULA of 9 elements with 1198731 = 4 and1198732 = 5 One can clearly see in Figure 4 that the performanceof the proposed algorithm is better for sidelobe suppressionas compared to the other two algorithms

Example 2 This example compares the performance of theproposed algorithm with INS by decoupling real weightswhen the second null is at 70∘ and also when it is shifted to75∘ while positions of all the other nulls remain unchangedIn Figure 5 again one can see that the proposed schemeoutperforms Realwt no matter whether the second null is at70∘ or is shifted to 75∘ Similar behavior can be observed if theposition of multiple nulls is changed

Example 3 This example demonstrates the effect of numberof coefficients of AF2 that is 1198732 on the sidelobe level Forthis purpose sidelobe levels are compared for the same setof signals as in Example 1 when 1198732 = 4 6 and 8 It is clearfrom Figure 6 that the sidelobe suppression capability of theproposed algorithm increases as1198732 is increased

Example 4 This example shows the capability of the pro-posed algorithm to decouple complex weights of AF1 tosteer nulls independently For this purpose main beam istaken along 95∘ and four interferences are placed at 20∘ 70∘120∘ and 160∘ Then the position of interferences is changedone by one and the results are shown in Table 1 where thechanged position of the nulls and corresponding weights areshown by bold entries It can easily be deduced that whenany interference changes its position only the correspondingweight is changed while the rest of the weights remainunaffected


Table 1 Decoupling of weights by proposed algorithm for INS when 120579119904= 95∘ 1198731 = 4 1198732 = 5

S number 120579119894

1199111 1199112 1199113 1199114

1 20∘ 70∘ 120∘ 160∘ minus09964 minus 00842119894 02207 + 09753119894 02704 minus 09627119894 minus08946 minus 04469119894

2 25∘ 70∘ 120∘ 160∘ minus09998 + 00205i 02207 + 09753119894 02704 minus 09627119894 minus08946 minus 04469119894

3 20∘ 65∘ 120∘ 160∘ minus09964 minus 00842119894 minus00307 + 09995i 02704 minus 09627119894 minus08946 minus 04469119894

4 20∘ 70∘ 125∘ 160∘ minus09964 minus 00842119894 02207 + 09753119894 00426 minus 09991i minus08946 minus 04469119894

5 20∘ 70∘ 120∘ 155∘ minus09964 minus 00842119894 02207 + 09753119894 02704 minus 09627119894 minus08429 minus 05381i

Table 2 Effect of change in 120579119904on different parameters

S number 120579119904

Parameters Value

1 95∘1198870 minus 1198874 00989 + 00098119894 02426 + 00120119894 03171 02426 minus 00120119894 00989 minus 00098119894

1199111 minus 1199114 minus09964 minus 00842119894 02207 + 09753119894 02704 minus 09627119894 minus08946 minus 04469119894

120572 02738

2 90∘1198870 minus 1198874 00987 02427 03172 02427 00987


120572 0

0 20 40 60 80 100 120 140 160 180

0

Angle of arrival (deg)

Out

put p

ower

(dB)

ProposedRealwt [3]Cmplxwt [6]

minus100

minus90

minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

Figure 4 Performance comparison of the proposed algorithm withINS by decoupling real weights (Realwt) and INS by decouplingcomplex weights (Cmplxwt)

Example 5 This example shows the effect of change indirection of main beam on 119911

119894 b and 120572 For this purpose

initiallymain beam is taken along 95∘ while four interferencesare placed at 20∘ 70∘ 120∘ and 160∘ Then main beam isshifted at 90∘ while the interferences do not change theirposition as given in Table 2 for both the cases It is clear thatthese parameters change with the change in 120579

119904

6 Conclusion

A method for uniform linear array is proposed to steerthe nulls by independent weight control and to suppresssidelobes It uses the progressive phase shift for beam steeringand product of two polynomials for independent null steering

0 20 40 60 80 100 120 140 160 180Angle of arrival (deg)

Out

put p

ower

(dB)

Proposed-75Realwt [3]-75


0

minus100

minus90

minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

Figure 5 Performance comparison of proposed algorithmwith INSby decoupling real weights [3] when secondnull changes its positionfrom 70∘ to 75∘

with sidelobe suppression The zeros of independent nullsteering polynomial are forced to lie on the unit circle inthe complex plane Similarly the coefficients for sidelobesuppression are determined by convex optimization Then astructure is proposed that incorporates the product of the twopolynomials In order to steer1198731 nulls independently and touse 1198732 weights for sidelobe suppression a ULA of 1198731 + 1198732elements is required In the structure the number of weightsfor independent null steering and the total number of weightswill be 051198731(1198731 + 21198732 minus 1) and 051198731(1198731 + 21198732 minus 1) + 1198732respectively Traditionally this array can steer 1198731 + 1198732 minus 1nulls but the proposed technique steers only 1198731 nulls withsuppressed sidelobes at the cost of 051198731(1198731 + 21198732 minus 1) +1198732



Out

put p

ower

(dB)

0

minus100

minus110

minus90

minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

N2 = 4

N2 = 6

N2 = 8

Figure 6 Performance of proposed algorithm for 1198732= 4 119873

2= 6

and1198732= 8

complex weights 051198731(1198731 + 21198732 minus 1) + 1 adders and1198732 minus 1nulls

The coefficients for the sidelobe suppression are notaffected by the change in position of one or more nullswith the condition that the main beam does not change itsdirection On the other hand if the main beam changes itsposition the whole set of weights of the structure along withthe progressive phase shift will be changedThe sidelobes canbe suppressed more by increasing the number of coefficientsdedicated for this purpose

In future one can use the same approach for two- andthree-dimensional arrays

Conflict of Interests

All the authors of this paper declare that there is no conflictof interests regarding the publication of this paper

References

[1] B Friedlander and B Porat ldquoPerformance analysis of a null-steering algorithm based on direction-of-arrival estimationrdquoIEEE Transactions on Acoustics Speech and Signal Processingvol v pp 461ndash466 1992

[2] T B Vu ldquoMethod of null steering without using phase shiftersrdquoIEE ProceedingsHMicrowavesOptics andAntennas vol 131 no4 pp 242ndash246 1984

[3] HM Ibrahim ldquoNull steering by real-weight controlmdashamethodof decoupling the weightsrdquo IEEE Transactions on Antennas andPropagation vol 39 no 11 pp 1648ndash1650 1991

[4] H Steyskal ldquoSimple method for pattern nulling by phaseperturbationrdquo IEEE Transactions on Antennas and Propagationvol 31 no 1 pp 163ndash166 1983

[5] M Mouhamadou P Vaudon and M Rammal ldquoSmartantenna array patterns synthesis null steering and multi-user

beamforming by phase controlrdquo Progress in ElectromagneticsResearch vol 60 pp 95ndash106 2006

[6] Z U Khan A Naveed I M Qureshi and F Zaman ldquoInde-pendent null steering by decoupling complex weightsrdquo IEICEElectronics Express vol 8 no 13 pp 1008ndash1013 2011

[7] 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

[8] J A Hejres ldquoArray pattern nulling using the elevations ofselected elements of phased antenna arraysrdquo in Proceedingsof the IEEE Antennas and Propagation Society InternationalSymposium vol 2 pp 68ndash71 July 2005

[9] 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

[10] F Tokan and F Gunes ldquoInterference suppression by optimisingthe positions of selected elements using generalised patternsearch algorithmrdquo IET Microwaves Antennas and Propagationvol 5 no 2 pp 127ndash135 2011

[11] A F Morabito A Massa P Rocca and T Isernia ldquoAn effectiveapproach to the synthesis of phase-only reconfigurable lineararraysrdquo IEEETransactions onAntennas andPropagation vol 60no 8 pp 3622ndash3631 2012

[12] J Liu A B Gershman Z-Q Luo and K M Wong ldquoAdaptivebeamforming with sidelobe control a second-order cone pro-gramming approachrdquo IEEE Signal Processing Letters vol 10 no11 pp 331ndash334 2003

[13] Y Zhang B P Ng and Q Wan ldquoSidelobe suppression foradaptive beamforming with sparse constraint on beam patternrdquoElectronics Letters vol 44 no 10 pp 615ndash616 2008

[14] S A Schelkunoff ldquoA mathematical theory of linear arraysrdquoTheBell System Technical Journal vol 22 pp 80ndash107 1943

[15] S Boyd andLVandenbergheConvexOptimization CambridgeUniversity Press 2004

[16] M Grant and S Boyd ldquoCVX a system for disciplined convexprogramming cvx version 121rdquo 2011

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



sidelobes by utilizing remaining weights The coefficients ofAF1 are determined by forcing the zeros of AF1 to lie onunit circle in the complex plane [6] while the coefficients ofAF2 are evaluated using convex optimization Furthermore astructure is presented to implement this array factor wherethe sidelobe suppression weights are decoupled from theweights meant for independent null steering Therefore thecoefficients of AF2 will not be required to change with thechange in position of any interference

The rest of the work is organized as follows Section 2presents problem formulation while proposed technique andstructure are discussed in Sections 3 and 4 respectivelySimulation results are given in Section 5 Finally conclusionand future work recommendation are given in Section 6

2 Problem Formulation

Consider a uniform linear array (ULA) of119873 omnidirectionalelements Let 119889 be the element spacing 120572 the progressivephase and 120579 the angle of arrival of the plane wave impingingon the array Figure 1 shows the path difference of this waveon the adjacent elements of the ULA that is 119909 = 119889cos120579The output of the array elements is multiplied by the properlyselected coefficients1198600 1198601 119860119873minus1 to steer the nulls in thedirections of interferencesWe take the first element (element0) of the array as reference and 120595 = 120572 + (2120587120582)119889cos120579 where119896 = 2120587120582 is known as wave numberThe signal at element 2 isthe delayed version of the signal at element 1 and is expressedas 119890119895120595 Let 119890119895120595 = 119911 then the array factor can be given as

AF1 = 1198600 +1198601119911 + sdot sdot sdot +119860119873minus1119911(119873minus1)

(1)

This AF1 is the 119873 minus 1-degree polynomial and has 119873 minus 1roots (nulls) dependent on coefficients 1198600 1198601 119860119873minus1 Infactorized form (1) becomes

AF1 = (119911 minus 1199111) (119911 minus 1199112) sdot sdot sdot (119911 minus 119911119873minus1) (2)

where 1199111 1199112 119911119873minus1 are zeros forced to exist on the unitcircle in the complex plane [2 3 6 14] and their position onthe circle depends on coefficients 1198600 1198601 119860119873minus1

To steer the main beam in the desired direction 120579119904 the

progressive phase shift is given by

120572 = minus(

2120587120582

)119889 cos 120579119904 (3)

In order to incorporate expression (1) factors of (2) are tobe multiplied iteratively by combining the coefficients of thesame powers of 119911 at each step This process of multiplicationis complicated and time consuming for large arrays Thealgorithms proposed in [3 6] avoid this multiplication bydecoupling null steering weights while the current workpresents a method to add sidelobe suppression capability tothese algorithms by adjusting and decoupling the weightsof spare elements of the array from those of null steeringweights

d d

x

3

2 1N

Output

120579

A0A2 A1ANminus1

Σ

Figure 1 Uniform linear array of 119873 elements with 119873 minus 1 steerablenulls

3 Proposed Algorithm for Independent NullSteering (INS) with Suppressed Sidelobes

According to Schelkunoff [14] the product of two or morepolynomials results in a polynomial and there exists a lineararray corresponding to this resultant polynomial Using thesame fact consider the desired array factor AF given below

AF = AF1 sdotAF2 (4)

where AF1 is meant for independent steering of1198731 nulls andcan be given as

AF1 =1198731

prod

119894=1(119911 minus 119911

119894) (5)

The purpose of AF2 is to suppress sidelobes and it can beexpressed by

AF2 = 1198870 + 1198871119911 + sdot sdot sdot + 1198871198732minus11199111198732minus1

= s1198732b (6)

where b = [1198870 1198871 1198871198732minus1]119879 represents the complex weight

vector and s1198732=[1 119911 1199111198732minus1] is the steering vector The

condition on the vector b is that it guarantees the sidelobesuppression and steers the main beam in the same directionas that of AF1 To find out b consider a set 119878

119877that is union

of two sets containing angles from the left and right sideloberegions of the main beam with beam-width 120579mb that is

119878119877= 0lt 120579le 120579

119904minus

120579mb2

cup120579119904+

120579mb2

lt 120579le 180∘ (7)

Then 119901 discrete angles are selected from 119878119877and placed in

another set 119860119878119877given by

119860119878119877= 120579119894| 120579119894isin 119878119877and 120579119894= 1205790 + 119894120575120579 sube 119878

119877 (8)


0 20 40 60 80 100 120 140 160 180 Angle (deg)

Out

put p

ower

(dB)

t

minus50

minus40

minus30

minus20

minus10

0

y

HSLL = t

(a)

0 20 40 60 80 100 120 140 160 180Angle (deg)

Out

put p

ower

(dB)

t

minus40

minus30

minus20

minus10

0

y

HSLL = t

(b)





119878119877 that is

A = [s1198732


(9)

where s1198732(120579119894) = [1 119911

119894 119911

1198732minus1119894

] with 119911119894

= exp(119895(120572 +




119904


119894isin 119860119878119877is the same


minb

max (10038161003816100381610038161003816s1198732

(120579119894) b1003816100381610038161003816

1003816) 120579

119894isin 119860119878119877

subject to s1198732

(120579119904) b = 1

(10)


minb

max |Ab|

subject to s1198732

(120579119904) b = 1

(11)


Minimize 119905


s1198732

(120579119904) b = 1

(12)





1199101119895 = 119911119895minus1

(119911 minus 1199111) for 119895 = 1 1198731 + 1198732 minus 1 (13)


1199102119895 = 119911119895minus1

(119911 minus 1199111) (119911 minus 1199112)

for 119895 = 1 1198731 + 1198732 minus 2(14)


1199101198731119895

= 119911119895minus11198731

prod

119894=1(119911 minus 119911

119894) = 119911119895minus1AF1

for 119895 = 1 1198732

(15)


[1199101198731 1 1199101198731 2 11991011987311198732]

119879

= AF1 [1 119911 1199111198732minus1

]

119879

(16)



z6

z5

z4

z3

z2

z

minusz1

minusz2

minusz3

minusz4

minusz5

minusz5

minusz5

minusz4

minusz4

minusz4

minusz3

minusz3

minusz3

minusz3

minusz2

minusz2

minusz2

minusz2

minusz2

minusz1

minusz1

minusz1

minusz1

minusz1

minusz1 +

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+ +

+

+

+

+

+ y17

y16

y15

y14

y13

y12

y11

y26

y25

y24

y23

y22

y21

y35

y34

y33

y32

y31

y44

y43

y42

y41

y53

y52

y51

y

b2

b1

b01




119910 = AF1 [1 119911 1199111198732minus1

] [1198870 1198871 1198871198732minus1]119879

= AF1AF2(17)











S number 120579119894

1199111 1199112 1199113 1199114







S number 120579119904

Parameters Value



120572 02738

2 90∘1198870 minus 1198874 00987 02427 03172 02427 00987


120572 0

0 20 40 60 80 100 120 140 160 180

0


Out

put p

ower

(dB)


minus100

minus90

minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10





119904

6 Conclusion



Out

put p

ower

(dB)



0

minus100

minus90

minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10





Out

put p

ower

(dB)

0

minus100

minus110

minus90

minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

N2 = 4

N2 = 6

N2 = 8


2= 6

and1198732= 8






References




















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 20 40 60 80 100 120 140 160 180 Angle (deg)

Out

put p

ower

(dB)

t

minus50

minus40

minus30

minus20

minus10

0

y

HSLL = t

(a)

0 20 40 60 80 100 120 140 160 180Angle (deg)

Out

put p

ower

(dB)

t

minus40

minus30

minus20

minus10

0

y

HSLL = t

(b)





119878119877 that is

A = [s1198732


(9)

where s1198732(120579119894) = [1 119911

119894 119911

1198732minus1119894

] with 119911119894

= exp(119895(120572 +




119904


119894isin 119860119878119877is the same


minb

max (10038161003816100381610038161003816s1198732

(120579119894) b1003816100381610038161003816

1003816) 120579

119894isin 119860119878119877

subject to s1198732

(120579119904) b = 1

(10)


minb

max |Ab|

subject to s1198732

(120579119904) b = 1

(11)


Minimize 119905


s1198732

(120579119904) b = 1

(12)





1199101119895 = 119911119895minus1

(119911 minus 1199111) for 119895 = 1 1198731 + 1198732 minus 1 (13)


1199102119895 = 119911119895minus1

(119911 minus 1199111) (119911 minus 1199112)

for 119895 = 1 1198731 + 1198732 minus 2(14)


1199101198731119895

= 119911119895minus11198731

prod

119894=1(119911 minus 119911

119894) = 119911119895minus1AF1

for 119895 = 1 1198732

(15)


[1199101198731 1 1199101198731 2 11991011987311198732]

119879

= AF1 [1 119911 1199111198732minus1

]

119879

(16)



z6

z5

z4

z3

z2

z

minusz1

minusz2

minusz3

minusz4

minusz5

minusz5

minusz5

minusz4

minusz4

minusz4

minusz3

minusz3

minusz3

minusz3

minusz2

minusz2

minusz2

minusz2

minusz2

minusz1

minusz1

minusz1

minusz1

minusz1

minusz1 +

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+ +

+

+

+

+

+ y17

y16

y15

y14

y13

y12

y11

y26

y25

y24

y23

y22

y21

y35

y34

y33

y32

y31

y44

y43

y42

y41

y53

y52

y51

y

b2

b1

b01




119910 = AF1 [1 119911 1199111198732minus1

] [1198870 1198871 1198871198732minus1]119879

= AF1AF2(17)











S number 120579119894

1199111 1199112 1199113 1199114







S number 120579119904

Parameters Value



120572 02738

2 90∘1198870 minus 1198874 00987 02427 03172 02427 00987


120572 0

0 20 40 60 80 100 120 140 160 180

0


Out

put p

ower

(dB)


minus100

minus90

minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10





119904

6 Conclusion



Out

put p

ower

(dB)



0

minus100

minus90

minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10





Out

put p

ower

(dB)

0

minus100

minus110

minus90

minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

N2 = 4

N2 = 6

N2 = 8


2= 6

and1198732= 8






References




















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











z6

z5

z4

z3

z2

z

minusz1

minusz2

minusz3

minusz4

minusz5

minusz5

minusz5

minusz4

minusz4

minusz4

minusz3

minusz3

minusz3

minusz3

minusz2

minusz2

minusz2

minusz2

minusz2

minusz1

minusz1

minusz1

minusz1

minusz1

minusz1 +

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+ +

+

+

+

+

+ y17

y16

y15

y14

y13

y12

y11

y26

y25

y24

y23

y22

y21

y35

y34

y33

y32

y31

y44

y43

y42

y41

y53

y52

y51

y

b2

b1

b01




119910 = AF1 [1 119911 1199111198732minus1

] [1198870 1198871 1198871198732minus1]119879

= AF1AF2(17)











S number 120579119894

1199111 1199112 1199113 1199114







S number 120579119904

Parameters Value



120572 02738

2 90∘1198870 minus 1198874 00987 02427 03172 02427 00987


120572 0

0 20 40 60 80 100 120 140 160 180

0


Out

put p

ower

(dB)


minus100

minus90

minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10





119904

6 Conclusion



Out

put p

ower

(dB)



0

minus100

minus90

minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10





Out

put p

ower

(dB)

0

minus100

minus110

minus90

minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

N2 = 4

N2 = 6

N2 = 8


2= 6

and1198732= 8






References




















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











S number 120579119894

1199111 1199112 1199113 1199114







S number 120579119904

Parameters Value



120572 02738

2 90∘1198870 minus 1198874 00987 02427 03172 02427 00987


120572 0

0 20 40 60 80 100 120 140 160 180

0


Out

put p

ower

(dB)


minus100

minus90

minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10





119904

6 Conclusion



Out

put p

ower

(dB)



0

minus100

minus90

minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10





Out

put p

ower

(dB)

0

minus100

minus110

minus90

minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

N2 = 4

N2 = 6

N2 = 8


2= 6

and1198732= 8






References




















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











Out

put p

ower

(dB)

0

minus100

minus110

minus90

minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

N2 = 4

N2 = 6

N2 = 8


2= 6

and1198732= 8






References




















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Research Article Sidelobe Suppression with Null Steering...

Documents

Transcript of Research Article Sidelobe Suppression with Null Steering...