Review Article Mutual Coupling in Phased Arrays: A Reviewdownloads.hindawi.com/journals/ijap/2013/348123.pdf · Mutual Coupling in Phased Arrays: A Review HemaSingh,H.L.Sneha,andR.M.Jha

Hindawi Publishing CorporationInternational Journal of Antennas and PropagationVolume 2013 Article ID 348123 23 pageshttpdxdoiorg1011552013348123

Review ArticleMutual Coupling in Phased Arrays A Review

Hema Singh H L Sneha and R M Jha

Centre of Electromagnetics CSIR-National Aerospace Laboratories Bangalore 560 017 India

Correspondence should be addressed to Hema Singh ishihema30gmailcom

Received 18 February 2013 Accepted 20 March 2013

Academic Editor Ahmed A Kishk

Copyright copy 2013 Hema Singh 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

The mutual coupling between antenna elements affects the antenna parameters like terminal impedances reflection coefficientsand hence the antenna array performance in terms of radiation characteristics output signal-to-interference noise ratio (SINR)and radar cross section (RCS) This coupling effect is also known to directly or indirectly influence the steady state and transientresponse the resolution capability interference rejection and direction-of-arrival (DOA) estimation competence of the arrayResearchers have proposed several techniques and designs for optimal performance of phased array in a given signal environmentcounteracting the coupling effect This paper presents a comprehensive review of the methods that model and mitigate the mutualcoupling effect for different types of arrays The parameters that get affected due to the presence of coupling thereby degrading thearray performance are discussedThe techniques for optimization of the antenna characteristics in the presence of coupling are alsoincluded

1 Introduction

A phased array comprises of definitely arranged finite sizedantenna elements which are fed by an appropriate feednetwork In such an array the fields radiated from oneantenna might be received by the other elements Further-more this signal might get reflected reradiated or scatteredThe properties of these signals depend on the power of thesignal reflection coefficients and the additional electricalphase introduced due to the propagation delay from oneelement to the other This kind of interaction between theantenna elements will lead to coupling effect and hence canalter the array characteristics

In other words in a phased array the electromagnetic(EM) characteristics of a particular antenna element influ-ence the other elements and are themselves influenced bythe elements in their proximity This interelement influenceor mutual coupling between the antennas is dependent onvarious factors namely number and type of antenna elements(A) interelement spacing relative orientation of elementsradiation characteristics of the radiators scan angle band-width direction of arrival (DOA) of the incident signals andthe components of the feed network (Figure 1) that is phaseshifters (P) and couplers (C)

The presence of coupling in an array changes the terminalimpedances of the antennas reflection coefficients and thearray gain These being the fundamental properties of thearray have a greater influence on their radiation character-istics output signal-to-interference plus noise ratio (SINR)and radar cross section (RCS) Furthermore it affects thesteady state response transient response speed of responseresolution capability interference rejection ability and DOAestimation competence of the array

Several researchers have studied the effect of mutual cou-pling on different types of adaptive arraysThese include Yagiarray [1] LMS and Applebaumarrays [2] power inversionarray [3] circular array of isotropic elements and semicirculararray of printed dipoles [4] microstrip patch antenna arrays[5] linear arrays of dipole sleeve dipole and spiral antennas[6] conformal dipole arrays [7] helical arrays [8] and arraysof arbitrary geometry [9]

The parameters governing the array performance areobtained using various techniques like method of moments(MoM) multiple signal classification (MUSIC) estimationof signal parameters via rotational invariance techniques(ESPRIT) scheme for spatial multiplexing of local elements(SMILE) and direct data domain (DDD) algorithms Thealgorithms used to estimate the coupling can be extended

2 International Journal of Antennas and Propagation

Weight updating

Feed network

Scattered signalScattered signal

Reflected signal

Reradiated signal

A1 A2 A119873

P2P1 P119873

C1 C2 C119873

middot middot middot



Figure 1 Schematic of adaptive antenna array

towards the compensation of its effect Moreover optimiza-tion techniques such as genetic algorithm (GA) particleswarm optimization (PSO) and linear programming (LP)can be used in conjuncture with these techniques towardsthe enhancement of array efficiency This paper presents aunified review of the techniques and the designs proposedfor mitigating mutual coupling effect in phased arrays Theperformance parameters that get affected due to the couplingeffect in antenna array are discussed The work reportedin open domain towards efficient array design mitigatingmutual coupling effect is reviewed and compared

The subsequent sections describe the analysis and com-pensation of mutual coupling effect in phased arrays usingthese techniques The effect of mutual coupling on the arrayparameters such as antenna impedance and steering vectorwhich further affects the radiation pattern resolution andinterference suppression ability DOA estimation outputSINR response speed and RCS is described in Section 2Section 3 reviews the methods for efficient antenna designand the use of optimization techniques to counteract theadverse effects of mutual coupling on the array performanceIn Section 4 the mutual coupling effects and the techniquesto compensate it for the phased arrays conforming to thesurface are discussed Section 5 presents the cases wherethe presence of coupling proves to be advantageous This isfollowedby a brief summary on the reviewofmutual couplingin Section 6

2 Parameters Affected due to MutualCoupling Effect

21 Antenna Impedance The antenna radiation patterndepends mainly on the impedance at the antenna terminalsHowever the antenna impedance of phased array is signifi-cantly different in comparison to that of an isolated elementThis variation in impedance is due to the presence of couplingbetween the array elements In general for an array of 119873-elements the impedance matrix is given by [2]

119885 =

[[[[

[

11988511+ 119885119871

11988512

sdot sdot sdot 1198851119873

11988521

11988522+ 119885119871sdot sdot sdot 119885

2119873

d

1198851198731

1198851198732

sdot sdot sdot 119885119873119873

+ 119885119871

]]]]

]

(1)

0 1 2 3 4 5 60

10

20

30

40

50

60

70

80

0306090120150180

MagnitudePhase

Mag

nitu

de (119885

)

minus180minus150minus120minus90minus60minus30 Ph

ase (119885

)

119863(120582)

Figure 2 Variation of mutual impedance between two half-wavelength center-fed dipoles with interelement spacing

where 119885119898119899

represents the self (119898 = 119899) and mutual (119898 = 119899)impedances of the antennas and 119885

119871is the terminating load

This impedance matrix yields the net impedance at theantenna terminal [10] that is

119885119909=

119873

sum

119910=1

119885119909119910

(

119868119910

119868119909

) (2)

where 119868119901represents the current at the terminals of 119901th

antenna element and 119885119909119910

is the impedance between 119909th and119910th antenna elements

In general the mutual impedance matrix is a squarematrix with the order corresponding to the array size Thisimplies that the computations involved in arriving at thematrix coefficients increases with the size of array Howeverit is possible to exploit certain properties of coupling forreducing the computation complexity One such propertyis the inverse dependence of coupling coefficients on thedistance between the array elements [11] as shown inFigure 2 It is apparent that farther the elements in the arrayleast will be the mutual impedance and hence the couplingeffect In uniform linear arrays the couplingmatrix possessesrotational symmetry and hence has a symmetric Toeplitzmatrix structure Similarly for a uniform circular array thecoupling matrix is circulant

Furthermore the self- and mutual impedances of (1) aredependent on the type and configuration of the antennaarray The estimation of these impedances is essentially aproblem of finding the current distribution on the antennasurface Among the available methods for estimation of self-and mutual impedances of wire-type antennas the integralequation-moment method [12] and the induced electromo-tive force (EMF) method [13] are the most popular Both ofthese methods are based on the integral forms of inducedcurrent and voltages at the antenna terminals The methodof induced EMF is advantageous over the integral equation-moment method as it facilitates the simplifiedeasier designof antennas by providing the closed form solutions Howeverit is applicable only to the simplified models of straight

International Journal of Antennas and Propagation 3

wire antennas with smaller radii and the antenna arrayswith typical geometries Moreover this method does notaccount for the radius of the antenna wire and the gapsat the feed of the antennas accurately On the other handintegral equation-moment method is valid for both largerradii dipoles and for the arrays with complex geometriesincluding skewed arrangements of elements

Carter [13] determined the mutual impedance betweentwo half-wave dipoles in echelon configuration using inducedEMF method King [14] extended the method for twoantennas of arbitrary and unequal lengths however themethod was found computationally extensiveThis drawbackwas overcome by using a concise formula by Hansen [15]assuming a sinusoidal current distribution for the dipolesin echelon configuration It was shown that the convergentiterative algorithms could be used to analyze the couplingeffects in linear dipole array [16 17]

Ehrlich and Short [18] analyzed the effect of couplingin slot array on a finite ground plane fed by two resonantwaveguides It was shown that the input slot admittance ofthe driven slot changes due to the presence of matched ter-minated parasitic slot waveguide An experimental methodof determining the coupling between themicrostrip antennaswas presented by Jedlicka et al [19]

The mutual impedance between the microstrip dipolesof arbitrary configuration printed on a grounded substratewas determined using integral equation-based method [20]It was shown that the mutual coupling depends on themagnitude of surface wave and hence on the substratethickness for different configurations The surface wavesenhance the mutual coupling to a greater extent in a collineararrangement of the printed dipoles as compared to the broad-side configuration Furthermore for a single propagatingmode the mutual coupling in broadside arrangement wasfound to be mainly due to the direct higher order andleaky waves In broadside configuration the coupling effectis mainly due to TEmode of surface waves while in collinearconfiguration it is due to TMmode

Inami et al [21] estimated the mutual impedancesbetween the rectangular slot antennas located on the concaveside of a spherical surface The surface fields excited by ahorizontal magnetic source located on a conducting concavespherical surface were formulated using the combinationof ray-optical fields and simplified integrals The mutualcoupling between the slots was determined based on theassumption that aperture field is the dominant mode field inthe waveguides connected to slots It is shown that themutualcoupling decays rapidly on the convex side of the surfaceOn the other hand its magnitude exhibits an oscillatorybehaviour when the arc distance is increased on the concaveside

Similar integral equation-based method was employedby Eleftheriades and Rebeiz [22] to estimate the mutualimpedance of arbitrarily placed parallel-aligned narrow slotson a semi-infinite dielectric substrate The electric vectorpotential was determined in terms of the equivalentmagneticcurrents in the slots These currents were expanded usingfast converging basis functions The admittance matrix wasobtained by Galerkinrsquos method in the spectral domain along

the slot length and point matching across their widthsPorter and Gearhart [23] extended this technique to calculatethe mutual coupling in arbitrarily placed perpendicularlyaligned slots on an infinite dielectric substrate In thismethod the admittance matrix was used to determine thecoupling between the slots present in different dielectricsnamely free-space fused quartz and GaAs The copolarizedand cross-polarized radiation patterns were obtained usingthe Fourier transform of the electric fields in the slots Thepeak cross-polarization levels were found strongly correlatedto the level of coupling which in turn depends on bothdielectric constant and the slot geometry This MoM-basedtechnique is useful to determine the mutual impedancebetween slot antennas operating in different polarizations

Pozar [24] proposed the technique of computing the inputand mutual impedances of rectangular microstrip antennasfor various configurations The exact Greenrsquos function for anisotropic grounded dielectric substrate was evaluated usingthe moment method The analysis considered the effects ofboth surface waves and the coupling due to nearby antennasFor similar antenna structures the mutual impedance canbe obtained by solving the reaction integral equation usingMoM [25] Another technique is based on transmission linemodel [26] which neglects the effect of surface waves andapproximates each rectangular resonator in terms of twoequivalent radiating slots

Hansen and Patzold [27] used spectral domain approachto analyze both input and mutual impedance of a rectangularmicrostrip antenna with a dielectric superstrateThismethodis based on the Richmondrsquos reaction and assumes isotropicsubstrates and superstrates with the conducting patcheslocated on the same dielectric substrate The techniqueof computing the mutual coupling between the microstripdipoles in multielement arrays using the dyadic Greenrsquosfunction was presented [28] The dipoles were excited byan electromagnetically coupled transmission line in theisotropic substrates

The reaction theorem was used to calculate the mutualimpedance between two printed antennas of a groundedisotropic dielectric substrate [29] The impedances of amicrostrip array were calculated by replacing the elementsby equivalent magnetic current sources Similar approach ofreplacing the edge aperture field by an equivalent magneticline source has been used [30] The method accounts forthe effect of both isotropic substrate and superstrate onthe edge-fed rectangular microstrip antennas placed on thesame dielectric layer Terret et al [31] analyzed the mutualcoupling in stackedmicrostrip antennas using a combinationof spectral domain Greenrsquos function [24] and the reciprocitytheorem This approach is unique it considers the couplingbetween two printed antennas located on different isotropiclayers

The mutual coupling in arbitrarily located parallel cylin-drical dipoles [32] was determined using a numerical tech-nique based on the method of auxiliary sources (MAS) Inthis method a set of fictitious sources like infinitesimaldipoles were assumed to be located inside the spatiallyoverlapped unequal sized dipoles The EM fields generatedfrom the fictitious sources were subjected to boundary


conditions to obtain the current distribution over the dipoleelements and then the self- and mutual admittances of arrayelements can be estimated Although increasing the numberof fictitious sources enhances the accuracy of the proposedmethod it also has an adverse effect It cannot be deniedthat the method proposed is simple and easily extendable tocomplex antenna arrays such as curtain arrays and Yagi-Udaantennas or for predicting coupling phenomena in closelyspaced transceivers Moreover the method is computation-ally less intense when compared to MoM which involvessingle or double numerical integrations

The fuzzymodelling technique can be employed to calcu-late the input impedance of two coupled monopole antennas[33] or coupled dipoles [34] First the knowledge base for thetwo-coupled dipoles in parallel and collinear configurationswas obtained usingMoMThis knowledge base together withthe concept of spatial membership functions was used topredict the input impedance of two coupled dipole antennasin echelon formThis fuzzy-inference-based method is fasterthan MoM where the computational complexity increaseswith the array size and the accuracy desired

The mutual coupling in a nonlinearly loaded antennaarray [35] was determined using power series expansion[36 37] The method of nonlinear currents [38] was usedto treat every port of a microwave circuit describing theantenna input terminals of an array [39] This method isadvantageous as it provides physical interpretations of thescattering mechanisms within the array structure at differentharmonics The method has no restriction on the antennatype and also accounts for higher order coupling establishingthat the higher order couplings have negligible effect

22 Steering Vector The response of an array towards theincident signal is expressedmathematically as steering vectorThis vector is dependent on the antenna element positionsinter-element spacing radiation characteristics of each ele-ment and the polarization of the incident wave The steeringvector of an array can be obtained by direct measurementof complex array element patterns however it proves to bedifficult Moreover it demands for a huge memory to savethe measured data and hence does not present a practicalapproach [40] The popular method to obtain the steeringvector without storing the data of all searching directionsis to use numerical techniques like MoM [41] Furthermorethe limitation of huge memory requirement can be overcomeby using the distortion matrix with a limited size [42] Thedistortion matrix is derived by comparing the actual arraymanifold (steering vector) with the ideal one in limitednumber of directions

The presence of coupling between the array elementsaffects the steering vector and hence the array response[43] The analysis of the steering vector of an array in thepresence of coupling can be done analytically [44] Katoand Kuwahara [45] studied the effect of mutual couplingon a planar dipole antenna array of vertically polarizedidentical elements with a back reflector Assuming reciprocalantenna elements the steering vector of 119899th driven element isequivalent to the complex array element pattern provided all

CSVUSV

Angle of arrival (deg)

Nor

mal

ized

radi

atio

n pa

ttern

(dB)

minus180 minus120 minus60

0

60 120 180minus50

minus40

minus30

minus20

minus10

0

Figure 3 Synthesized pattern of a 2-element dipole array Desiredsignal (green arrow) 0∘ 40 dB 1 jammer (red arrow) 60∘ 0 dB

other elements are short circuited (zero voltage) The mutualcoupling effect on the array performance was analyzed usingboth conventional steering vector (CSV) and induced EMFmethod Mutual coupling effects in the vertical plane wereshown to be less than that in the horizontal plane for an inter-element spacing of more than half wavelength

As the CSV does not account for coupling effects thecompensation of received voltages becomes mandatory forthe analysis of a practical array To overcome this difficultyYuan et al [42] proposed the method of obtaining the arraymanifold directly using universal steering vector (USV)

The CSV and USV of an array are related to each other bythe following relation

[119860119906(120579 Φ)] = [119862 (120579 Φ)]

minus1[119860119888(120579 Φ)]

with [119862(120579 Φ)]minus1= 119885119871 [119884] [119879 (120579 Φ)]

(3)

where119860119906 and119860119888 representUSV andCSVof the array respec-tively 119879 is the transformation matrix 119884 is the admittancematrix and 119885

119871is the termination impedance of the antenna

The matrix 119862 in (3) depends on the angle and polarizationunder consideration unlike the impedance matrix of Guptaand Ksienski [2]

Thepresence of coupling in between the elements changesthe impedance and hence the radiation pattern of the phasedarray This indicates that the accurate calculation of theradiation pattern of an array is feasible only if (i) CSV withan appropriate compensation technique or (ii) USV is usedfor the calculation This is apparent from Figure 3 The USVis shown to provide the desired radiation pattern by directingthe main beam in the direction of desired signal (at 0∘) andsimultaneously nulling the jammer at 50∘

Zhang et al [46] used MoM for analyzing the perfor-mance of a power inversion adaptive array in the presenceof coupling The quiescent array patterns and the output


SINR depend on the accurate choice of steering vector Theweighting function in the proposed technique is expressed as

[119882] = [119880] + 119892 [Φ]minus1

[1198820] (4)

where 119892 is the loop gain and [1198820] is the steering vector

In order to form a desired receiving array pattern properweights with and without mutual coupling are needed Foran array of omnidirectional antenna elements the steeringvector in the absence of mutual coupling is given by

[11988200] = [1 0 0 0]

119879 (5)

However when mutual coupling is considered the steer-ing vector is modified as

[1198820] = [119885

119871]minus1

[119885] [11988200] (6)

where 119885 and 119885119871indicate the impedance and load matrices

respectivelyLiao and Chan [47] proposed an adaptive beamforming

algorithm based on an improved estimate of calibrated signalsteering vector with a diagonally loaded robust beamformerThe angularly independent mutual coupling was modeled interms of angularly dependent array gain and phase uncertain-ties The mutual coupling matrix used was in the form of abanded symmetric Toeplitz matrix [48] for a uniform lineararray and a circulant matrix for a uniform circular array Theinverse relationship between the mutual coupling betweentwo sensor elements and their relative separation was used

23 Radiation Pattern The effect of coupling on the patternsynthesis of an array can be analyzed using classical tech-niques (like pattern multiplication) numerical techniques(like MoM) and active element patterns (AEP) method [49]The choice of a particular pattern synthesis method for agiven array type needs a tradeoff depending on (i) the type ofarray element (ii) spacing between the elements (iii) arraysize (iv) level of acceptable accuracy and (v) the resourcesavailable for the computation In particular the radiationcharacteristic of an array is highly influenced by the currentfed at the antenna terminals

An adaptive antenna is expected to produce a patternwhich has its main beam towards the desired signal andnulls towards the undesired signals It is also required thatpatterns have sufficiently low sidelobe level (SLL) Althoughthe standard pattern synthesis methods [50] yield lowerSLL in the antenna pattern the elements were assumedto be isotropic in nature Moreover the mutual couplingeffect was ignored in the pattern synthesis which leadsto pattern errors in practical situations The corrections tothe pattern errors for a dipole array were proposed [51]incorporating the mutual coupling effect Two methodsnamely (i) characteristicmode approach and (ii) arraymodewith point matching were presented The characteristicmode-based technique exploits the orthogonality propertiesof characteristic modes to compute the array pattern Thismethod provides volumetric correction at the expense ofcomplex computations This demerit may be overcome by

using point matching for the array modes which need notbe orthogonalThese methods are valid for both uniform andnonuniform arrays provided that array can be expressed inthe form of moment method matrix

The principle of pattern multiplication uses the knowl-edge of currents at the feed terminals of individual elementsto arrive at the complete array pattern This classical tech-nique is applicable only to the array with similar elementsas it assumes identical element patterns for all individualarray elements The radiation pattern of an array is expressedas the product of an element and array factor However ina practical antenna array the presence of coupling resultsin the variation of individual element patterns Moreoverfor an array with electrically large elements which differ insize shape andor orientation the patterns of the individualelements differ considerably This introduces a noise floor inthe array pattern thereby increasing SLL and degrading thequality of the result

Such practical situations for which classical approachesbecome unsuitable can be analyzed using the numerical tech-niques such as MoM or FDTD These numerical techniquesdirectly estimate the current distributions over the antennaelements The voltage and current expansion coefficients arerelated in terms of self- and mutual impedance matricesThese numerical techniques yield accurate results howeverthe size of thematrices increaseswith the array size and beamscanning Moreover these numerical techniques cannot beused for the arrays that are located in highly complexinhomogeneous media

Kelley and Stutzman [49]proposed the method of activeelement patterns (AEP) for analyzing the array pattern inthe situations where both classical and numerical techniquesfail In this method the pattern of complete array is obtainedusing the calculated ormeasured individual element patternsAEP method is faster than the other methods provided thedata of individual element pattern is available in advanceAEP techniques can be broadly classified as nonapproximateAEP methods (unit-excitation AEP method and phase-adjusted unit-excitation AEPmethod) and approximate AEPmethods (average AEP method and hybrid AEP method)

In unit excitation AEP method the individual elementalpatterns are computed assuming the elements excited bya feed voltage of unit magnitude These active elementpatterns represent the pattern of entire array consideringdirect excitation of a single element and parasitic exci-tation of others Furthermore these individual patternsare superimposedsummed-up and scaled by a factor ofcomplex-valued feed voltage applied at the terminals to arriveat the complete array patternThismethod is advantageous asit needs to compute the pattern of individual array elementsonly once Moreover this method is valid for arrays of bothsimilar andor dissimilar elements located in inhomoge-neous linear media

The dependence of the element pattern on the arraygeometry can be explicitly mentioned using an exponentialterm Such approach in which individual element patternsvary due to the presence of additional spatial phase infor-mation factor is called phase-adjusted unit-excitation AEPmethod This method considers the spatial translation of


the array elements unlike the unit-excitation element pat-tern which refers only to the origin of the array coordinatesystem Although the phase-adjusted element patterns differfor different array geometries the concept is useful in thedevelopment of approximate array analysis methods

The computational complexity of both unit-excitationand phase-adjusted unit-excitation methods increases withthe array size due to the need of the active element patterndata for every array element However if the uniform arrayis infinitely large then the phase-adjusted active elementpatterns of individual elements will become identical In suchscenarios the complete array pattern can be expressed inthe form of an average active element pattern which will bethe active element pattern of a typical interior element [52]Although this method yields accurate results for very largeequally spaced arrays it fails for arrays with smaller sizeThisis because the individual elemental patterns of a small arrayvary widely due to edge and mutual coupling effects

Hybrid active element pattern method proposed by Kel-ley and Stutzman [49] can be used for array with moderatenumber of elements In this method the array is dividedinto two groups namely an interior element group and anedge element group The pattern of the interior group isdetermined using average AEP method while the pattern ofedge element group is calculated using phase-adjusted unit-excitation AEP method The total antenna pattern is the sumof the two patterns obtainedThismethod is advantageous asit requires less memory than the exact methods Moreover ityields accurate results for arrays of any antenna type

In a large array almost all the elements experience similarEM environment unlike small array with prominent edgeeffectsThis causes a considerable difference in the individualelement patterns of the array leading to higher SLL More-over small arrays require exceedingly fine control over bothmagnitude and phase of each element for accurate beamsteering Darwood et al [53] analyzed the mutual couplingeffect in a small linear dipole array using the adaptive array-based technique [54] The pattern synthesis yields low SLL inboth sum and difference beams in the presence of couplingeffect In small array [55] voltages at the antenna terminals(with coupling) 119881

119888can be expressed in terms of voltages (no

coupling) 119881119899and coupling matrix 119862 as

119881119888= 119862119881119899 (7)

The mutual coupling compensation can be done bymultiplying the inverse of coupling matrix and 119881

119888as

119881119899= 119862minus1119881119888 (8)

This process is simple for an array with single mode ele-ments the coupling compensation matrix 119862minus1 is scan inde-pendent However an array of multimode elements requiresthe scan-dependent coupling compensation Although thedetermination of the compensation matrix is a difficult taskit is easily realized in a digital beamforming (DBF) antennasystem The performance of large arrays can be predictedfrom the mutual admittance matrices of small arrays ofsimilar lattice [56ndash58]

The coupling matrix can be estimated [55] using eitherFourier decomposition of the measured element patternsor coupling measurements between the array ports Fourierdecomposition method is based on the fact that the couplingcoefficients 119888

119898119899 are the Fourier coefficients of the complex

voltage patterns of the array elements 119892119898(119906) and the isolated

element pattern 119891119894(119906) as

119888119898119899

=1

2120587int

120587119896119889

minus120587119896119889

119892119898(119906)

119891119894 (119906)119890minus119895119899119896119889119906

119889119906 (9)

This solution is based on the assumptions that 119891119894(119906) isindependent of nulls in the integration interval and that theinter-element spacing is larger than half-wavelength Thismethod is applicable only for the nonreciprocal antennasystems as it requires the antennas to be driven in a singlemode either transmit or receive Moreover the derivedcoupling coefficient matrix accounts for channel imbalanceslike the differences in insertion amplitude and phase betweenthe aperture and the output terminal of the antenna element

The second method for calculating coupling coefficientsis based on the scattering matrix 119878 of an array of uniformlyspaced waveguide elements fed by matched generators Thescattering matrix as measured from a reference plane coin-ciding with the plane of element apertures is given by

119862 = 119868 + 119878 (10)

where 119868 denotes the identity matrix In general the scatteringmatrix is measured from a reference plane which is at acertain distance away from the aperture plane Transmissionlines with different insertion loss are included between theplanes of aperture and reference Assuming these feed linesto be matched and reciprocal the modified scattering matrix1198781015840 is given by

1198781015840= 119879119878119879 (11)

where 119879 is the diagonal matrix of transmission coefficientsThis yields the modified coupling matrix 1198621015840 at the referenceplane as

1198621015840= 119879 + 119878

1015840119879minus1 (12)

The measurement of the network parameters becomesdifficult when the feed lines between the element aperturesand output terminals are not matched Furthermore thismethod requires each element to be driven in both transmitand receivemodesThe information about the reference planecorresponding to the phase center of each radiating elementis required which is not feasible for a real array composed ofnonideal elements and complex feed network [59]

Darwood et al [60] showed that the method of SteyskalandHerd [55] is unsuitable for small planar arrays of arbitrarygeometry The compensation of mutual coupling effect wasdone by multiplying active element pattern with the inversecoupling matrix This improves the performance of smallplanar dipole array by reducing SLL and increasing thedirectivity of sum and difference beams This method winsover the Fourier decomposition method as it yields a more


robust compensation for the coupling effects Moreover theestimation of the couplingmatrix is to be carried on only oncefor a given array geometry

The mutual coupling compensation requires the knowl-edge of coupling coefficients which can be waived by usingthe experimental method of applying retrodirective beams[61] This technique yields low SLL patterns by applyingdetermined set of complex weights to each antenna elementSu and Ling [62] compared the approaches thatmodelmutualcoupling as coupling matrix In Gupta and Ksienski [2]approach the terminal voltage of an isolated antenna is takento be the same as that in an array provided all other elementsare open circuited This assumption is impractical as theopen circuit condition does not imply zero current on theantenna elements This approach is valid only if very smallcurrent is induced on half-wave dipole elements On theother hand Friedlander and Weiss [11] approach is validonly if the relationship between the active element patternand the stand-alone element pattern is angle independentThis condition is achieved when (i) all antennas are verticalwires with all incident directions having the same elevationangle [63] and (ii) array elements operate near resonancethat is when the shape of stand-alone current distributionis the same for all incident angles Although this methodoutperforms the approach of Gupta and Ksienski [2] itfails for complex structures Su and Ling [62] employed anextended approach of couplingmatrix formulation for a Yagi-Uda array This technique includes the coupling effect due toboth active and parasitic elements However this method isbound by the conditions of standard approach for parasiticelements and requires the knowledge of large number ofincident angles for unique solution

In general the coupling matrix-based methods assumethat the coupling matrix is an averaged effect of the angle-dependent relationship between the active element patternsand the stand-alone element patterns In such scenariothe minimum mean-square error (MMSE) matching of thetwo pattern sets for a few known incident angle yieldsthe coupling matrix [64 65] The coupling matrix methodthough capable of modeling both coupling and calibrationeffects fails to determine the array pattern accurately [66]This is because these methods neglect the effect of structuralscattering which is significant especially for an antennaconforming to the surface Apart from structural scatteringthe array pattern gets affected due to the calibration errorsThe effects of such calibration issues can be dealt by using theautocalibrationmethods [67ndash70]These techniques are basedon the common criteria that every source of error can berepresented through coupling matrix The calibration tech-nique proposed by Hung [70] was verified experimentally fora narrow-band array in the presence of coupling [71] Thisrobust auto-calibrationmethod is shown to improve the arrayperformance by compensating for the sources of errors

In certain scenarios the antenna arrays are loaded withnonlinear devices in view of protection from the externalpower The analysis of such arrays is complex due to thenonlinear characteristics of each array element For sucharrays Lee [72] approximated the mutual coupling effects bythe infinite periodic array method [73ndash75] This method is

suitable only for large periodic arrays due to infinite periodicGreenrsquos function Poisson sum technique was employed toreduce the analysis of an infinite array into that of a singleantenna element Moreover this approach ignores the edgeeffects

Themutual coupling in a finite array of printed dipoles fedby a corporate feed network was studied by Lee and Chu [76]The analysis was based on the variation in the mismatcheswithin the feed network array pattern and gain due topresence of coupling The self- and mutual impedances for agiven excitation were determined using the spectral domaintechnique [77] This method is applicable for both forcedand free excitations that is radiation impedances with andwithout feed network can be obtained It was shown that theedge effects gain more and more prominence as the arraysize decreases causing greater variations in the impedancevaluesThis in turn leads to the enhancedmultiple reflectionswithin the feed network causing ripples in the amplitude andphase distributions across the antenna aperture The arrayperformance gets degraded especially for large scan angles

The effects of coupling for amicrostrip GSMphased arrayfed by a Butler feed network were analyzed [78]The couplingdistorts the array pattern with higher SLL due to increase inthe reflected power towards the feed network [79]

24 Resolution The presence of mutual coupling amongstthe array elements affects the array resolution adverselyManikas and Fistas [80] proposed a complexmutual couplingmatrix (MCM) given by

119862 = 119860Θ119871Θ119890119895ΦΘ119866Θ119890

119895120587119863 (13)

whereΘ denotes Hadamard product matrix119860 represents therms values of direct and reradiated signals 119871 is the free spacepropagation loss matrix Φ represents the random phasesintroduced to the re-radiated signal by the elements 119866 is thematrix of gain and phase of the elements and119863 is the matrixof inter-element distances Alternatively the MCM can beexpressed as

119862 = (1 sdot 119887119879minus diag (diag (1 sdot 119887119879)) + 119868)Θ119866Θ (119878

119903+ 119868)

(14)

where 119878119903is the matrix whose columns are source position

vectors (SPV) of the re-radiated signals and 119887 is the vectorobtained by pre- and postprocessing of the matrix (119862 sdot 119878 sdot

119878119867sdot 119862119867) The computed MCM is independent of angle

of incidence However it depends on the geometry andthe electrical characteristics of array The associated signalcovariance matrix is given by

119877119909119909= 119862 sdot 119878 sdot 119878

119867sdot 119862119867+ 1205902sdot 119868 (15)

The mutual coupling worsens the array performancefurther if the signals are wideband This is because the arrayloses its ability to match the desired signals or to null thejammers [6 81] over a broad frequency range The couplingis compensated by correcting the actual voltage matrix usingeffectiveweights computed based on the terminal impedance


matrix derived fromMoM impedance matrixThe proposedmethod fails to compensate the loss in antenna gain due towideband signals

The resolution capability of an array is also affected dueto array calibration errors similar to that of its radiation pat-tern Such effects in eigenstructure-based method MUSICpresented by Friedlander [82] are seen to be independent ofsignal-to-noise ratio (SNR) Pierre and Kaveh [68] showedthat the array fails to resolve the signal sources when proneto calibration errors in spite of having high SNR

25 Interference Suppression An adaptive array is expected toaccurately place sufficiently deep nulls towards the impingingunwanted signals The presence of mutual coupling betweenthe antenna elements affects both the positioning and thedepth of the nulls The interference rejection ability dependson array geometry direction of arrivals (DOA) of the signalsand weight adaptation Riegler and Compton [83] analyzedthe performance of an adaptive array prone to mutualcoupling in rejecting the interfering signals using minimummean-square error technique It is shown that for high powerincoming or desired signals the array responds readily as thecorresponding weight vector is updated faster minimizingthe error signal

Adve and Sarkar [84] proposed a numerical techniqueto account for andor to eliminate the coupling effects inlinear array in the presence of near field scatterers SimilarMoM-based approach was used [85] to analyze the couplingeffect on the interference suppression of direct data domain(DDD) algorithms The reported results present an insightof the signal recovery problems in case of a linear array ofequispaced centrally loaded thin half-wavelength dipolesArray analysis in contrast to Gupta and Ksienskirsquos approach[2] uses multiple basis functions for each element Thismethod successfully nulls the strong interferences and iscomputationally simple and efficient However it requires theincoming elevation angles of the signals and interferences tobe equal and known a priori

An improvement over the technique of open circuitvoltage method so as to include the scattering effect ofantenna elements was presented [86] The estimated mutualimpedance matrix was shown to improve the performanceof DDD techniques in nulling the interference Anothertechnique to suppress the interfering signals at the basestations of mobile communication system using normal-mode helical antenna array was proposed by Hui et al [87]This method neither requires the current distributions overthe antenna elements [6] nor the incoming elevation angleof the desired and interfering signals [85] Single estimatedcurrent distribution for every antenna element is requiredto estimate the mutual coupling matrix This method canreduce the coupling effect to an extent but not eliminatingit completely

The current distribution of a small helical antenna isshown to be independent of azimuth angle of the incidentfield if it impinges from horizontal direction [87]This yieldsa reasonably accurate estimate of the current distributionson the antenna elements and hence the mutual impedanceterms This new technique shows an improvement in the

adaptive nulling capability for an array of helical antennaThe improvement of adaptive nulling in dipole array waspresented by Hui [88]The calculation of mutual impedanceswas based on an estimated current distribution with phasecorrections instead of an equal-phase sinusoidal currentdistribution This method performs satisfactorily for stronginterfering signals and for the elevation angles (of the signalsand interferences) which are not too far from the horizontaldirection This method is less sensitive to the variation inelevation angle and the intensity of signal of interest (SOI) Itwas further used to compensate the mutual coupling effect inuniform circular array [89] estimating the DOA using maxi-mum likelihood (ML) algorithm Although optimization oflog-likelihood function of the ML method overcomes itscomplexity this method is used mostly for ULAs Moreoverthis method does not assure global convergence in generalcases [90 91]

Some special techniques were proposed for small andultrawideband arrays Darwood et al [60] used MoM tocompute the coupling coefficients of small planar array ofprinted dipoles The depth of nulls was improved by mutualcoupling compensation The performance of LMS array ofdipoles for wideband signal environment was analyzed byZhang et al [92]The array size was shown to aid the effectivesuppression of the wideband interfering signal Adaptivenulling in small circular and semicircular arrays in the pres-ence of coupling and edge effects was presented [4] Broadernulls were achieved incorporating multiple constraints overa small angular region However the constraints increaseSLL in the array pattern as more number of degrees offreedom was used in null placementThe ability of linear andcircular dipole array to reject the interfering signal in thepresence of mutual coupling effect was studied by Durraniand Bialkowski [93] The signal-to-interference ratio (SIR)was shown to improve with array size up to certain limitHowever SIR of linear array gets degraded as onemove fromthe broad side to the end fire unlike circular array This isbecause of the coupling effect which is more pronouncedat the broadside of a linear array than that in the case of acircular array of half-wavelength spaced dipoles

26 Direction of Arrival (DOA) An adaptive array needs toestimate accurately the emitter location (DOA) and otherdetails so as to suppress it effectively DOA estimationdepends on the array parameters determined by various tech-niques These techniques are either spectral based or para-metric based [94] In spectral-based approach the locationsof the highest peaks of the spectrum are recorded as the DOAestimates On the other hand parametric techniques performsimultaneous multidimensional search for all parameters ofinterest

MUSIC and ESPRIT algorithms are among the popularmethods for DOA estimation The sensitivity of the MUSICalgorithm to the system errors in the presence of couplingwas studied by Friedlander [82] The effect of phase errors isless in linear arrays while the gain errors affect the sensitivityof both linear and nonlinear arrays identically The lineararrays do not fail to resolve the sources due to mergingof spectral peaks The sensitivity of MUSIC algorithm in


case of a non-linear array is inversely proportional to thesource separation The optimal design of an array requiresa tradeoff between the cost required for accurate calibrationand the cost incurred due to an increase in the arrayaperture Friedlander and Weiss [11] proposed the couplingmatrix inclusion in DOA estimation This eigenstructure-based method uses only signals of opportunity to yield thecalibrated array parameters A banded Toeplitz couplingmatrix for linear arrays and banded circulant couplingmatrixfor circular arrays were used This method neither requiresthe knowledge of array manifold nor the locations of theelements a priori Roller and Wasylkiwskyj [95] attempted toanalyze the effect of both coupling and terminal impedancemismatches on the DOA estimation capacity of an isotropicarrayThis approach unlike Friedlander andWeissrsquos approach[11] does not change theMUSIC algorithmThe improvementin the angle of arrival (AOA) estimation is achieved byvarying the impedances at the antenna terminations Thisindicates that the array recalibration might not be requiredif the error is essentially due to mutual coupling between theantenna elements

A preprocessing technique for accurate DOA estimationin coherent signal environment was proposed for uniformcircular array [43] This method in conjunction with spatialsmoothing is capable of tackling the effects of coupling andarray geometry imperfections in narrowband signal environ-ment The received signal vector of the array is transformedto a virtual vector on which spatial smoothing technique isapplied The eigenstructure of the signal covariance matrixvaries by a factor of inversed normalized impedance matrixdue to the presence of coupling The coupling effect canbe compensated either by multiplying the search vector byinverse of the normalized impedance matrix or by resolvingthe coherent signals using spatial smoothing technique [63]A similar method of transforming search vector to nullifycoupling effect is proposed for a circular dipole array [96]

The coupling affects the phase vectors of radiationsources which in turn varies the signal covariance matrixand its eigenvalues affecting the array performance [80]The direct application of spatial smoothing schemes is notfeasible when the coupling effects are prominent This isbecause in such situations the phase vectors at the sourceand subarray do not differ only in phase This necessitatesadditional computations to reconstruct the signal and noisesubspaces and hence to improve the estimation capability

Pasala and Friel [6] analyzed the accuracy of DOAestimation using MUSIC algorithm for signals distributedover a broad frequency range Results were presented forlinear arrays comprising of dipole sleeve dipole and spiralantenna The method of moments was used for calculatingthe induced current and the actual voltages The couplingeffect is least for spiral antenna as compared to the dipoleand the sleeve dipole Although the spiral antenna elementcan mitigate the coupling effect it fails in accurate DOAestimation owing to its grating lobes

The coupling effect on the performance of ESPRITalgorithm for a uniform linear array was studied by HimedandWeiner [97] usingmodified steering vector Swindlehurstand Kailath [98] employed statistical approach to study the

first-order effects of gain and phase perturbations sensorposition errors mutual coupling effects and channel pertur-bations onMUSIC algorithmA similar performance analysiswas carried formultidimensional subspace-fitting algorithmslike deterministic ML multidimensional MUSIC weightedsubspace fitting (WSF) and ESPRIT [99]

Fletcher and Darwood [4] analyzed the beam synthesisand DOA estimation capacity of small circular and semicir-cular arrays It is shown that the smaller circular arrays are lessaffected due to the effect of mutual coupling when comparedto that of small linear or planar or semicircular arraysThis isbecause the linear or planar or semicircular arrays have edgeeffects which lead to the failure of conventional beamformingalgorithms

The coupling effect inDOA estimation capacity of a smartarray of dipoles was studied using Numerical Electromagnet-ics Code (NEC) The NEC simulation considers the couplingeffect by using compensated steering vectors [41] beforeapplying any DOA estimation algorithm such as Bartlett orMUSIC Su et al [100] employed coupling matrix method forDOA estimation in circular arrays This approach makes useof both full-wave electromagnetic solver NEC and MUSICalgorithm for direction finding The technique is effective forsimple antenna structures provided the array calibration datais accurate

Inoue et al [40] analyzed the performance of MUSICand ESPRIT algorithms for the cases of uniformly spaceddipole and sleeve antenna arrays It is shown that theadverse effects of coupling and position errors gain moreand more prominence for larger angles of arrival Similarto uniform arrays the performance of non-uniform arraysalso degrades due to mutual coupling [101]The conventionaladaptive algorithms for analyzing these non-uniform arraysare inadequate [68 102]

Another approach based on the concept of interpolatedarrays was proposed [103] for non-linear arrays A semicircu-lar array composed of half-wave thin wire centrally loadeddipole antennas prone to coupling near-field scattererscoherent jammers clutter and thermal noise was consideredThe approach used Galerkin method in conjunction withMoM The method transforms non-uniform array into avirtual array of omnidirectional isotropic sources beforeapplying direct data domain least squares algorithm

Accurate DOA estimation in the presence of couplingfor a normal mode helical antenna and dipole array waspresented [88 104]Thismethod is based on receivingmutualimpedance [87] and does not rely on the current distributions[6] or the elevation angles of the incoming signals [85] Theuncoupled voltage vector is related to the coupled voltagevector by the relation [105]

[[[[

[

1198811198991198881

1198811198991198882

119881119899119888119873

]]]]

]

=

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

[

1 minus11988512

119905

119885119871

sdot sdot sdot minus1198851119873

119905

119885119871

minus11988521

119905

119885119871

1 sdot sdot sdot minus1198852119873

119905

119885119871

d

minus1198851198731

119905

119885119871

minus1198851198732

119905

119885119871

sdot sdot sdot 1

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

]

[[[[

[

1198811198881

1198811198882

119881119888119873

]]]]

]

(16)


where [119881119899119888] and [119881

119888] represent uncoupled and coupled

voltage vectors respectively 119885119898119899119905

is the receiving mutualimpedance between the 119898th and 119899th antenna elements and119885119871represents the terminating load This equation is readily

comparable with the expression given by Gupta and Ksienski[2] that relates the open circuited voltages 119881

119900 with the

coupled voltage vectors as

[[[[

[

1198811199001

1198811199002

119881119900119873

]]]]

]

=

[[[[[[[[[[[

[

1 +11988511

119885119871

11988512

119885119871

sdot sdot sdot1198851119873

119885119871

11988521

119885119871

1 +11988522

119885119871


119885119871

d

1198851198731

119885119871

1198851198732

119885119871

sdot sdot sdot 1 +119885119873119873

119885119871

]]]]]]]]]]]

]

[[[[

[

1198811198881

1198811198882

119881119888119873

]]]]

]

(17)

where 119885119898119899

represents the conventional mutual impedancebetween119898th and 119899th elements

The mutual impedance terms in (16) are calculated interms of phase-corrected current distribution instead ofequal-phase sinusoidal currents used in (17)This enables theproposed method to overcome the over-simplified assump-tion of current distribution [2] and arrive at a closer estimateof compensated voltage However the method is valid onlyfor receiving antennas due to the immature technology incase of transmitters

As already mentioned an array response towards theincident field is accurately expressed by USV and not byCSV Thus the accuracy of DOA estimation algorithms canbe improved using USV [42] The method employs MoM toarrive at USV and MUSIC algorithms for DOA estimationThe results were shown for the arrays of dipoles monopolesand planar inverted-F antenna (PIFA) mounted on mobilehandsets The method does not require any further compen-sation for the received voltages to remove the coupling effects

Any conventional method assumes a ULA for mutualcoupling compensation Lindmark [59] proposed the exten-sion of the method of Gupta et al [66] to account for bothco- and crosscoupling in dual polarized arrays The arrayresponse using Vandermonde structure was expressed as

119860ULA (120601) = [119886ULA (1206011) 119886ULA (1206012) 119886ULA (120601119889)](18a)

with

119886ULA (120601) = 119890minus119895119896119889(119873minus1) sin(1206012)

[[[[

[

1

119890119895119896119889 sin120601

119890119895119896119889(119873minus1) sin120601

]]]]

]

(18b)

where119873 is the number of array elements 119896 is wave number120601 is the angle of incidence and 119889 is the inter-element spacing

05 1

SIN

R (d

B)

0

8

16

24

32

minus1 minus05 0sin(120601119889)

120585119889 = 5dB120585119889 = 10dB

120585119889 = 20dBNo mutual coupling

Figure 4 Effect of 120585119889on output SINR of a 6-element array of half-

wavelength center-fed dipoles

The array response with nonunit amplitudes was used tomimic the behavior of a real array that is

119886 (120601) = cos119899 (120601) 119886ULA (120601) (19)

where 119899 is an exponent chosen to best fit the elements in useA mutual coupling compensation technique is employed foran array rotated around a point off its phase center

The techniques of coupling compensation proposed byCoetzee and Yu [106] and Chua and Coetzee [107] are basedon decoupling the input ports of the feeding networks Yu andHui [105] presented the design of coupling compensation net-work for a small and compact-size receiving monopole arrayThe proposed method provides the coupling-free voltages atthe antenna terminals and is more suitable for applicationssuch as beamforming and direction finding

27 Output SINR and Response Speed The presence of cou-pling between the antenna elements affects both steady stateand transient response of an array In general the outputSINR represents the steady-state performance of the arraywhile the transient response is expressed in terms of speedof array response Figure 4 shows the change in output SINR(steady-state performance) of 6-element uniform dipolearray with and without mutual coupling effect The effect ofratio of the desired signal power to thermal noise power 120585

119889

on the array performance is analyzedThe output SINR of a least mean square (LMS) adaptive

array in the presence of multiple interfering signals is givenby [2 108]

SINR = 120585119889119880119879

119889119877minus1

119899119880lowast

119889 (20)

where 119877119899is the covariance matrix of the undesired signals

(interference signals and thermal noise) 120585119889the ratio of

desired signal power to thermal noise power 119879 denotestranspose and 119880

119889is the desired signal vector ofthe array


1 15

Dipole

Feed point middot middot middot

2

1205822 1205822 1205822 1205822

119889 119889

119873 = 16

(a)

0

6

12

18

24Varying interelement spacing

05 1

SIN

R (d

B)

minus1 minus05 0sin(120601119889)

No mutual coupling119889 = 025120582

119889 = 05120582

119889 = 120582

(b)

Figure 5 Effect of inter-element spacing on output SINR of a 16-element array of half-wavelength center-fed dipoles 120585119889= 10 dB 120579

119889= 90∘

(a) Schematic of dipole array (b) Output SINR

For narrowband signals uniformly distributed over(0 2120587) 119877

119899and the steady state weight vector are expressed

as [108]

119877119899= 119868 +

119898

sum

119896=1

120585119894119896119880lowast

119894119896119880119879

119894119896

119882 = 119870119877minus1

119899119880lowast

119889

(21)

where 119870 is the constant 119868 is an identity matrix of array-size 119898 is the number of jammers 120585

119894119896is the ratio of the 119896th

jammer power to the thermal noise power and 119880119894119896is the 119896th

jammer vector The coupling between the elements changesthe covariance matrix and the weight vector as [2]

119877119899= 119885lowast

0119885119879

0+

119898

sum

119896=1

120585119894119896119880lowast

119894119896119880119879

119894119896

119882 = 119870119885119879

0119877minus1

119899119880lowast

119889

(22)

where 1198850

is the normalized characteristic impedanceobtained from (1) Equations (20) and (22) give insight of theoutput SINR in the presence of mutual coupling Figure 5shows the dependence of output SINR on inter-elementspacing and hence the coupling factor A reduction in

inter-element spacing will reduce the antenna apertureand hence the incident energy due to the desired signalHowever the noise being internal to the receiver array willnot be affected causing reduced output SINR Moreover theperformance of output SINR degrades due to the additionof more elements into the fixed array aperture (Figures 6and 7) This is because in such cases total thermal noise ofthe array increases while the available signal power remainsconstant Further when mutual coupling is considered theoutput SINR of an array would also depend on incidentangle of the desired signal (Figure 4) However this is notobserved for no mutual coupling case

The output SINR is proportional to the gain of adaptivesystem based on its input SINR The presence of couplingaffects both input and output SINRs especially if the inter-element spacing is less [109] Although the gain of thesystem reduces as both input and output SINRs degradedue to coupling the adaptive processing remains unaffectedMoreover the insertion of invertible compensation matrixwould not improve output SINR as it has no effect on theoutput signal

One of the desired characteristics of the adaptive arrayis its ability to adapt to the changes in signal environmentinstantlyThis requires a quick updating process of the weightvector of an array which in turn is a function of feedback loop

12 International Journal of Antennas and PropagationSI

NR

(dB)

0

12

24

36

48

05 1minus1 minus05 0

119873 = 6

119873 = 16

119873 = 32

119873 = 64

119873 = 256

119873 = 512

119873 = 1024

119863(120582)

Figure 6 Comparison of output SINR in the presence of mutualcoupling by varying the number of antenna elements

0 2 4 6 8 10 12

SIN

R (d

B)

0

4

8

12

16

Number of elements

No mutual couplingWith mutual coupling

Figure 7 Variation of output SINR of an array of half-wavelengthcenter-fed dipoles of fixed aperture with array size 120585

119889= 5 dB 119885

119871=

119885lowast

119894119894 (120579119889 120601119889) = (90

∘ 0∘) total aperture = 2120582

gain steering vector and signal covariancematrixThe signalcovariance matrix of an adaptive array is expressed as [2]

119877119909119909= 1205902[119868 +

119898

sum

119896=1

120585119894119896(119885minus1

119900119880119894119896) lowast (119885

minus1

119900119880119894119896)119879

+ 120585119889(119885minus1

119900119880119889) lowast (119885

minus1

119900119880119889)119879

]

(23)

where 1205902 is the thermal noise power 120585119889is the ratio of desired

signal power to the thermal noise power 120585119894119896represent the

ratio of 119896th jammer power to the thermal noise power119880119889and

119880119894119896are the desired and 119896th jammer vectors and119898 represents

the number of jammersSince the covariance matrix depends on the impedance

of antenna elements themutual coupling affects the transientresponse of the arrayThe coupling changes the eigenvalues ofthe signal covariance matrix Smaller inter-element spacingcauses greater coupling effect lowering the eigenvalues andhence longer transients This reduces the speed of responseof an array resulting in delayed suppression of jammers Theperformance analysis of an adaptive array in the presence ofcoupling by Dinger [110] Gupta and Ksienski [2] Leviatan etal [1] andZhang et al [3] holds only for narrow-band signalsThe effect of mutual coupling on the array performance inwideband signal environment was analyzed usingMoM [92]The output SINR in the presence of mutual coupling showsmore oscillations than in no mutual coupling case Althoughthe coupling effect is similar to that for narrow band signalsit is more pronounced in small arrays This indicates that thecoupling effect can be mitigated by increasing the array size

28 Radar Cross Section (RCS) Themajor focus for strategicapplications is towards the reduction of radar cross section(RCS) of antenna array while maintaining an adequate arrayfunctionality in terms of gain beam steering and interferencerejection This necessitates the analysis and compensationof mutual coupling in array system The RCS of an arrayis affected by coupling effects mutual coupling changes theterminal impedance of the antenna elements and hence thereflection coefficients within the feed network The couplingeffect depends on the type of antenna element array geom-etry scan angle and the nature of feed network Figure 8shows the schematic of series-fed dipole array which canhave different configurations namely collinear parallel-in-echelon and side-by-side configurations Figures 9 through11 show that the coupling effect is least with collinearconfiguration of series-fed dipole array as compared to side-by-side and parallel-in-echelon arrays Beam scanning overthe large angle has greater coupling effect on the RCS patternirrespective of its configuration

Abdelaziz [111] tried to improve the performance of amicrostrip patch antenna array by using an absorbing radarcover The mutual coupling between the microstrip patchantennas is due to both space and surface waves In particularthe surface wave contributes to the coupling and scattering[112 113] It is shown that the surface waves is reducedconsiderably by using a radar absorbing cover The couplingfactor between the elements is given by

119862119898119899

=119884119898119899

119884119899119899+ 119884119892

(24)

where 119884119898119899

is the mutual admittance between 119898th and 119899thantenna elements 119884

119899119899is the self admittance of 119899th antenna

and 119884119892is the generator or feed line admittance


21

Receiveport

Terminating load

Dipoles

Couplers Matched

termination

λ2

Phaseshifters

119889 = 01120582

1205822 1205822 1205822

119873 minus 1 119873 = 30119889 = 01120582


Figure 8 Schematic for 30-element series-fed dipole array

0 20 40 60 80

0

10

20

minus60

minus50

minus40

minus30

minus20

minus10

minus80 minus60 minus40 minus20

Monostatic angle 120579 (deg)

1205901205822

(dB)

Without mutual couplingWith mutual coupling

(a)

0 20 40 60 80

0

10

20

minus60

minus50

minus40

minus30

minus20

minus10



1205901205822

(dB)


(b)

Figure 9 Effect of mutual coupling on RCS of series-fed linear collinear dipole array of 119873 = 30 120595 = 1205872 119889 = 01120582 119897 = 05120582 119886 = 10minus5120582

1198850= 75Ω and 119885

119897= 150Ω unit amplitude uniform distribution (a) 120579

119904= 0∘ (b) 120579

119904= 85∘

Knowing the coupling factor actual excited voltages at theantenna terminals are obtained as

119881119899= 119881

app119899

minus

119873

sum

119898=1

119898 = 119899

119862119898119899119881

app119899

(25)

where 119881app119899

is the applied voltage and 119873 is the number ofelements in the arrayThe proposed method shows reductionin mutual coupling as well as RCS over a wide band offrequencies without affecting the antenna parameters

Zhang et al [114 115] presented both the radiation andscattering patterns of the linear dipole array in the presenceof couplingThe radiation and scattered fields are determinedin terms of self- and mutual impedance and terminal loadimpedance The array performance is improved by optimiz-ing the position of array elements so as to have a low sideloberadiation and scattering pattern It does not account for the

effect of secondary scattering The method can be used forplanar arrays as well

3 Optimization Techniques for Reduction inCoupling Effect

The array performance can be further improved by opti-mizing the array parameters These optimization techniquescan be either global [55 71] or local in nature The globaloptimization techniques optimize uniformly over all thedirections As a result they will be inefficient in calibratingevery direction properly While the array suppresses theerrors in pilot signal directions the residual errors increasein other directions On the other hand the local optimizationtechniques are based on the application of array calibrationfor each and every direction separately


0 20 40 60 80

0

10

20

minus60

minus50

minus40

minus30

minus20

minus10



1205901205822

(dB)


(a)

0 20 40 60 80

0

10

20

minus60

minus50

minus40

minus30

minus20

minus10



1205901205822

(dB)


(b)

Figure 10 Effect of mutual coupling on RCS of series-fed linear parallel-in-echelon dipole array of 119873 = 30 120595 = 1205872 119889 = 01120582 119897 = 05120582119886 = 10

minus5120582 1198850= 125Ω and 119885


119904= 0∘ (b) 120579

119904= 85∘

0 20 40 60 80

0

10

20

minus60

minus50

minus40

minus30

minus20

minus10



1205901205822

(dB)


minus80

(a)

0 20 40 60 80

0

10

20

minus60

minus50

minus40

minus30

minus20

minus10



1205901205822

(dB)


minus80

(b)

Figure 11 Effect of mutual coupling on RCS of series-fed linear side-by-side dipole array of119873 = 30 120595 = 1205872 119889 = 01120582 119897 = 05120582 119886 = 10minus5120582

1198850= 150Ω and 119885


119904= 0∘ (b) 120579

119904= 85∘

31 Array Design In the preceding sections the effects of thecoupling on the array performance and their compensationwere discussed It should be noted that the source of errorsthat hinder the array performance are due to improperantenna designs In other words a careful and efficient designof an antenna system can effectively minimize the mutualcoupling between the array elements

Lindmark et al [116] proposed a design of a dual-polarized 12 times 12 planar array for a spatial division multipleaccess (SDMA) system The performance ability in terms ofDOA estimation was analyzed using least squares estimationof signal parameters via rotational invariance techniques

(TLS-ESPRIT) The proposed array design showed betterresults than that of Pan and Wolff [117] The improvementwas due to theminimization of coupling effects in corrugatedarray design However cross polarization was more and thedesign was cost-inefficient [59]

Another design technique to mitigate the effects ofcoupling is to use dummy columns terminated with matchedloads on each side of the array This is effective as it pseudoequalizes the environment around the outer columns of thearray to that at its inner columns Although such an arraydesign shows an improved performance [118] it is not cost-effective


A wideband folded dipole array in the presence of mutualcoupling was analyzed [119] using MoM and closed form ofGreenrsquos function The proposed method is valid only for thinsubstrates or the substrates with a relative permittivity andpermeability close to unity All themetallic surfaces are takenas perfect electric conductors (PEC) The array performanceimproves by using metallic walls to prevent inter-elementcoupling via parallel-plate waveguide modes

In general patch antenna is designed on a thick substratefor wideband performance and higher data rates Howeverthis enhances the coupling effect as thicker substrate sup-ports higher amount of current flow in the form of surfacewaves Fredrick et al [79] presented the coupling compensa-tion for such smart antenna fed by a corporate feed networkThe method is based on the fact that the reduction in flow ofsurface current on the adjacent elements reduces the couplingeffect The magnitude of surface currents on the antennaelements depends on the terminating load impedances ofthe elements These terminal impedances are varied from amatched load condition to an open circuit condition using aswitching PIN diode multiplexer The coupling effect on thearray performance is compensated by locating the switch at aproper location along the feed line of the element

Blank andHutt [120] presented an empirical optimizationalgorithm in two versions The method considered the effectof both mutual coupling and scattering between the arrayelements and nearby environment The method is basedon the measured or calculated element-pattern data andoptimizes the design using an iterative technique The firstversion of the proposed optimization algorithm is used ifinter-element spacing is less than half-wavelength and thecoupling effects do not vary rapidly as a function of elementlocations Although this version considers the effect of passiveelements in the vicinity of the array it cannot optimizetheir locations In the second version induced EMF methodis used to compute the admittance matrix of the arrayconsidering the effect of both active and passive elementsThis helps to arrive at the active element scan impedances andis applicable to an array of arbitrary geometry Moreover itcan optimize the array in terms of both inter-element spacingand element excitations and has a higher rate of convergence

Many attempts have been made to compensate the effectof coupling in microstrip antennas The finite differencetime domain (FDTD) method was used to analyze the arrayof electromagnetic band gap structures composing printedantennas on a single isotropic dielectric substrate [121] Theanalysis of mutual coupling between a two-element array ofcircular patch antennas on an isotropic dielectric substrate ispresented by Chair et al [122]

Yousefzadeh et al [123] designed a linear array of uni-formly fed microstrip patch antennas by iterating the fractalgeometries The performance of such an array surpasses thatof an array with ordinary rectangular microstrip patches asbothmutual coupling effects and the return loss at the input ofeach patch get reduced The array performance is dependenton the fractal characteristics (type size and relative spacing)feed point location and the number of parasitic elementsBuell et al [124] proposed themutual coupling compensationbetween the elements in a densely packed array by using

metamaterial isolation walls The proposed design enhancesthe array performance in controlled beam scanning andnull steering The mutual coupling in an array operating inreceive mode differ from that of a transmitting array A leastsquare approach is used to compensate the coupling effect[125 126] The method is valid only when the number ofemitting sources is greater than the number of elements inthe receiving array

Yang and Rahmat-Samii [121] proposed the technique ofreducing the mutual coupling between two collinear orthog-onal or parallel planar inverted-F antennas (PIFAs) abovea single ground plane with air substrate It is based on theconcept of suppressing the surface wave propagation by usingmushroom-like electromagnetic band gap (EBG) structuresThe fabrication of such structure is complicated On theother hand the slitted ground plane structure suggested byChiu et al [127] is simple economical and can be fabricatedeasilyThe slitted ground plane reduces the coupling betweenthe radiators and thus improves the isolation between themThe method proposed is applicable for nonplanar radiatingelements and large number of array elements The size ofslits and strips required to reduce mutual coupling differsbased on the resonant frequency and the type of antennasin use Other decoupling networks comprise of transmissionline decouplers [128] and capacitively loaded loop (CLL)magnetic resonators [129]

Bait-Suwailam et al [130] used single-negative magnetic(MNG) metamaterials to suppress the electromagnetic cou-pling between closely spaced high-profile monopoles Inthis method the single-negative magnetic inclusions realizedusing broadside coupled split-ring resonators (SRRs) act asantenna decouplersThese structures satisfy the property thattheir mutual impedance should be purely reactive at the res-onance frequency in order to decouple the antenna elementsArrays of such structures when properly arranged and excitedwith a specific polarization mimic the behavior of magneticdipole arrays and have a negative magnetic permeability overa frequency range This prevents the existence of real propa-gating modes within the MNG metamaterials thus avoidingthe coupling between the array elements The analysis of theproposed method in terms of scattering parameters is shownto increase the performance of MIMO system by reducingthe correlation between its array elements Moreover theproposed technique increases the system gain in the desireddirection by reducing back radiation and thus helps inachieving quasiorthogonal patterns

32 Other Optimization Techniques Digital beamforming(DBF) of an array is preferred over analog beamformingowing to low sidelobe beamforming adaptive interferencecancellation high-resolutionDOAestimation and easy com-pensation for coupling and calibration errors [131 132] Atechnique of combined optimization method (COM) basedon the concept of space equalization was proposed [133]Thismethod is a combination of global and local optimisation andhence minimizes local errors in the desired signal directionswhile maintaining the global errors at a tolerable level Theperformance analysis of COMcalibrated uniform linear arrayshows better performance than the one calibrated using


global optimization technique Moreover COM eliminatesthe coupling errors channel (both gain andphase) errors overa broad angular region and compensates for errors due toboth manufacture and nonuniform antenna material

Demarcke et al [134] presented a technique of accu-rate beamforming for a uniform circular microstrip patchantenna subjected to mutual coupling and platform effectsThis DOA-based method of beamforming using AEPs usesthe received information to construct a transmitting weightvector Basically it concentrates the energy in the desiredmain beamdirection(s) minimizing the total radiated powerThis maximizes the gain in the specified main beam direc-tion(s) and hence yields maximum SIR at the receiver Herethe beamforming is treated as a constrained minimizationproblem with complex valued steering vector being itsargument

In general beamforming is viewed as a constrainedoptimization problemThus the evolutionary algorithms andrelated swarm-based techniques useful for solving uncon-strained optimization problems are not applicable readilyfor beamforming The improvement can be achieved byoptimizing the system parameters using algorithms likeparticle swarm optimization (PSO) Basu and Mahanti [135]used modified PSO for reconfiguring the beam of a lineardipole array with or without ground plane In the dual-beam switching technique the self andmutual impedances ofparallel half-wavelength dipoles are obtained using inducedEMF method Multiple beams are generated by switchingthrough real excitation voltages leading to a simple designof the feed network

4 Conformal Array

The mutual coupling in conformal arrays is dependent onthe curvature of surface on which antennas are mounted Amajority of techniques used to analyze the conformal arrays[136ndash138] attempt to reduceavoid or simplify couplingbetween the antenna and platform on which it is mountedThis is not feasible in every scenario as it is not possible toisolate array from its platform or from the environment ofnear-field scatterers Moreover current minimization doesnot always assure an optimal solution as the performanceimproves by reinforcing the currents induced due to cou-pling Thus it is necessary to consider rather than neglect theeffect of platform coupling

Pathak andWang [139] used uniform theory of diffraction(UTD) to compute coupling between the slots on conductingconvex surface This method relies on the surface geodesicsobtained by ray tracing The coupling between the antennasmounted on general paraboloid of revolution (GPOR) canalso be determined using geometrical theory of diffraction(GTD) This ray-based method however requires the geo-metric parameters like arc length Fock parameter and soforth associated with all the geodesics to be known a priori[140] This approach can be extended for quadric cylindersellipsoids paraboloids and for composite quadric surfaceslike cone cylinder parabolic cylinder and so forth [141]

Wills [142] analyzed the effects of creeping fields andcoupling on double curved conformal arrays of waveguide

elements The theoretical far-field patterns were comparedwith the measured results for an ellipsoidal array Perssonet al [143] used a hybrid UTD-MoM method to calculatethe mutual coupling between the circular apertures on asingly and doubly curved perfectly conducting surface [144]The mutual coupling between the circular waveguide fedapertures on the curved surface is shown to be heavilydependent on the polarization This hybrid method dealswith isolated coupling valuesThe accuracy of the parametersinvolved in the UTD formulation relies on the geodesicconstant and can be further improved by including higherorder modes

The radiation pattern of a dipole array mounted on areal complex conducting structure [145] was obtained byincluding coupling between elements coupling due to near-field objects and the coupling between the array and thesurface and the feed network The excitation coefficients ofthe impedance and radiation matrices were obtained using aMoM-based electromagnetic code NEC-2 An optimizationprocedure based on the pattern synthesis algorithm [146]was employed to obtain radiation pattern with low sidelobelevel

Obelleiro et al [147] analyzed the conformal monopoleantenna array considering both the mutual coupling betweenthe array elements and their interaction with the mountingplatform This method modeled the currents induced onthe platform and the antennas using the surface-wire MoMformulation A global-optimization procedure was used toarrive at the optimal excitation coefficients for the arrayelements This approach is applicable for both PEC anddielectric platforms

A method of finite element-boundary integral (FE-BI)was used to determine the mutual impedance between con-formal cavity-backed patch antennas [148] mounted on PECcylinder The vector-edge-based elliptic-shell element basisfunctions were used to describe the field within the cavityregion The field external to the cylinder was representedby elliptic-cylinder dyadic Greenrsquos function The method ofweighted residuals was used to enforce the field continuityacross the cavity aperture and obtain amatrix equation for thebasis function amplitudes The fields within the cavity regionwere obtained using bi-conjugate gradient method The self-andmutual impedances of the patcheswere determined usinginduced EMF method The H-plane coupling exists due toTE-surface wave and is inversely proportional to the surfacecurvature The E-plane coupling is stronger than H-planecoupling due to TM surface-wave mode and is maximum fora specific curvature

A finite array of microstrip patch antennas loaded withdielectric layers on a cylindrical structure was studied byVegni and Toscano [149] The mutual coupling coefficientswere determined using method of lines (MoL) in terms ofcylindrical coordinates The effect of inter-element spacingand the superstrates on the coupling was analyzed It wasshown that the mutual coupling does not decay monotoni-cally with the increasing patch separations Instead it exhibitsoscillatory behaviour depending on the dielectric layer Thesuperstrates used can efficiently reduce the effect of mutualcoupling in the antenna array The proposed method can be


readily extended to multiple stacked layers and patches withsuperstrates with an acceptable increase in computationalcomplexity unlike MoM

The effect of human coupling on the performance of tex-tile antenna like a log periodic folded dipole array (LPFDA)antenna was analyzed [150] The coupling due to humanoperator affects the antenna input impedance by shifting itsresonant frequency and by increasing the return loss Theseadverse effects can be compensated by designing an antennain free space that exhibits a good impedance match over abandwidth say a linear array of wire folded dipoles

The mutual coupling between the apertures of dielectric-covered PEC circular cylinders was determined in terms oftangential magnetic current sources of the waveguide-fedaperture antennasarrays [151] The method involves closedform of Greenrsquos function (CFGF) representations and two-level generalized pencil of function (GPOF) The methodis valid for both electrically small and large cylinders overwider source-field regions Yang et al [152] synthesized thepattern of a vertically polarized rectangular microstrip patchconformal array by decomposing the embedded elementpatterns as a product of a characteristic matrix and a Van-dermonde structured matrix The weights of the modes wereoptimized using a modified PSO technique The synthesizedpattern showed low SLL copolarization beamwith amainlobeconstraint The method being simple in terms of bothmemorystorage requirement and computation can be easilyimplemented for any antenna array provided their embeddedelement patterns differ greatly

The discussion presented so far shows that the conformalarrays can be analyzed using various techniques each withtheir own merits and demerits This indicates that a few ofthem when carefully chosen and combined [24 153] mightresult in a hybrid technique with most of the merits Sipuset al [154] presented such a hybrid technique that combinesspectral domain method with UTD to analyze a waveguidearray embedded in a multilayer dielectric structure It isrobust method and capable of analyzing multilayer electri-cally large structures

Raffaelli et al [155] analyzed an array of arbitrarily rotatedperfectly conducting rectangular patches embedded in amul-tilayered dielectric grounded circular cylinder The cylinderwas assumed to be of infinite length in the axial directionwith infinitesimally thin patches The method involves 2DFourier transform to model fields and currents and MoM tosolve the electric field integral equations (EFIE)The couplingeffect was analyzed in terms of frequency and edge spacingbetween the array elements The embedded element patternvaries more in E-plane than in H-plane due to the strongermutual coupling along E-plane This method can deal withthe structures with larger radii unlike the spectral approach

A conformal conical slot array was optimally synthesizedtaking coupling effect into account [156]The synthesis aimedat maximizing the directivity with minimum SLL and nullingthe interfering signal considering least number of activeelement patternsThis constrained optimization problem canbe readily extended to quadratic constraints other than justlinear ones such as the constraints on main beam radiationefficiency and power in the cross-polarization component

The effect of coupling is compensated using the technique ofSteyskal and Herd [55]

Thors et al [157] presented the scattering propertiesof dielectric coated waveguide aperture antennas mountedon circular cylinders The proposed hybrid technique is acombination of MoM and asymptotic techniques It relies onMoM to solve the integral equation for the aperture fieldsand on asymptotic techniques to compute the coupling Themutual coupling between the rectangular patch antennasin nonparaxial region was computed using the asymptoticsolution [143 158]The self- and themutual coupling terms inthe paraxial region were determined using spectral domaintechnique This constrains the ability of the technique toconsider smaller radii cylinders

5 Advantages of Mutual Coupling

The performance of the phased array deteriorates due to thepresence of coupling however this cannot be generalizedThe presence of coupling is reported to be advantageous incertain cases The coupling has positive effect on the channelcapacity of multiple element antenna (MEA) systems [159]The channel capacity in terms of couplingmatrices of receiverand transmitter is expressed as

1198671015840= 119862119877119867119862119879 (26)

where 1198671015840 and 119867 are the channel capacity with and withoutmutual coupling 119862

119877 and 119862

119879are the coupling matrices at

the receiver and transmitter respectively The coupling effectreduces the correlation between the channel coefficientsfor closely spaced MIMO systems It also compensates thepropagation phase difference of the frequencies lying withinthe range [93] This increases the channel capacity whichis contradictory to the statements [160] Mutual couplingis also reported to increase the efficiency by decreasing thebit-error rate of a Nakagami fading channel Nonradiativecoupling between high-frequency circuits within their near-field zone is found to aid in the efficient transfer of power inwireless applications This kind of biofriendly power transferis free from electromagnetic interference (EMI) caused byscattering and atmospheric absorption unlike radiative EMenergy transfer [161]

The presence of coupling between the array elements isshown to be desirable if arrays are to bemade self-calibrating[162] Furthermore Yuan et al [109] showed that the presenceof coupling improves the convergence of adaptive algorithmThe rate of convergence of an algorithm is shown to decreaseif the eigenvalues of the array covariance matrix differ widelyOn the other hand the convergence becomes faster when theratio of themaximum eigenvalue to theminimum eigenvalueof the covariance matrix decreases This is due to the factthat closer array elements and hence high coupling yield asmaller maximum eigenvalue with approximately the sameminimum eigenvalue This indicates an improvement in theeigenvalue behavior for an array in the presence of couplingevenwhen coherent sources are considered [163]However inmost of the cases the compensation of the mutual couplingeffects is required to obtain optimal array performance


6 Summary

This paper presents an overview of the methods that modelmutual coupling effect in terms of impedance matrix fordifferent arrays Researchers have extended the conventionalmethods based on self- and mutual impedance matrix toinclude the effects of calibration errors and near field scat-terers These methods aimed at compensating the effectsof mutual coupling by including the coupling matrix inthe pattern generation There are autocalibration methodswhich mitigate the effects of structural scattering along withthe mutual coupling mitigation such methods facilitate theanalysis of conformal phased arrays

The trend moved towards developing the techniqueswhich estimate the parameters affecting the performanceof real-scenario arrays accurately in extreme conditionsincluding that for coherent signals with minimum numberof inputs This reduced the computational complexity andfacilitated easy experimental verification and parametricanalysis It has been shown that the accurate array patternsynthesis is feasible only if the effect of coupling on the arraymanifold is considered Thus the USV is used instead of CSVin the coupling analysis of phased arrays mounted over theplatform such as aircraft wings or mobile phones

Further hybrid techniques like UTD-MoM spectraldomain method with UTD were shown to be better in termsof both accuracy and computation Most of these techniquesare suitable only for uniform and infinitely large arraysin narrowband scenarios Therefore adaptive array basedtechnique was developed to deal with the small non-uniformplanar and circular arrays operating over a wide frequencyrange

The effect of mutual coupling on the parameters liketerminal impedances and eigenvalues has been discussedconsidering the radiation pattern steady state response tran-sient response and the RCS of the array Many compensationtechniques are analyzed to mitigate the adverse effects ofcoupling on array performance issues such as high resolutionDOA estimation and interference suppression These havebeen further simplified due to the optimization of antennadesign parameters It is suggested that a good and efficientdesign of an antenna system would compensate for themutual coupling effects However in few cases the presenceof coupling has been proved advantageous as well

References

[1] Y Leviatan A T Adams P H Stockmann and D R MiedanerldquoCancellation performance degradation of a fully adaptive Yagiarray due to inner-element couplingrdquo Electronics Letters vol 19no 5 pp 176ndash177 1983

[2] I J Gupta and A A Ksienski ldquoEffect of mutual coupling on theperformance of adaptive arraysrdquo IEEETransactions onAntennasand Propagation vol 31 no 5 pp 785ndash791 1983

[3] Y Zhang KHirasawa andK Fujimoto ldquoPerformance of powerinversion adaptive array with the effect of mutual couplingrdquo inProceedings of the International Symposium on Antennas andPropagation pp 803ndash806 Kyoto Japan August 1985

[4] P N Fletcher and P Darwood ldquoBeamforming for circularand semicircular array antennas for low cost wireless LAN

data communication systemsrdquo IEE Proceedings on MicrowaveAntennas and Propagation vol 145 pp 153ndash158 1998

[5] M M Dawoud and M K Amjad ldquoAnalytical solution formutual coupling in microstrip patch antenna arraysrdquo ArabianJournal for Science and Engineering vol 31 no 1 pp 47ndash602006

[6] K M Pasala and E M Friel ldquoMutual coupling effects and theirreduction in wideband direction of arrival estimationrdquo IEEETransactions on Aerospace and Electronic Systems vol 30 no4 pp 1116ndash1122 1994

[7] J L Guo and J Y Li ldquoPattern synthesis of conformal arrayantenna in the presence of platform using differential evolutionalgorithmrdquo IEEE Transactions on Antennas and Propagationvol 57 no 9 pp 2615ndash2621 2009

[8] H T Hui K Y Chan and E K N Yung ldquoCompensating forthe mutual coupling effect in a normal-mode helical antennaarray for adaptive nullingrdquo IEEE Transactions on VehicularTechnology vol 52 no 4 pp 743ndash751 2003

[9] B Strait and K Hirasawa ldquoArray design for a specified patternby matrix methodsrdquo IEEE Transactions on Antennas and Prop-agation vol 17 pp 237ndash239 1969

[10] C A Balanis AntennaTheory Analysis and Design JohnWileyamp Sons Hoboken NJ USA 3rd edition 2005

[11] B Friedlander and A J Weiss ldquoDirection finding in thepresence of mutual couplingrdquo IEEE Transactions on Antennasand Propagation vol 39 pp 273ndash284 1991

[12] G A Thiele ldquoAnalysis of Yagi-Uda type antennasrdquo IEEETransactions on Antennas and Propagation vol 17 pp 24ndash311969

[13] P S Carter ldquolsquoCircuit relations in radiating systems and appli-cations to antenna problemsrdquo Proceedings of IRE vol 20 pp1004ndash1041 1932

[14] H E King ldquoMutual impedance of unequal length antennas inechelonrdquo IRE Transactions on Antennas Propagation vol 5 pp306ndash313 1957

[15] R C Hansen ldquoFormulation of echelon dipole mutual impe-dance for computerrdquo IEEE Transactions on Antennas andPropagation vol 20 pp 780ndash781 1972

[16] J A G Malherbe ldquoCalculator program for mutual impedancerdquoMicrowave Journal vol 27 no 2 p 82 1984

[17] J A G Malherbe ldquoAnalysis of a linear antenna array includingthe effects ofmutual couplingrdquo IEEE Transactions on Educationvol 32 no 1 pp 29ndash34 1989

[18] M J Ehrlich and J Short ldquoMutual coupling considerations inlinear-slot array designrdquo Proceedings of the IRE vol 42 pp 956ndash961 1954

[19] R P Jedlicka M T Poe and K R Carver ldquoMeasured mutualcoupling between microstrip antennasrdquo IEEE Transactions onAntennas and Propagation vol 29 no 1 pp 147ndash149 1981

[20] N G Alexopoulos and I E Rana ldquoMutual impedance compu-tation between printed dipolesrdquo IEEE Transactions on Antennasand Propagation vol 29 no 1 pp 106ndash111 1981

[21] K Inami K Sawaya and Y Mushiake ldquoMutual couplingbetween rectangular slot antennas on a conducting concavespherical surfacerdquo IEEE Transactions on Antennas and Propa-gation vol 30 no 5 pp 927ndash933 1982

[22] G V Eleftheriades and G M Rebeiz ldquoSelf and mutual admit-tance of slot antennas on a dielectric half-spacerdquo InternationalJournal of Infrared and Millimeter Waves vol 14 no 10 pp1925ndash1946 1993

[23] B G Porter and S S Gearhart ldquoTheoretical analysis ofcoupling and cross polarization of perpendicular slot antennas


on a dielectric half-spacerdquo IEEE Transactions on Antennas andPropagation vol 46 no 3 pp 383ndash390 1998

[24] D M Pozar ldquoInput impedance and mutual coupling of rectan-gular microstrip antennasrdquo IEEE Transactions on Antennas andPropagation vol 30 no 6 pp 1191ndash1200 1982

[25] E H Newman J H Richmond and B W Kwan ldquoMutualimpedance computation between microstrip antennasrdquo IEEETransactions on Microwave Theory and Techniques vol 31 no11 pp 941ndash945 1983

[26] E H Van Lil and A R van de Capelle ldquoTransmission linemodel for mutual coupling betweenmicrostrip antennasrdquo IEEETransactions on Antennas and Propagation vol 32 no 8 pp816ndash821 1984

[27] V Hansen and M Patzold ldquoInput impedance and mutualcoupling of rectangular microstrip patch antennas with adielectric coverrdquo in Proceedings of the 16th European MicrowaveConference pp 643ndash648 Dublin Ireland October 1986

[28] P B Katehi ldquoMutual coupling between microstrip dipolesin multielement arraysrdquo IEEE Transactions on Antennas andPropagation vol 37 no 3 pp 275ndash280 1989

[29] A H Mohammadian N M Martin and D W GriffinldquoTheoretical and experimental study of mutual coupling inmicrostrip antenna arraysrdquo IEEE Transactions on Antennas andPropagation vol 37 no 10 pp 1217ndash1223 1989

[30] A Benalla K C Gupta and R Chew ldquoMutual couplingbetween rectangular microstrip patches covered with a dielec-tric cover layerrdquo in Proceedings of the Antennas and PropagationSociety International Symposium Merging Technologies for the90rsquosrsquo Digest pp 358ndash361 May 1990

[31] C Terret S Assailly K Mahdjoubi and M Edimo ldquoMutualcoupling between stacked square microstrip antennas fed ontheir diagonalrdquo IEEE Transactions on Antennas and Propaga-tion vol 39 no 7 pp 1049ndash1051 1991

[32] P J Papakanellos and C N Capsalis ldquoStudy of two arbitrarilylocated parallel cylindrical dipoles based on an auxiliary sourcestechniquerdquo Electromagnetics vol 23 no 5 pp 399ndash416 2003

[33] M Tayarani and Y Kami ldquoFuzzy inference in engineeringelectromagnetics an application to conventional and angledmonopole-antennardquo IEICETransactions onElectronics vol E83no 1 pp 85ndash95 2000

[34] S R Ostadzadeh M Soleimani and M Tayarani ldquoA fuzzymodel for computing input impedance of two coupled dipoleantennas in the echelon formrdquo Progress in ElectromagneticsResearch vol 78 pp 265ndash283 2008

[35] K C Lee and T H Chu ldquoMutual coupling mechanismswithin arrays of nonlinear antennasrdquo IEEE Transactions onElectromagnetic Compatibility vol 47 no 4 pp 963ndash970 2005

[36] D Westreich ldquoEvaluating the matrix polynomialI+A+A2++A119873minus1rdquo IEEE Transactions on Circuits and Systemsvol 36 no 1 pp 162ndash164 1989

[37] K C Lee and T H Chu ldquoA circuit model for mutual couplinganalysis of a finite antenna arrayrdquo IEEE Transactions on Electro-magnetic Compatibility vol 38 no 3 pp 483ndash489 1996

[38] S A Maas Nonlinear Microwave Circuits Artech HouseNorwood Mass USA 1988

[39] K C Lee and T H Chu ldquoAnalysis of injection-locked antennaarray including mutual coupling effectsrdquo IEEE Transactions onAntennas and Propagation vol 52 no 11 pp 2885ndash2890 2004

[40] Y Inoue K Mori and H Arai ldquoDOA estimation in consider-ation of the array element patternrdquo in Proceedings of the IEEE55th Vehicular Technology Conference (VTC rsquo02) vol 2 pp 745ndash748 May 2002

[41] K R Dandekar H Ling and G Xu ldquoEffect of mutual couplingon direction finding in smart antenna applicationsrdquo ElectronicsLetters vol 36 no 22 pp 1889ndash1891 2000

[42] Q Yuan Q Chen and K Sawaya ldquoAccurate DOA estimationusing array antenna with arbitrary geometryrdquo IEEE Transac-tions on Antennas and Propagation vol 53 no 4 pp 1352ndash13572005

[43] M Wax and J Sheinvald ldquoDirection finding of coherentsignals via spatial smoothing for uniform circular arraysrdquo IEEETransactions on Antennas and Propagation vol 42 no 5 pp613ndash620 1994

[44] Y Kuwahara ldquoSteering vector of MSPA array taking mutualcoupling effects into considerationrdquo in Proceedings of the Inter-national Symposium on Antennas and Propagation pp 1614ndash1617 ACROS Fukuoka Japan August 2000

[45] A Kato and Y Kuwahara ldquoSteering vector taking mutualcoupling effects into considerationrdquo in Proceedings of the IEEEAntennas and Propagation Society International Symposium vol2 pp 918ndash921 Salt Lake City Utah USA July 2000

[46] Y Zhang K Hirasawa and K Fujimoto ldquoAnalysis of powerinversion adaptive array performance by moment methodrdquoTransactions of IEICE vol 71 pp 600ndash606 1988

[47] B Liao and S C Chan ldquoAdaptive beamforming for uniformlinear arrays with unknown mutual couplingrdquo IEEE Antennasand Wireless Propagation Letters vol 11 pp 464ndash467 2012

[48] Z Ye and C Liu ldquoNon-sensitive adaptive beamforming againstmutual couplingrdquo IET Signal Processing vol 3 no 1 pp 1ndash62009

[49] D F Kelley and W L Stutzman ldquoArray antenna patternmodeling methods that include mutual coupling effectsrdquo IEEETransactions on Antennas and Propagation vol 41 no 12 pp1625ndash1632 1993

[50] R S Elliot Antenna Theory and Design Prentice-Hall Engle-wood Cliffs NJ USA 1st edition 1981

[51] Y W Kang and D M Pozar ldquoCorrection of error in reducedsidelobe synthesis due to mutual couplingrdquo IEEE Transactionson Antennas and Propagation vol 33 no 9 pp 1025ndash1028 1985

[52] W Wasylkiwskyj and W K Kahn ldquoElement patterns andactive reflection coefficient in uniform phased arraysrdquo IEEETransactions on Antennas and Propagation vol 22 no 2 pp207ndash212 1974

[53] P Darwood P M Fletcher and G S Hilton ldquoPattern synthesisin small phased arrays using adaptive array theoryrdquo ElectronicsLetters vol 33 no 4 pp 254ndash255 1997

[54] E C Dufort ldquoPattern synthesis based on adaptive array theoryrdquoIEEE Transactions on Antennas and Propagation vol 37 no 8pp 1011ndash1018 1989

[55] H Steyskal and J S Herd ldquoMutual coupling compensationin small array antennasrdquo IEEE Transactions on Antennas andPropagation vol 38 no 12 pp 1971ndash1975 1990

[56] S Karimkashi and A A Kishk ldquoFocused microstrip arrayantenna using a Dolph-Chebyshev near-field designrdquo IEEETransactions on Antennas and Propagation vol 57 no 12 pp3813ndash3820 2009

[57] A A Kishk and S L Ramaraju ldquoDesign of different config-urations of finite microstrip arrays of circular patchesrdquo Inter-national Journal of Microwave and Millimeter-Wave Computer-Aided Engineering vol 4 no 1 pp 6ndash17 1994

[58] A A Kishk ldquoPrediction of large array characteristics fromsmall array parametersrdquo in Proceedings of the 2nd EuropeanConference on Antennas and Propagation (EuCAP rsquo07) pp 584ndash588 Edinburgh UK November 2007


[59] B Lindmark ldquoComparison of mutual coupling compensationto dummy columns in adaptive antenna systemsrdquo IEEETransac-tions on Antennas and Propagation vol 53 no 4 pp 1332ndash13362005

[60] P Darwood P N Fletcher and G S Hilton ldquoMutual couplingcompensation in small planar array antennasrdquo IEE ProceedingsonMicrowave Antennas and Propagation vol 145 pp 1ndash6 1998

[61] PN Fletcher andMDean ldquoApplication of retrodirective beamsto achieve low sidelobe levels in small phased arraysrdquoElectronicsLetters vol 32 no 6 pp 506ndash508 1996

[62] T Su and H Ling ldquoOn modeling mutual coupling in antennaarrays using the coupling matrixrdquoMicrowave Optical Technolo-gies Letters vol 28 pp 231ndash237 2001

[63] C C YehM L Leou andD R Ucci ldquoBearing estimations withmutual coupling presentrdquo IEEE Transactions on Antennas andPropagation vol 37 no 10 pp 1332ndash1335 1989

[64] C M S See ldquoA method for array calibration in parametricsensor array processingrdquo in Proceedings of the IEEE SingaporeInternational Conference on Communications Systems pp 915ndash919 Singapore November 1994

[65] A N Lemma E F Deprettere and A J Veen ldquoExperimentalanalysis of antenna coupling for high-resolution DOA estima-tion algorithmsrdquo in Proceedings of the IEEEWorkshop on SignalProcessing Advances in Wireless Communications pp 362ndash365Annapolis Md USA May 1999

[66] I J Gupta J R Baxter S W Ellingson H G Park H S Ohand M G Kyeong ldquoAn experimental study of antenna arraycalibrationrdquo IEEE Transactions on Antennas and Propagationvol 51 no 3 pp 664ndash667 2003

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[68] J Pierre and M Kaveh ldquoExperimental performance of cal-ibration and direction-finding algorithmsrdquo in Proceedings ofthe International Conference on Acoustics Speech and SignalProcessing (ICASSP rsquo91) pp 1365ndash1368 Toronto Canada May1991

[69] CM S See ldquoSensor array calibration in the presence of mutualcoupling and unknown sensor gains and phasesrdquo ElectronicsLetters vol 30 no 5 pp 373ndash374 1994

[70] E K L Hung ldquoComputation of the coupling matrix among theelements of an array antennardquo inProceedings of the InternationalConference on Radar (Radar rsquo94) pp 703ndash706 Paris FranceMay 1994

[71] E K L Hung ldquoMatrix-construction calibration method forantenna arraysrdquo IEEE Transactions on Aerospace and ElectronicSystems vol 36 no 3 pp 819ndash828 2000

[72] K C Lee ldquoTwo efficient algorithms for the analyses of anonlinearly loaded antenna and antenna array in the frequencydomainrdquo IEEE Transactions on Electromagnetic Compatibilityvol 42 no 4 pp 339ndash346 2000

[73] A A Oliner and R G Malech ldquoMutual coupling in infi-nite scanning arraysrdquo in Microwave Scanning Antennas R CHansen Ed chapter 3 pp 195ndash335 Academic Press New YorkNY USA 1966

[74] D M Pozar and D H Schaubert ldquoScan blindness in infinitephased arrays of printed dipolesrdquo IEEE Transactions on Anten-nas and Propagation vol 32 no 6 pp 602ndash610 1984

[75] L Stark ldquoRadiation impedance of a dipole in an infinite planarphased arrayrdquo Radio Science vol 1 pp 361ndash377 1966

[76] K M Lee and R S Chu ldquoAnalysis of mutual coupling betweena finite phased array of dipoles and its feed networkrdquo IEEETransactions on Antennas and Propagation vol 36 no 12 pp1681ndash1699 1988

[77] DM Pozar ldquoAnalysis of finite phased arrays of printed dipolesrdquoIEEE Transactions on Antennas and Propagation vol 33 no 10pp 1045ndash1053 1985

[78] F Najib R Mohamad and V Patrick ldquoEffects of mutual cou-pling in phased arrays with butler network feedrdquo in Proceedingsof the 17th International Conference on Applied Electromagneticsand Communications pp 374ndash379 Dubrovnik Croatia Octo-ber 2003

[79] J D Fredrick Y Wang and T Itoh ldquoSmart antennas basedon Spatial Multiplexing of Local Elements (SMILE) for mutualcoupling reductionrdquo IEEE Transactions on Antennas and Prop-agation vol 52 no 1 pp 106ndash114 2004

[80] A Manikas and N Fistas ldquoModelling and estimation of mutualcoupling between array elementsrdquo in Proceedings of the IEEEInternational Conference on Acoustics Speech and Signal Pro-cessing (ICASSP rsquo94) vol 4 pp 553ndash556 April 1994

[81] R A Monzingo and T W Miller Introduction to AdaptiveArrays John Wiley amp Sons 1980

[82] B Friedlander ldquoA sensitivity analysis of the MUSIC algorithmrdquoIEEE Transactions on Acoustics Speech and Signal Processingvol 38 no 10 pp 1740ndash1751 1990

[83] R L Riegler and R T Compton Jr ldquoAn adaptive array forinterference rejectionrdquo Proceedings of the IEEE vol 61 no 6pp 748ndash758 1973

[84] R S Adve and T K Sarkar ldquoElimination of the effects ofmutual coupling in an adaptive nulling system with a lookdirection constraintrdquo in Proceedings of the IEEE Antennas andPropagation Society International Symposium vol 2 pp 1164ndash1167 July 1996

[85] R S Adve and T K Sarkar ldquoCompensation for the effects ofmutual coupling on direct data domain adaptive algorithmsrdquoIEEE Transactions on Antennas and Propagation vol 48 no 1pp 86ndash94 2000

[86] H T Hui ldquoReducing the mutual coupling effect in adaptivenulling using a re-defined mutual impedancerdquo IEEEMicrowaveand Wireless Components Letters vol 12 no 5 pp 178ndash1802002

[87] H T Hui H P Low T T Zhang and Y L Lu ldquoAntennaDesignerrsquos Notebook receivingmutual impedance between twonormal-mode helical antennas (NMHAs)rdquo IEEE Antennas andPropagation Magazine vol 48 no 4 pp 92ndash96 2006

[88] H T Hui ldquoA practical approach to compensate for the mutualcoupling effect in an adaptive dipole arrayrdquo IEEE TransactionsonAntennas and Propagation vol 52 no 5 pp 1262ndash1269 2004

[89] T T Zhang Y L Lu and H T Hui ldquoCompensation for themutual coupling effect in uniform circular arrays for 2D DOAestimations employing the maximum likelihood techniquerdquoIEEE Transactions on Aerospace and Electronic Systems vol 44no 3 pp 1215ndash1221 2008

[90] P Stoica and A B Gershman ldquoMaximum-likelihoodDOA esti-mation by data-supported grid searchrdquo IEEE Signal ProcessingLetters vol 6 no 10 pp 273ndash275 1999

[91] T Zhang andW Ser ldquoRobust beampattern synthesis for antennaarrays with mutual coupling effectrdquo IEEE Transactions onAntennas and Propagation vol 59 pp 2889ndash2895 2011

[92] Y Zhang K Hirasawa and K Fujimoto ldquoSignal bandwidthconsideration of mutual coupling effects on adaptive arrayperformancerdquo IEEE Transactions on Antennas and Propagationvol 35 no 3 pp 337ndash339 1987


[93] S Durrani and M E Bialkowski ldquoEffect of mutual couplingon the interference rejection capabilities of linear and circulararrays in CDMA systemsrdquo IEEE Transactions on Antennas andPropagation vol 52 no 4 pp 1130ndash1134 2004

[94] HKrim andMViberg ldquoTwodecades of array signal processingresearch the parametric approachrdquo IEEE Signal ProcessingMagazine vol 13 no 4 pp 67ndash94 1996

[95] C Roller and W Wasylkiwskyj ldquoEffects of mutual couplingon super-resolution DF in linear arraysrdquo in Proceedings of theIEEE International Conference on Acoustics Speech and SignalProcessing vol 5 pp 257ndash260 1992

[96] J Litva and M Zeytinoglu ldquoApplication of high-resolutiondirection finding algorithms to circular arrays with mutualcoupling presentrdquo Final Report Part II Defence ResearchEstablishment Ottawa Canada 1990

[97] B Himed and D D Weiner ldquoCompensation for mutualcoupling effects in direction findingrdquo in Proceedings of theIEEE International Conference on Acoustics Speech and SignalProcessing (ICASSP rsquo90) vol 5 pp 2631ndash2634 AlbuquerqueNM USA April 1990

[98] A L Swindlehurst and T Kailath ldquoA performance analysis ofsubspace-basedmethods in the presence ofmodel errorsmdashI theMUSIC algorithmrdquo IEEE Transactions on Signal Processing vol40 no 7 pp 1758ndash1774 1992

[99] A L Swindlehurst and T Kailath ldquoPerformance analysis ofsubspace-base methods in the presence of model errors partIImdashmultidimensional algorithmsrdquo IEEE Transactions on SignalProcessing vol 41 pp 1277ndash1308 1993

[100] T Su KDandekar andH Ling ldquoSimulation ofmutual couplingeffect in circular arrays for direction-finding applicationsrdquoMicrowave andOptical Technologies Letters vol 26 pp 331ndash3362000

[101] K R Dandekar H Ling and G Xu ldquoExperimental study ofmutual coupling compensation in smart antenna applicationsrdquoIEEE Transactions onWireless Communications vol 1 no 3 pp480ndash487 2002

[102] B Friedlander ldquoThe root-MUSIC algorithm for direction find-ing with interpolated arraysrdquo Signal Processing vol 30 no 1 pp15ndash29 1993

[103] K Kim T K Sarkar and M S Palma ldquoAdaptive processingusing a single snapshot for a nonuniformly spaced array in thepresence of mutual coupling and near-field scatterersrdquo IEEETransactions on Antennas and Propagation vol 50 no 5 pp582ndash590 2002

[104] H T Hui ldquoImproved compensation for the mutual couplingeffect in a dipole array for direction findingrdquo IEEE TransactionsonAntennas andPropagation vol 51 no 9 pp 2498ndash2503 2003

[105] Y Yu andH THui ldquoDesign of amutual coupling compensationnetwork for a small receiving monopole arrayrdquo IEEE Transac-tions on Microwave Theory and Techniques vol 59 pp 2241ndash2245 2011

[106] J C Coetzee and Y Yu ldquoPort decoupling for small arrays bymeans of an eigenmode feed networkrdquo IEEE Transactions onAntennas and Propagation vol 56 no 6 pp 1587ndash1593 2008

[107] P T Chua and J C Coetzee ldquoMicrostrip decoupling networksfor low-order multiport arrays with reduced element spacingrdquoMicrowave and Optical Technology Letters vol 46 no 6 pp592ndash597 2005

[108] I J Gupta and A A Ksienski ldquoPrediction of adaptive arrayperformancerdquo IEEE Transactions on Aerospace and ElectronicSystems vol 19 no 3 pp 380ndash388 1983

[109] Q Yuan Q Chen and K Sawaya ldquoPerformance of adaptivearray antennawith arbitrary geometry in the presence ofmutualcouplingrdquo IEEE Transactions on Antennas and Propagation vol54 no 7 pp 1991ndash1996 2006

[110] R J Dinger ldquoReactively steered adaptive array using microstrippatch elements at 4MHzrdquo IEEE Transactions on Antennas andPropagation vol 32 no 8 pp 848ndash856 1984

[111] A A Abdelaziz ldquoImproving the performance of an antennaarray by using radar absorbing coverrdquo Progress In Electromag-netics Research Letters vol 1 pp 129ndash138 2008

[112] G Dubost ldquoInfluence of surface wave upon efficiency andmutual coupling between rectangular microstrip antennasrdquo inProceedings of the IEEE International Symposium Digest onAntennas and Propagation vol 2 pp 660ndash663 Dallas TexUSA May 1990

[113] G E Antilla and N G Alexopoulos ldquoSurface wave andrelated effects on the RCS of microstrip dipoles printed onmagnetodielectric substratesrdquo IEEE Transactions on Antennasand Propagation vol 39 no 12 pp 1707ndash1715 1991

[114] S Zhang S X Gong Y Guan J Ling and B Lu ldquoA newapproach for synthesizing both the radiation and scattering pat-terns of linear dipole antenna arrayrdquo Journal of ElectromagneticWaves and Applications vol 24 no 7 pp 861ndash870 2010

[115] S Zhang S X Gong Y Guan and B Lu ldquoOptimized elementpositions for prescribed radar cross section pattern of lineardipole arraysrdquo International Journal of RF and MicrowaveComputer-Aided Engineering vol 21 pp 622ndash628 2011

[116] B Lindmark S Lundgren J R Sanford and C BeckmanldquoDual-polarized array for signal-processing applications inwireless communicationsrdquo IEEE Transactions on Antennas andPropagation vol 46 no 6 pp 758ndash763 1998

[117] S G Pan and I Wolff ldquoComputation of mutual couplingbetween slot-coupled microstrip patches in a finite arrayrdquo IEEETransactions on Antennas and Propagation vol 40 no 9 pp1047ndash1053 1992

[118] S Lundgren ldquoStudy of mutual coupling effects on the directionfinding performance of ESPRIT with a linear microstrip patcharray using the method of momentsrdquo in Proceedings of the IEEEAntennas and Propagation Society International SymposiumDigest pp 1372ndash1375 Baltimore Md USA July 1996

[119] M C V Beurden A B Smolders and M E J Jeuken ldquoDesignof wide-band phased array antennasrdquo in Perspectives on RadioAstronomy Technologies for Large Antenna Arrays Proceedingsof the Conference pp 64ndash70 ASTRON Institute DwingelooThe Netherlands 1999

[120] S J Blank and M F Hutt ldquoOn the empirical optimization ofantenna arraysrdquo IEEE Antennas and PropagationMagazine vol47 no 2 pp 58ndash67 2005

[121] F Yang and Y Rahmat-Samii ldquoMicrostrip antennas integratedwith electromagnetic band-gap (EBG) structures a low mutualcoupling design for array applicationsrdquo IEEE Transactions onAntennas and Propagation vol 51 no 10 pp 2936ndash2946 2003

[122] R Chair A A Kishk and K F Lee ldquoComparative study on themutual coupling between different sized cylindrical dielectricresonators antennas and circular microstrip patch antennasrdquoIEEE Transactions on Antennas and Propagation vol 53 no 3pp 1011ndash1019 2005

[123] N Yousefzadeh C Ghobadi and M Kamyab ldquoConsiderationof Mutual Coupling in a microstrip patch array using fractalelementsrdquo Progress in Electromagnetics Research vol 66 pp 41ndash49 2006


[124] K Buell H Mosallaei and K Sarabandi ldquoMetamaterial insula-tor enabled superdirective arrayrdquo IEEE Transactions on Anten-nas and Propagation vol 55 no 4 pp 1074ndash1085 2007

[125] H S Lui and H T Hui ldquoImproved mutual coupling compensa-tion in compact antenna arraysrdquo IETMicrowaves Antennas andPropagation vol 4 no 10 pp 1506ndash1516 2010

[126] H S Lui and H T Hui ldquoMutual coupling compensation fordirection-of-arrival estimations using the receiving-mutual-impedance methodrdquo International Journal of Antennas andPropagation vol 2010 Article ID 373061 7 pages 2010

[127] C Y Chiu C H Cheng R D Murch and C R RowellldquoReduction ofmutual coupling between closely-packed antennaelementsrdquo IEEE Transactions on Antennas and Propagation vol55 no 6 pp 1732ndash1738 2007

[128] S C Chen Y SWang and S J Chung ldquoA decoupling techniquefor increasing the port isolation between two strongly coupledantennasrdquo IEEE Transactions on Antennas and Propagation vol56 no 12 pp 3650ndash3658 2008

[129] P J Ferrer J M Gonzalez-Arbesu and J Romeu ldquoDecorrela-tion of two closely spaced antennas with a metamaterial AMCsurfacerdquoMicrowave and Optical Technology Letters vol 50 no5 pp 1414ndash1417 2008

[130] M M Bait-Suwailam M S Boybay and O M RamahildquoElectromagnetic coupling reduction in high-profile monopoleantennas using single-negative magnetic metamaterials forMIMO applicationsrdquo IEEE Transactions on Antennas and Prop-agation vol 58 no 9 pp 2894ndash2902 2010

[131] R Schmidt ldquoMultiple emitter location and signal parameterestimationrdquo in Proceedings of the RADC Spectrum EstimationWorkshop pp 243ndash258 1979

[132] T J Shan M Wax and T Kailath ldquoOn spatial smoothingfor direction-of-arrival estimation of coherent signalsrdquo IEEETransactions on Acoustics Speech and Signal Processing vol 33no 4 pp 806ndash811 1985

[133] Y Wang and S Xu ldquoMutual coupling calibration of DBF arraywith combined optimization methodrdquo IEEE Transactions onAntennas and Propagation vol 51 no 10 pp 2947ndash2952 2003

[134] P Demarcke H Rogier R Goossens and P De JaegerldquoBeamforming in the presence of mutual coupling based onconstrained particle swarmoptimizationrdquo IEEETransactions onAntennas and Propagation vol 57 no 6 pp 1655ndash1666 2009

[135] B Basu and G K Mahanti ldquoBeam reconfiguration of lineararray of parallel dipole antennas through switching with realexcitation voltage distributionrdquo Annals of Telecommunicationsvol 67 pp 285ndash293 2012

[136] O M Bucci A Capozzoli and G D Elia ldquoReconfigurableconformal array synthesis with near-field constraintsrdquo in Pro-ceedings of the Progress in Electromagnetics Research Symposium(PIERS rsquo00) pp 864ndash870 Cambridge Mass USA July 2000

[137] L Landesa F Obelleiro J L Rodrıguez J A Rodrıguez FAres and A G Pino ldquoPattern synthesis of array antennas withadditional isolation of near field arbitrary objectsrdquo ElectronicsLetters vol 34 no 16 pp 1540ndash1542 1998

[138] H Steyskal ldquoSynthesis of antenna patterns with imposed near-field nullsrdquo Electronics Letters vol 30 no 24 pp 2000ndash20011994

[139] P H Pathak and N Wang ldquoRay analysis of mutual couplingbetween antennas on a convex surfacerdquo IEEE Transactions onAntennas and Propagation vol 29 pp 911ndash922 1981

[140] R M Jha V Sudhakar and N Balakrishnan ldquoRay analysis ofmutual coupling between antennas on a general paraboloid ofrevolution (GPOR)rdquo Electronics Letters vol 23 no 11 pp 583ndash584 1987

[141] R M Jha and W Wiesbeck ldquoGeodesic constant method anovel approach to analytical surface-ray tracing on convexconducting bodiesrdquo IEEE Antennas and Propagation Magazinevol 37 no 2 pp 28ndash38 1995

[142] R W Wills ldquoMutual coupling and far field radiation fromwaveguide antenna elements on conformal surfacesrdquo in Pro-ceedings of the International Conference (RADAR rsquo87) pp 515ndash519 London UK October 1987

[143] P Persson L Josefsson and M Lanne ldquoInvestigation of themutual coupling between apertures on doubly curved convexsurfaces theory and measurementsrdquo IEEE Transactions onAntennas and Propagation vol 51 no 4 pp 682ndash692 2003

[144] P Persson and L Josefsson ldquoCalculating the mutual couplingbetween apertures on a convex circular cylinder using a hybridUTD-MoM methodrdquo IEEE Transactions on Antennas andPropagation vol 49 no 4 pp 672ndash677 2001

[145] L Landesa J M Taboada F Obelleiro and J L RodriguezldquoDesign of on-board array antennas by pattern optimizationrdquoMicrowave and Optical Technology Letters vol 21 pp 446ndash4481999

[146] L Landesa FObelleiro J L Rodrıguez andAG Pino ldquoPatternsynthesis of array antennas in presence of conducting bodies ofarbitrary shaperdquo Electronics Letters vol 33 no 18 pp 1512ndash15131997

[147] F Obelleiro L Landesa J M Taboada and J L RodrıguezldquoSynthesis of onboard array antennas including interactionwith the mounting platform and mutual coupling effectsrdquo IEEEAntennas and Propagation Magazine vol 43 no 2 pp 76ndash822001

[148] C W Wu L C Kempel and E J Rothwell ldquoMutual couplingbetween patch antennas recessed in an elliptic cylinderrdquo IEEETransactions on Antennas and Propagation vol 51 no 9 pp2489ndash2492 2003

[149] L Vegni and A Toscano ldquoThe method of lines for mutualcoupling analysis of a finite array of patch antennas on acylindrical stratified structurerdquo IEEE Transactions on Antennasand Propagation vol 51 no 8 pp 1907ndash1913 2003

[150] H J Visser and A C F Reniers ldquoTextile antennas a practicalapproachrdquo in Proceedings of the 2nd European Conference onAntennas and Propagation (EuCAP rsquo07) pp 1ndash8 EdinburghUK November 2007

[151] M S Akyuz V B Erturk and M Kalfa ldquoClosed-formgreenrsquos function representations for mutual coupling calcula-tions between apertures on a perfect electric conductor circularcylinder covered with dielectric layersrdquo IEEE Transactions onAntennas and Propagation vol 59 pp 3094ndash3098 2011

[152] K Yang Z Zhao Z Nie J Ouyang and Q H Liu ldquoSynthesisof conformal phased arrays with embedded element patterndecompositionrdquo IEEE Transactions on Antennas and Propaga-tion vol 59 pp 2882ndash2888 2011

[153] V B Erturk and R G Rojas ldquoEfficient analysis of inputimpedance and mutual coupling of microstrip antennasmounted on large coated cylindersrdquo IEEE Transactions onAntennas and Propagation vol 51 no 4 pp 739ndash749 2003

[154] Z Sipus P Persson and M Bosiljevac ldquoModeling conformalantennas using a hybrid spectral domain-UTD methodrdquo inProceedings of the Nordic Antenna Symposium (Antenn rsquo06)Linkoping Sweden May 2006

[155] S Raffaelli Z Sipus and P S Kildal ldquoAnalysis and measure-ments of conformal patch array antennas onmultilayer circularcylinderrdquo IEEE Transactions on Antennas and Propagation vol53 no 3 pp 1105ndash1113 2005


[156] T E Morton and K M Pasala ldquoPerformance analysis of con-formal conical arrays for airborne vehiclesrdquo IEEE Transactionson Aerospace and Electronic Systems vol 42 no 3 pp 876ndash8902006

[157] BThors L Josefsson and R G Rojas ldquoTheRCS of a cylindricalarray antenna coated with a dielectric layerrdquo IEEE Transactionson Antennas and Propagation vol 52 no 7 pp 1851ndash1858 2004

[158] P Persson and R G Rojas ldquoHigh-frequency approximation formutual coupling calculations between apertures on a perfectelectric conductor circular cylinder covered with a dielectriclayer nonparaxial regionrdquo Radio Science vol 38 no 4 pp 181ndash1814 2003

[159] T Svantesson and A Ranheim ldquoMutual coupling effects on thecapacity of multielement antenna systemsrdquo in Proceedings ofthe IEEE Conference on Acoustics Speech and Signal Processing(ICASSP rsquo01) pp 2485ndash2488 Salt Lake City Utah USA May2001

[160] G J Foschini and M J Gans ldquoOn limits of wireless communi-cations in a fading environment when usingmultiple antennasrdquoWireless Personal Communications vol 6 no 3 pp 311ndash3351998

[161] Y Urzhumov and D R Smith ldquoMetamaterial-enhanced cou-pling between magnetic dipoles for efficient wireless powertransferrdquo Physical Review B vol 83 no 20 Article ID 2051142011

[162] H M Aumann A J Fenn and F G Willwerth ldquoPhasedarray antenna calibration and pattern prediction using mutualcoupling measurementsrdquo IEEE Transactions on Antennas andPropagation vol 37 no 7 pp 844ndash850 1989

[163] C Yang and Y Z Ruan ldquoEigenvalues of covariance matrix ofadaptive array withmutual coupling and two correlated sourcespresentrdquo in Proceedings of the IEEE Antennas and PropagationSociety International Symposium pp 706ndash709 July 1993

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



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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



Weight updating

Feed network

Scattered signalScattered signal

Reflected signal

Reradiated signal

A1 A2 A119873

P2P1 P119873

C1 C2 C119873




Figure 1 Schematic of adaptive antenna array

towards the compensation of its effect Moreover optimiza-tion techniques such as genetic algorithm (GA) particleswarm optimization (PSO) and linear programming (LP)can be used in conjuncture with these techniques towardsthe enhancement of array efficiency This paper presents aunified review of the techniques and the designs proposedfor mitigating mutual coupling effect in phased arrays Theperformance parameters that get affected due to the couplingeffect in antenna array are discussed The work reportedin open domain towards efficient array design mitigatingmutual coupling effect is reviewed and compared

The subsequent sections describe the analysis and com-pensation of mutual coupling effect in phased arrays usingthese techniques The effect of mutual coupling on the arrayparameters such as antenna impedance and steering vectorwhich further affects the radiation pattern resolution andinterference suppression ability DOA estimation outputSINR response speed and RCS is described in Section 2Section 3 reviews the methods for efficient antenna designand the use of optimization techniques to counteract theadverse effects of mutual coupling on the array performanceIn Section 4 the mutual coupling effects and the techniquesto compensate it for the phased arrays conforming to thesurface are discussed Section 5 presents the cases wherethe presence of coupling proves to be advantageous This isfollowedby a brief summary on the reviewofmutual couplingin Section 6

2 Parameters Affected due to MutualCoupling Effect

21 Antenna Impedance The antenna radiation patterndepends mainly on the impedance at the antenna terminalsHowever the antenna impedance of phased array is signifi-cantly different in comparison to that of an isolated elementThis variation in impedance is due to the presence of couplingbetween the array elements In general for an array of 119873-elements the impedance matrix is given by [2]

119885 =

[[[[

[

11988511+ 119885119871

11988512

sdot sdot sdot 1198851119873

11988521

11988522+ 119885119871sdot sdot sdot 119885

2119873

d

1198851198731

1198851198732

sdot sdot sdot 119885119873119873

+ 119885119871

]]]]

]

(1)

0 1 2 3 4 5 60

10

20

30

40

50

60

70

80

0306090120150180

MagnitudePhase

Mag

nitu

de (119885

)

minus180minus150minus120minus90minus60minus30 Ph

ase (119885

)

119863(120582)

Figure 2 Variation of mutual impedance between two half-wavelength center-fed dipoles with interelement spacing

where 119885119898119899

represents the self (119898 = 119899) and mutual (119898 = 119899)impedances of the antennas and 119885

119871is the terminating load

This impedance matrix yields the net impedance at theantenna terminal [10] that is

119885119909=

119873

sum

119910=1

119885119909119910

(

119868119910

119868119909

) (2)

where 119868119901represents the current at the terminals of 119901th

antenna element and 119885119909119910

is the impedance between 119909th and119910th antenna elements

In general the mutual impedance matrix is a squarematrix with the order corresponding to the array size Thisimplies that the computations involved in arriving at thematrix coefficients increases with the size of array Howeverit is possible to exploit certain properties of coupling forreducing the computation complexity One such propertyis the inverse dependence of coupling coefficients on thedistance between the array elements [11] as shown inFigure 2 It is apparent that farther the elements in the arrayleast will be the mutual impedance and hence the couplingeffect In uniform linear arrays the couplingmatrix possessesrotational symmetry and hence has a symmetric Toeplitzmatrix structure Similarly for a uniform circular array thecoupling matrix is circulant

Furthermore the self- and mutual impedances of (1) aredependent on the type and configuration of the antennaarray The estimation of these impedances is essentially aproblem of finding the current distribution on the antennasurface Among the available methods for estimation of self-and mutual impedances of wire-type antennas the integralequation-moment method [12] and the induced electromo-tive force (EMF) method [13] are the most popular Both ofthese methods are based on the integral forms of inducedcurrent and voltages at the antenna terminals The methodof induced EMF is advantageous over the integral equation-moment method as it facilitates the simplifiedeasier designof antennas by providing the closed form solutions Howeverit is applicable only to the simplified models of straight



















CSVUSV


Nor

mal

ized

radi

atio

n pa

ttern

(dB)


0

60 120 180minus50

minus40

minus30

minus20

minus10

0





[119860119906(120579 Φ)] = [119862 (120579 Φ)]

minus1[119860119888(120579 Φ)]

with [119862(120579 Φ)]minus1= 119885119871 [119884] [119879 (120579 Φ)]

(3)








[119882] = [119880] + 119892 [Φ]minus1

[1198820] (4)



[11988200] = [1 0 0 0]

119879 (5)


[1198820] = [119885

119871]minus1

[119885] [11988200] (6)



















119881119888= 119862119881119899 (7)


119888as

119881119899= 119862minus1119881119888 (8)






119888119898119899

=1

2120587int

120587119896119889

minus120587119896119889

119892119898(119906)

119891119894 (119906)119890minus119895119899119896119889119906

119889119906 (9)



119862 = 119868 + 119878 (10)


1198781015840= 119879119878119879 (11)


1198621015840= 119879 + 119878

1015840119879minus1 (12)












119862 = 119860Θ119871Θ119890119895ΦΘ119866Θ119890

119895120587119863 (13)



119903+ 119868)

(14)





119877119909119909= 119862 sdot 119878 sdot 119878

119867sdot 119862119867+ 1205902sdot 119868 (15)

























[[[[

[

1198811198991198881

1198811198991198882

119881119899119888119873

]]]]

]

=

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

[

1 minus11988512

119905

119885119871


119905

119885119871

minus11988521

119905

119885119871


119905

119885119871

d

minus1198851198731

119905

119885119871

minus1198851198732

119905

119885119871

sdot sdot sdot 1

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

]

[[[[

[

1198811198881

1198811198882

119881119888119873

]]]]

]

(16)


where [119881119899119888] and [119881





119900 with the


[[[[

[

1198811199001

1198811199002

119881119900119873

]]]]

]

=

[[[[[[[[[[[

[

1 +11988511

119885119871

11988512

119885119871


119885119871

11988521

119885119871

1 +11988522

119885119871


119885119871

d

1198851198731

119885119871

1198851198732

119885119871

sdot sdot sdot 1 +119885119873119873

119885119871

]]]]]]]]]]]

]

[[[[

[

1198811198881

1198811198882

119881119888119873

]]]]

]

(17)

where 119885119898119899





119860ULA (120601) = [119886ULA (1206011) 119886ULA (1206012) 119886ULA (120601119889)](18a)

with


[[[[

[

1

119890119895119896119889 sin120601

119890119895119896119889(119873minus1) sin120601

]]]]

]

(18b)


05 1

SIN

R (d

B)

0

8

16

24

32

minus1 minus05 0sin(120601119889)

120585119889 = 5dB120585119889 = 10dB





119886 (120601) = cos119899 (120601) 119886ULA (120601) (19)




119889



SINR = 120585119889119880119879

119889119877minus1

119899119880lowast

119889 (20)






1 15

Dipole


2

1205822 1205822 1205822 1205822

119889 119889

119873 = 16

(a)

0

6

12

18


05 1

SIN

R (d

B)

minus1 minus05 0sin(120601119889)


119889 = 05120582

119889 = 120582

(b)


119889= 90∘




as [108]

119877119899= 119868 +

119898

sum

119896=1

120585119894119896119880lowast

119894119896119880119879

119894119896

119882 = 119870119877minus1

119899119880lowast

119889

(21)





119877119899= 119885lowast

0119885119879

0+

119898

sum

119896=1

120585119894119896119880lowast

119894119896119880119879

119894119896

119882 = 119870119885119879

0119877minus1

119899119880lowast

119889

(22)

where 1198850






NR

(dB)

0

12

24

36

48


119873 = 6

119873 = 16

119873 = 32

119873 = 64

119873 = 256

119873 = 512

119873 = 1024

119863(120582)


0 2 4 6 8 10 12

SIN

R (d

B)

0

4

8

12

16

Number of elements



119889= 5 dB 119885

119871=

119885lowast

119894119894 (120579119889 120601119889) = (90



119877119909119909= 1205902[119868 +

119898

sum

119896=1

120585119894119896(119885minus1

119900119880119894119896) lowast (119885

minus1

119900119880119894119896)119879

+ 120585119889(119885minus1

119900119880119889) lowast (119885

minus1

119900119880119889)119879

]

(23)









119862119898119899

=119884119898119899

119884119899119899+ 119884119892

(24)

where 119884119898119899





21

Receiveport

Terminating load

Dipoles

Couplers Matched

termination

λ2

Phaseshifters

119889 = 01120582

1205822 1205822 1205822

119873 minus 1 119873 = 30119889 = 01120582



0 20 40 60 80

0

10

20

minus60

minus50

minus40

minus30

minus20

minus10



1205901205822

(dB)


(a)

0 20 40 60 80

0

10

20

minus60

minus50

minus40

minus30

minus20

minus10



1205901205822

(dB)


(b)


1198850= 75Ω and 119885


119904= 0∘ (b) 120579

119904= 85∘


119881119899= 119881

app119899

minus

119873

sum

119898=1

119898 = 119899

119862119898119899119881

app119899

(25)

where 119881app119899







0 20 40 60 80

0

10

20

minus60

minus50

minus40

minus30

minus20

minus10



1205901205822

(dB)


(a)

0 20 40 60 80

0

10

20

minus60

minus50

minus40

minus30

minus20

minus10



1205901205822

(dB)


(b)


minus5120582 1198850= 125Ω and 119885


119904= 0∘ (b) 120579

119904= 85∘

0 20 40 60 80

0

10

20

minus60

minus50

minus40

minus30

minus20

minus10



1205901205822

(dB)


minus80

(a)

0 20 40 60 80

0

10

20

minus60

minus50

minus40

minus30

minus20

minus10



1205901205822

(dB)


minus80

(b)


1198850= 150Ω and 119885


119904= 0∘ (b) 120579

119904= 85∘



















4 Conformal Array




















1198671015840= 119862119877119867119862119879 (26)


119877 and 119862





6 Summary





References













































































































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation



























CSVUSV


Nor

mal

ized

radi

atio

n pa

ttern

(dB)


0

60 120 180minus50

minus40

minus30

minus20

minus10

0





[119860119906(120579 Φ)] = [119862 (120579 Φ)]

minus1[119860119888(120579 Φ)]

with [119862(120579 Φ)]minus1= 119885119871 [119884] [119879 (120579 Φ)]

(3)








[119882] = [119880] + 119892 [Φ]minus1

[1198820] (4)



[11988200] = [1 0 0 0]

119879 (5)


[1198820] = [119885

119871]minus1

[119885] [11988200] (6)



















119881119888= 119862119881119899 (7)


119888as

119881119899= 119862minus1119881119888 (8)






119888119898119899

=1

2120587int

120587119896119889

minus120587119896119889

119892119898(119906)

119891119894 (119906)119890minus119895119899119896119889119906

119889119906 (9)



119862 = 119868 + 119878 (10)


1198781015840= 119879119878119879 (11)


1198621015840= 119879 + 119878

1015840119879minus1 (12)












119862 = 119860Θ119871Θ119890119895ΦΘ119866Θ119890

119895120587119863 (13)



119903+ 119868)

(14)





119877119909119909= 119862 sdot 119878 sdot 119878

119867sdot 119862119867+ 1205902sdot 119868 (15)

























[[[[

[

1198811198991198881

1198811198991198882

119881119899119888119873

]]]]

]

=

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

[

1 minus11988512

119905

119885119871


119905

119885119871

minus11988521

119905

119885119871


119905

119885119871

d

minus1198851198731

119905

119885119871

minus1198851198732

119905

119885119871

sdot sdot sdot 1

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

]

[[[[

[

1198811198881

1198811198882

119881119888119873

]]]]

]

(16)


where [119881119899119888] and [119881





119900 with the


[[[[

[

1198811199001

1198811199002

119881119900119873

]]]]

]

=

[[[[[[[[[[[

[

1 +11988511

119885119871

11988512

119885119871


119885119871

11988521

119885119871

1 +11988522

119885119871


119885119871

d

1198851198731

119885119871

1198851198732

119885119871

sdot sdot sdot 1 +119885119873119873

119885119871

]]]]]]]]]]]

]

[[[[

[

1198811198881

1198811198882

119881119888119873

]]]]

]

(17)

where 119885119898119899





119860ULA (120601) = [119886ULA (1206011) 119886ULA (1206012) 119886ULA (120601119889)](18a)

with


[[[[

[

1

119890119895119896119889 sin120601

119890119895119896119889(119873minus1) sin120601

]]]]

]

(18b)


05 1

SIN

R (d

B)

0

8

16

24

32

minus1 minus05 0sin(120601119889)

120585119889 = 5dB120585119889 = 10dB





119886 (120601) = cos119899 (120601) 119886ULA (120601) (19)




119889



SINR = 120585119889119880119879

119889119877minus1

119899119880lowast

119889 (20)






1 15

Dipole


2

1205822 1205822 1205822 1205822

119889 119889

119873 = 16

(a)

0

6

12

18


05 1

SIN

R (d

B)

minus1 minus05 0sin(120601119889)


119889 = 05120582

119889 = 120582

(b)


119889= 90∘




as [108]

119877119899= 119868 +

119898

sum

119896=1

120585119894119896119880lowast

119894119896119880119879

119894119896

119882 = 119870119877minus1

119899119880lowast

119889

(21)





119877119899= 119885lowast

0119885119879

0+

119898

sum

119896=1

120585119894119896119880lowast

119894119896119880119879

119894119896

119882 = 119870119885119879

0119877minus1

119899119880lowast

119889

(22)

where 1198850






NR

(dB)

0

12

24

36

48


119873 = 6

119873 = 16

119873 = 32

119873 = 64

119873 = 256

119873 = 512

119873 = 1024

119863(120582)


0 2 4 6 8 10 12

SIN

R (d

B)

0

4

8

12

16

Number of elements



119889= 5 dB 119885

119871=

119885lowast

119894119894 (120579119889 120601119889) = (90



119877119909119909= 1205902[119868 +

119898

sum

119896=1

120585119894119896(119885minus1

119900119880119894119896) lowast (119885

minus1

119900119880119894119896)119879

+ 120585119889(119885minus1

119900119880119889) lowast (119885

minus1

119900119880119889)119879

]

(23)









119862119898119899

=119884119898119899

119884119899119899+ 119884119892

(24)

where 119884119898119899





21

Receiveport

Terminating load

Dipoles

Couplers Matched

termination

λ2

Phaseshifters

119889 = 01120582

1205822 1205822 1205822

119873 minus 1 119873 = 30119889 = 01120582



0 20 40 60 80

0

10

20

minus60

minus50

minus40

minus30

minus20

minus10



1205901205822

(dB)


(a)

0 20 40 60 80

0

10

20

minus60

minus50

minus40

minus30

minus20

minus10



1205901205822

(dB)


(b)


1198850= 75Ω and 119885


119904= 0∘ (b) 120579

119904= 85∘


119881119899= 119881

app119899

minus

119873

sum

119898=1

119898 = 119899

119862119898119899119881

app119899

(25)

where 119881app119899







0 20 40 60 80

0

10

20

minus60

minus50

minus40

minus30

minus20

minus10



1205901205822

(dB)


(a)

0 20 40 60 80

0

10

20

minus60

minus50

minus40

minus30

minus20

minus10



1205901205822

(dB)


(b)


minus5120582 1198850= 125Ω and 119885


119904= 0∘ (b) 120579

119904= 85∘

0 20 40 60 80

0

10

20

minus60

minus50

minus40

minus30

minus20

minus10



1205901205822

(dB)


minus80

(a)

0 20 40 60 80

0

10

20

minus60

minus50

minus40

minus30

minus20

minus10



1205901205822

(dB)


minus80

(b)


1198850= 150Ω and 119885


119904= 0∘ (b) 120579

119904= 85∘



















4 Conformal Array




















1198671015840= 119862119877119867119862119879 (26)


119877 and 119862





6 Summary





References













































































































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation















CSVUSV


Nor

mal

ized

radi

atio

n pa

ttern

(dB)


0

60 120 180minus50

minus40

minus30

minus20

minus10

0





[119860119906(120579 Φ)] = [119862 (120579 Φ)]

minus1[119860119888(120579 Φ)]

with [119862(120579 Φ)]minus1= 119885119871 [119884] [119879 (120579 Φ)]

(3)








[119882] = [119880] + 119892 [Φ]minus1

[1198820] (4)



[11988200] = [1 0 0 0]

119879 (5)


[1198820] = [119885

119871]minus1

[119885] [11988200] (6)



















119881119888= 119862119881119899 (7)


119888as

119881119899= 119862minus1119881119888 (8)






119888119898119899

=1

2120587int

120587119896119889

minus120587119896119889

119892119898(119906)

119891119894 (119906)119890minus119895119899119896119889119906

119889119906 (9)



119862 = 119868 + 119878 (10)


1198781015840= 119879119878119879 (11)


1198621015840= 119879 + 119878

1015840119879minus1 (12)












119862 = 119860Θ119871Θ119890119895ΦΘ119866Θ119890

119895120587119863 (13)



119903+ 119868)

(14)





119877119909119909= 119862 sdot 119878 sdot 119878

119867sdot 119862119867+ 1205902sdot 119868 (15)

























[[[[

[

1198811198991198881

1198811198991198882

119881119899119888119873

]]]]

]

=

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

[

1 minus11988512

119905

119885119871


119905

119885119871

minus11988521

119905

119885119871


119905

119885119871

d

minus1198851198731

119905

119885119871

minus1198851198732

119905

119885119871

sdot sdot sdot 1

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

]

[[[[

[

1198811198881

1198811198882

119881119888119873

]]]]

]

(16)


where [119881119899119888] and [119881





119900 with the


[[[[

[

1198811199001

1198811199002

119881119900119873

]]]]

]

=

[[[[[[[[[[[

[

1 +11988511

119885119871

11988512

119885119871


119885119871

11988521

119885119871

1 +11988522

119885119871


119885119871

d

1198851198731

119885119871

1198851198732

119885119871

sdot sdot sdot 1 +119885119873119873

119885119871

]]]]]]]]]]]

]

[[[[

[

1198811198881

1198811198882

119881119888119873

]]]]

]

(17)

where 119885119898119899





119860ULA (120601) = [119886ULA (1206011) 119886ULA (1206012) 119886ULA (120601119889)](18a)

with


[[[[

[

1

119890119895119896119889 sin120601

119890119895119896119889(119873minus1) sin120601

]]]]

]

(18b)


05 1

SIN

R (d

B)

0

8

16

24

32

minus1 minus05 0sin(120601119889)

120585119889 = 5dB120585119889 = 10dB





119886 (120601) = cos119899 (120601) 119886ULA (120601) (19)




119889



SINR = 120585119889119880119879

119889119877minus1

119899119880lowast

119889 (20)






1 15

Dipole


2

1205822 1205822 1205822 1205822

119889 119889

119873 = 16

(a)

0

6

12

18


05 1

SIN

R (d

B)

minus1 minus05 0sin(120601119889)


119889 = 05120582

119889 = 120582

(b)


119889= 90∘




as [108]

119877119899= 119868 +

119898

sum

119896=1

120585119894119896119880lowast

119894119896119880119879

119894119896

119882 = 119870119877minus1

119899119880lowast

119889

(21)





119877119899= 119885lowast

0119885119879

0+

119898

sum

119896=1

120585119894119896119880lowast

119894119896119880119879

119894119896

119882 = 119870119885119879

0119877minus1

119899119880lowast

119889

(22)

where 1198850






NR

(dB)

0

12

24

36

48


119873 = 6

119873 = 16

119873 = 32

119873 = 64

119873 = 256

119873 = 512

119873 = 1024

119863(120582)


0 2 4 6 8 10 12

SIN

R (d

B)

0

4

8

12

16

Number of elements



119889= 5 dB 119885

119871=

119885lowast

119894119894 (120579119889 120601119889) = (90



119877119909119909= 1205902[119868 +

119898

sum

119896=1

120585119894119896(119885minus1

119900119880119894119896) lowast (119885

minus1

119900119880119894119896)119879

+ 120585119889(119885minus1

119900119880119889) lowast (119885

minus1

119900119880119889)119879

]

(23)









119862119898119899

=119884119898119899

119884119899119899+ 119884119892

(24)

where 119884119898119899





21

Receiveport

Terminating load

Dipoles

Couplers Matched

termination

λ2

Phaseshifters

119889 = 01120582

1205822 1205822 1205822

119873 minus 1 119873 = 30119889 = 01120582



0 20 40 60 80

0

10

20

minus60

minus50

minus40

minus30

minus20

minus10



1205901205822

(dB)


(a)

0 20 40 60 80

0

10

20

minus60

minus50

minus40

minus30

minus20

minus10



1205901205822

(dB)


(b)


1198850= 75Ω and 119885


119904= 0∘ (b) 120579

119904= 85∘


119881119899= 119881

app119899

minus

119873

sum

119898=1

119898 = 119899

119862119898119899119881

app119899

(25)

where 119881app119899







0 20 40 60 80

0

10

20

minus60

minus50

minus40

minus30

minus20

minus10



1205901205822

(dB)


(a)

0 20 40 60 80

0

10

20

minus60

minus50

minus40

minus30

minus20

minus10



1205901205822

(dB)


(b)


minus5120582 1198850= 125Ω and 119885


119904= 0∘ (b) 120579

119904= 85∘

0 20 40 60 80

0

10

20

minus60

minus50

minus40

minus30

minus20

minus10



1205901205822

(dB)


minus80

(a)

0 20 40 60 80

0

10

20

minus60

minus50

minus40

minus30

minus20

minus10



1205901205822

(dB)


minus80

(b)


1198850= 150Ω and 119885


119904= 0∘ (b) 120579

119904= 85∘



















4 Conformal Array




















1198671015840= 119862119877119867119862119879 (26)


119877 and 119862





6 Summary





References













































































































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











[119882] = [119880] + 119892 [Φ]minus1

[1198820] (4)



[11988200] = [1 0 0 0]

119879 (5)


[1198820] = [119885

119871]minus1

[119885] [11988200] (6)



















119881119888= 119862119881119899 (7)


119888as

119881119899= 119862minus1119881119888 (8)






119888119898119899

=1

2120587int

120587119896119889

minus120587119896119889

119892119898(119906)

119891119894 (119906)119890minus119895119899119896119889119906

119889119906 (9)



119862 = 119868 + 119878 (10)


1198781015840= 119879119878119879 (11)


1198621015840= 119879 + 119878

1015840119879minus1 (12)












119862 = 119860Θ119871Θ119890119895ΦΘ119866Θ119890

119895120587119863 (13)



119903+ 119868)

(14)





119877119909119909= 119862 sdot 119878 sdot 119878

119867sdot 119862119867+ 1205902sdot 119868 (15)

























[[[[

[

1198811198991198881

1198811198991198882

119881119899119888119873

]]]]

]

=

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

[

1 minus11988512

119905

119885119871


119905

119885119871

minus11988521

119905

119885119871


119905

119885119871

d

minus1198851198731

119905

119885119871

minus1198851198732

119905

119885119871

sdot sdot sdot 1

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

]

[[[[

[

1198811198881

1198811198882

119881119888119873

]]]]

]

(16)


where [119881119899119888] and [119881





119900 with the


[[[[

[

1198811199001

1198811199002

119881119900119873

]]]]

]

=

[[[[[[[[[[[

[

1 +11988511

119885119871

11988512

119885119871


119885119871

11988521

119885119871

1 +11988522

119885119871


119885119871

d

1198851198731

119885119871

1198851198732

119885119871

sdot sdot sdot 1 +119885119873119873

119885119871

]]]]]]]]]]]

]

[[[[

[

1198811198881

1198811198882

119881119888119873

]]]]

]

(17)

where 119885119898119899





119860ULA (120601) = [119886ULA (1206011) 119886ULA (1206012) 119886ULA (120601119889)](18a)

with


[[[[

[

1

119890119895119896119889 sin120601

119890119895119896119889(119873minus1) sin120601

]]]]

]

(18b)


05 1

SIN

R (d

B)

0

8

16

24

32

minus1 minus05 0sin(120601119889)

120585119889 = 5dB120585119889 = 10dB





119886 (120601) = cos119899 (120601) 119886ULA (120601) (19)




119889



SINR = 120585119889119880119879

119889119877minus1

119899119880lowast

119889 (20)






1 15

Dipole


2

1205822 1205822 1205822 1205822

119889 119889

119873 = 16

(a)

0

6

12

18


05 1

SIN

R (d

B)

minus1 minus05 0sin(120601119889)


119889 = 05120582

119889 = 120582

(b)


119889= 90∘




as [108]

119877119899= 119868 +

119898

sum

119896=1

120585119894119896119880lowast

119894119896119880119879

119894119896

119882 = 119870119877minus1

119899119880lowast

119889

(21)





119877119899= 119885lowast

0119885119879

0+

119898

sum

119896=1

120585119894119896119880lowast

119894119896119880119879

119894119896

119882 = 119870119885119879

0119877minus1

119899119880lowast

119889

(22)

where 1198850






NR

(dB)

0

12

24

36

48


119873 = 6

119873 = 16

119873 = 32

119873 = 64

119873 = 256

119873 = 512

119873 = 1024

119863(120582)


0 2 4 6 8 10 12

SIN

R (d

B)

0

4

8

12

16

Number of elements



119889= 5 dB 119885

119871=

119885lowast

119894119894 (120579119889 120601119889) = (90



119877119909119909= 1205902[119868 +

119898

sum

119896=1

120585119894119896(119885minus1

119900119880119894119896) lowast (119885

minus1

119900119880119894119896)119879

+ 120585119889(119885minus1

119900119880119889) lowast (119885

minus1

119900119880119889)119879

]

(23)









119862119898119899

=119884119898119899

119884119899119899+ 119884119892

(24)

where 119884119898119899





21

Receiveport

Terminating load

Dipoles

Couplers Matched

termination

λ2

Phaseshifters

119889 = 01120582

1205822 1205822 1205822

119873 minus 1 119873 = 30119889 = 01120582



0 20 40 60 80

0

10

20

minus60

minus50

minus40

minus30

minus20

minus10



1205901205822

(dB)


(a)

0 20 40 60 80

0

10

20

minus60

minus50

minus40

minus30

minus20

minus10



1205901205822

(dB)


(b)


1198850= 75Ω and 119885


119904= 0∘ (b) 120579

119904= 85∘


119881119899= 119881

app119899

minus

119873

sum

119898=1

119898 = 119899

119862119898119899119881

app119899

(25)

where 119881app119899







0 20 40 60 80

0

10

20

minus60

minus50

minus40

minus30

minus20

minus10



1205901205822

(dB)


(a)

0 20 40 60 80

0

10

20

minus60

minus50

minus40

minus30

minus20

minus10



1205901205822

(dB)


(b)


minus5120582 1198850= 125Ω and 119885


119904= 0∘ (b) 120579

119904= 85∘

0 20 40 60 80

0

10

20

minus60

minus50

minus40

minus30

minus20

minus10



1205901205822

(dB)


minus80

(a)

0 20 40 60 80

0

10

20

minus60

minus50

minus40

minus30

minus20

minus10



1205901205822

(dB)


minus80

(b)


1198850= 150Ω and 119885


119904= 0∘ (b) 120579

119904= 85∘



















4 Conformal Array




















1198671015840= 119862119877119867119862119879 (26)


119877 and 119862





6 Summary





References













































































































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation
















119881119888= 119862119881119899 (7)


119888as

119881119899= 119862minus1119881119888 (8)






119888119898119899

=1

2120587int

120587119896119889

minus120587119896119889

119892119898(119906)

119891119894 (119906)119890minus119895119899119896119889119906

119889119906 (9)



119862 = 119868 + 119878 (10)


1198781015840= 119879119878119879 (11)


1198621015840= 119879 + 119878

1015840119879minus1 (12)












119862 = 119860Θ119871Θ119890119895ΦΘ119866Θ119890

119895120587119863 (13)



119903+ 119868)

(14)





119877119909119909= 119862 sdot 119878 sdot 119878

119867sdot 119862119867+ 1205902sdot 119868 (15)

























[[[[

[

1198811198991198881

1198811198991198882

119881119899119888119873

]]]]

]

=

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

[

1 minus11988512

119905

119885119871


119905

119885119871

minus11988521

119905

119885119871


119905

119885119871

d

minus1198851198731

119905

119885119871

minus1198851198732

119905

119885119871

sdot sdot sdot 1

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

]

[[[[

[

1198811198881

1198811198882

119881119888119873

]]]]

]

(16)


where [119881119899119888] and [119881





119900 with the


[[[[

[

1198811199001

1198811199002

119881119900119873

]]]]

]

=

[[[[[[[[[[[

[

1 +11988511

119885119871

11988512

119885119871


119885119871

11988521

119885119871

1 +11988522

119885119871


119885119871

d

1198851198731

119885119871

1198851198732

119885119871

sdot sdot sdot 1 +119885119873119873

119885119871

]]]]]]]]]]]

]

[[[[

[

1198811198881

1198811198882

119881119888119873

]]]]

]

(17)

where 119885119898119899





119860ULA (120601) = [119886ULA (1206011) 119886ULA (1206012) 119886ULA (120601119889)](18a)

with


[[[[

[

1

119890119895119896119889 sin120601

119890119895119896119889(119873minus1) sin120601

]]]]

]

(18b)


05 1

SIN

R (d

B)

0

8

16

24

32

minus1 minus05 0sin(120601119889)

120585119889 = 5dB120585119889 = 10dB





119886 (120601) = cos119899 (120601) 119886ULA (120601) (19)




119889



SINR = 120585119889119880119879

119889119877minus1

119899119880lowast

119889 (20)






1 15

Dipole


2

1205822 1205822 1205822 1205822

119889 119889

119873 = 16

(a)

0

6

12

18


05 1

SIN

R (d

B)

minus1 minus05 0sin(120601119889)


119889 = 05120582

119889 = 120582

(b)


119889= 90∘




as [108]

119877119899= 119868 +

119898

sum

119896=1

120585119894119896119880lowast

119894119896119880119879

119894119896

119882 = 119870119877minus1

119899119880lowast

119889

(21)





119877119899= 119885lowast

0119885119879

0+

119898

sum

119896=1

120585119894119896119880lowast

119894119896119880119879

119894119896

119882 = 119870119885119879

0119877minus1

119899119880lowast

119889

(22)

where 1198850






NR

(dB)

0

12

24

36

48


119873 = 6

119873 = 16

119873 = 32

119873 = 64

119873 = 256

119873 = 512

119873 = 1024

119863(120582)


0 2 4 6 8 10 12

SIN

R (d

B)

0

4

8

12

16

Number of elements



119889= 5 dB 119885

119871=

119885lowast

119894119894 (120579119889 120601119889) = (90



119877119909119909= 1205902[119868 +

119898

sum

119896=1

120585119894119896(119885minus1

119900119880119894119896) lowast (119885

minus1

119900119880119894119896)119879

+ 120585119889(119885minus1

119900119880119889) lowast (119885

minus1

119900119880119889)119879

]

(23)









119862119898119899

=119884119898119899

119884119899119899+ 119884119892

(24)

where 119884119898119899





21

Receiveport

Terminating load

Dipoles

Couplers Matched

termination

λ2

Phaseshifters

119889 = 01120582

1205822 1205822 1205822

119873 minus 1 119873 = 30119889 = 01120582



0 20 40 60 80

0

10

20

minus60

minus50

minus40

minus30

minus20

minus10



1205901205822

(dB)


(a)

0 20 40 60 80

0

10

20

minus60

minus50

minus40

minus30

minus20

minus10



1205901205822

(dB)


(b)


1198850= 75Ω and 119885


119904= 0∘ (b) 120579

119904= 85∘


119881119899= 119881

app119899

minus

119873

sum

119898=1

119898 = 119899

119862119898119899119881

app119899

(25)

where 119881app119899







0 20 40 60 80

0

10

20

minus60

minus50

minus40

minus30

minus20

minus10



1205901205822

(dB)


(a)

0 20 40 60 80

0

10

20

minus60

minus50

minus40

minus30

minus20

minus10



1205901205822

(dB)


(b)


minus5120582 1198850= 125Ω and 119885


119904= 0∘ (b) 120579

119904= 85∘

0 20 40 60 80

0

10

20

minus60

minus50

minus40

minus30

minus20

minus10



1205901205822

(dB)


minus80

(a)

0 20 40 60 80

0

10

20

minus60

minus50

minus40

minus30

minus20

minus10



1205901205822

(dB)


minus80

(b)


1198850= 150Ω and 119885


119904= 0∘ (b) 120579

119904= 85∘



















4 Conformal Array




















1198671015840= 119862119877119867119862119879 (26)


119877 and 119862





6 Summary





References













































































































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation


















119862 = 119860Θ119871Θ119890119895ΦΘ119866Θ119890

119895120587119863 (13)



119903+ 119868)

(14)





119877119909119909= 119862 sdot 119878 sdot 119878

119867sdot 119862119867+ 1205902sdot 119868 (15)

























[[[[

[

1198811198991198881

1198811198991198882

119881119899119888119873

]]]]

]

=

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

[

1 minus11988512

119905

119885119871


119905

119885119871

minus11988521

119905

119885119871


119905

119885119871

d

minus1198851198731

119905

119885119871

minus1198851198732

119905

119885119871

sdot sdot sdot 1

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

]

[[[[

[

1198811198881

1198811198882

119881119888119873

]]]]

]

(16)


where [119881119899119888] and [119881





119900 with the


[[[[

[

1198811199001

1198811199002

119881119900119873

]]]]

]

=

[[[[[[[[[[[

[

1 +11988511

119885119871

11988512

119885119871


119885119871

11988521

119885119871

1 +11988522

119885119871


119885119871

d

1198851198731

119885119871

1198851198732

119885119871

sdot sdot sdot 1 +119885119873119873

119885119871

]]]]]]]]]]]

]

[[[[

[

1198811198881

1198811198882

119881119888119873

]]]]

]

(17)

where 119885119898119899





119860ULA (120601) = [119886ULA (1206011) 119886ULA (1206012) 119886ULA (120601119889)](18a)

with


[[[[

[

1

119890119895119896119889 sin120601

119890119895119896119889(119873minus1) sin120601

]]]]

]

(18b)


05 1

SIN

R (d

B)

0

8

16

24

32

minus1 minus05 0sin(120601119889)

120585119889 = 5dB120585119889 = 10dB





119886 (120601) = cos119899 (120601) 119886ULA (120601) (19)




119889



SINR = 120585119889119880119879

119889119877minus1

119899119880lowast

119889 (20)






1 15

Dipole


2

1205822 1205822 1205822 1205822

119889 119889

119873 = 16

(a)

0

6

12

18


05 1

SIN

R (d

B)

minus1 minus05 0sin(120601119889)


119889 = 05120582

119889 = 120582

(b)


119889= 90∘




as [108]

119877119899= 119868 +

119898

sum

119896=1

120585119894119896119880lowast

119894119896119880119879

119894119896

119882 = 119870119877minus1

119899119880lowast

119889

(21)





119877119899= 119885lowast

0119885119879

0+

119898

sum

119896=1

120585119894119896119880lowast

119894119896119880119879

119894119896

119882 = 119870119885119879

0119877minus1

119899119880lowast

119889

(22)

where 1198850






NR

(dB)

0

12

24

36

48


119873 = 6

119873 = 16

119873 = 32

119873 = 64

119873 = 256

119873 = 512

119873 = 1024

119863(120582)


0 2 4 6 8 10 12

SIN

R (d

B)

0

4

8

12

16

Number of elements



119889= 5 dB 119885

119871=

119885lowast

119894119894 (120579119889 120601119889) = (90



119877119909119909= 1205902[119868 +

119898

sum

119896=1

120585119894119896(119885minus1

119900119880119894119896) lowast (119885

minus1

119900119880119894119896)119879

+ 120585119889(119885minus1

119900119880119889) lowast (119885

minus1

119900119880119889)119879

]

(23)









119862119898119899

=119884119898119899

119884119899119899+ 119884119892

(24)

where 119884119898119899





21

Receiveport

Terminating load

Dipoles

Couplers Matched

termination

λ2

Phaseshifters

119889 = 01120582

1205822 1205822 1205822

119873 minus 1 119873 = 30119889 = 01120582



0 20 40 60 80

0

10

20

minus60

minus50

minus40

minus30

minus20

minus10



1205901205822

(dB)


(a)

0 20 40 60 80

0

10

20

minus60

minus50

minus40

minus30

minus20

minus10



1205901205822

(dB)


(b)


1198850= 75Ω and 119885


119904= 0∘ (b) 120579

119904= 85∘


119881119899= 119881

app119899

minus

119873

sum

119898=1

119898 = 119899

119862119898119899119881

app119899

(25)

where 119881app119899







0 20 40 60 80

0

10

20

minus60

minus50

minus40

minus30

minus20

minus10



1205901205822

(dB)


(a)

0 20 40 60 80

0

10

20

minus60

minus50

minus40

minus30

minus20

minus10



1205901205822

(dB)


(b)


minus5120582 1198850= 125Ω and 119885


119904= 0∘ (b) 120579

119904= 85∘

0 20 40 60 80

0

10

20

minus60

minus50

minus40

minus30

minus20

minus10



1205901205822

(dB)


minus80

(a)

0 20 40 60 80

0

10

20

minus60

minus50

minus40

minus30

minus20

minus10



1205901205822

(dB)


minus80

(b)


1198850= 150Ω and 119885


119904= 0∘ (b) 120579

119904= 85∘



















4 Conformal Array




















1198671015840= 119862119877119867119862119879 (26)


119877 and 119862





6 Summary





References













































































































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation
































[[[[

[

1198811198991198881

1198811198991198882

119881119899119888119873

]]]]

]

=

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

[

1 minus11988512

119905

119885119871


119905

119885119871

minus11988521

119905

119885119871


119905

119885119871

d

minus1198851198731

119905

119885119871

minus1198851198732

119905

119885119871

sdot sdot sdot 1

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

]

[[[[

[

1198811198881

1198811198882

119881119888119873

]]]]

]

(16)


where [119881119899119888] and [119881





119900 with the


[[[[

[

1198811199001

1198811199002

119881119900119873

]]]]

]

=

[[[[[[[[[[[

[

1 +11988511

119885119871

11988512

119885119871


119885119871

11988521

119885119871

1 +11988522

119885119871


119885119871

d

1198851198731

119885119871

1198851198732

119885119871

sdot sdot sdot 1 +119885119873119873

119885119871

]]]]]]]]]]]

]

[[[[

[

1198811198881

1198811198882

119881119888119873

]]]]

]

(17)

where 119885119898119899





119860ULA (120601) = [119886ULA (1206011) 119886ULA (1206012) 119886ULA (120601119889)](18a)

with


[[[[

[

1

119890119895119896119889 sin120601

119890119895119896119889(119873minus1) sin120601

]]]]

]

(18b)


05 1

SIN

R (d

B)

0

8

16

24

32

minus1 minus05 0sin(120601119889)

120585119889 = 5dB120585119889 = 10dB





119886 (120601) = cos119899 (120601) 119886ULA (120601) (19)




119889



SINR = 120585119889119880119879

119889119877minus1

119899119880lowast

119889 (20)






1 15

Dipole


2

1205822 1205822 1205822 1205822

119889 119889

119873 = 16

(a)

0

6

12

18


05 1

SIN

R (d

B)

minus1 minus05 0sin(120601119889)


119889 = 05120582

119889 = 120582

(b)


119889= 90∘




as [108]

119877119899= 119868 +

119898

sum

119896=1

120585119894119896119880lowast

119894119896119880119879

119894119896

119882 = 119870119877minus1

119899119880lowast

119889

(21)





119877119899= 119885lowast

0119885119879

0+

119898

sum

119896=1

120585119894119896119880lowast

119894119896119880119879

119894119896

119882 = 119870119885119879

0119877minus1

119899119880lowast

119889

(22)

where 1198850






NR

(dB)

0

12

24

36

48


119873 = 6

119873 = 16

119873 = 32

119873 = 64

119873 = 256

119873 = 512

119873 = 1024

119863(120582)


0 2 4 6 8 10 12

SIN

R (d

B)

0

4

8

12

16

Number of elements



119889= 5 dB 119885

119871=

119885lowast

119894119894 (120579119889 120601119889) = (90



119877119909119909= 1205902[119868 +

119898

sum

119896=1

120585119894119896(119885minus1

119900119880119894119896) lowast (119885

minus1

119900119880119894119896)119879

+ 120585119889(119885minus1

119900119880119889) lowast (119885

minus1

119900119880119889)119879

]

(23)









119862119898119899

=119884119898119899

119884119899119899+ 119884119892

(24)

where 119884119898119899





21

Receiveport

Terminating load

Dipoles

Couplers Matched

termination

λ2

Phaseshifters

119889 = 01120582

1205822 1205822 1205822

119873 minus 1 119873 = 30119889 = 01120582



0 20 40 60 80

0

10

20

minus60

minus50

minus40

minus30

minus20

minus10



1205901205822

(dB)


(a)

0 20 40 60 80

0

10

20

minus60

minus50

minus40

minus30

minus20

minus10



1205901205822

(dB)


(b)


1198850= 75Ω and 119885


119904= 0∘ (b) 120579

119904= 85∘


119881119899= 119881

app119899

minus

119873

sum

119898=1

119898 = 119899

119862119898119899119881

app119899

(25)

where 119881app119899







0 20 40 60 80

0

10

20

minus60

minus50

minus40

minus30

minus20

minus10



1205901205822

(dB)


(a)

0 20 40 60 80

0

10

20

minus60

minus50

minus40

minus30

minus20

minus10



1205901205822

(dB)


(b)


minus5120582 1198850= 125Ω and 119885


119904= 0∘ (b) 120579

119904= 85∘

0 20 40 60 80

0

10

20

minus60

minus50

minus40

minus30

minus20

minus10



1205901205822

(dB)


minus80

(a)

0 20 40 60 80

0

10

20

minus60

minus50

minus40

minus30

minus20

minus10



1205901205822

(dB)


minus80

(b)


1198850= 150Ω and 119885


119904= 0∘ (b) 120579

119904= 85∘



















4 Conformal Array




















1198671015840= 119862119877119867119862119879 (26)


119877 and 119862





6 Summary





References













































































































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation





















[[[[

[

1198811198991198881

1198811198991198882

119881119899119888119873

]]]]

]

=

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

[

1 minus11988512

119905

119885119871


119905

119885119871

minus11988521

119905

119885119871


119905

119885119871

d

minus1198851198731

119905

119885119871

minus1198851198732

119905

119885119871

sdot sdot sdot 1

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

]

[[[[

[

1198811198881

1198811198882

119881119888119873

]]]]

]

(16)


where [119881119899119888] and [119881





119900 with the


[[[[

[

1198811199001

1198811199002

119881119900119873

]]]]

]

=

[[[[[[[[[[[

[

1 +11988511

119885119871

11988512

119885119871


119885119871

11988521

119885119871

1 +11988522

119885119871


119885119871

d

1198851198731

119885119871

1198851198732

119885119871

sdot sdot sdot 1 +119885119873119873

119885119871

]]]]]]]]]]]

]

[[[[

[

1198811198881

1198811198882

119881119888119873

]]]]

]

(17)

where 119885119898119899





119860ULA (120601) = [119886ULA (1206011) 119886ULA (1206012) 119886ULA (120601119889)](18a)

with


[[[[

[

1

119890119895119896119889 sin120601

119890119895119896119889(119873minus1) sin120601

]]]]

]

(18b)


05 1

SIN

R (d

B)

0

8

16

24

32

minus1 minus05 0sin(120601119889)

120585119889 = 5dB120585119889 = 10dB





119886 (120601) = cos119899 (120601) 119886ULA (120601) (19)




119889



SINR = 120585119889119880119879

119889119877minus1

119899119880lowast

119889 (20)






1 15

Dipole


2

1205822 1205822 1205822 1205822

119889 119889

119873 = 16

(a)

0

6

12

18


05 1

SIN

R (d

B)

minus1 minus05 0sin(120601119889)


119889 = 05120582

119889 = 120582

(b)


119889= 90∘




as [108]

119877119899= 119868 +

119898

sum

119896=1

120585119894119896119880lowast

119894119896119880119879

119894119896

119882 = 119870119877minus1

119899119880lowast

119889

(21)





119877119899= 119885lowast

0119885119879

0+

119898

sum

119896=1

120585119894119896119880lowast

119894119896119880119879

119894119896

119882 = 119870119885119879

0119877minus1

119899119880lowast

119889

(22)

where 1198850






NR

(dB)

0

12

24

36

48


119873 = 6

119873 = 16

119873 = 32

119873 = 64

119873 = 256

119873 = 512

119873 = 1024

119863(120582)


0 2 4 6 8 10 12

SIN

R (d

B)

0

4

8

12

16

Number of elements



119889= 5 dB 119885

119871=

119885lowast

119894119894 (120579119889 120601119889) = (90



119877119909119909= 1205902[119868 +

119898

sum

119896=1

120585119894119896(119885minus1

119900119880119894119896) lowast (119885

minus1

119900119880119894119896)119879

+ 120585119889(119885minus1

119900119880119889) lowast (119885

minus1

119900119880119889)119879

]

(23)









119862119898119899

=119884119898119899

119884119899119899+ 119884119892

(24)

where 119884119898119899





21

Receiveport

Terminating load

Dipoles

Couplers Matched

termination

λ2

Phaseshifters

119889 = 01120582

1205822 1205822 1205822

119873 minus 1 119873 = 30119889 = 01120582



0 20 40 60 80

0

10

20

minus60

minus50

minus40

minus30

minus20

minus10



1205901205822

(dB)


(a)

0 20 40 60 80

0

10

20

minus60

minus50

minus40

minus30

minus20

minus10



1205901205822

(dB)


(b)


1198850= 75Ω and 119885


119904= 0∘ (b) 120579

119904= 85∘


119881119899= 119881

app119899

minus

119873

sum

119898=1

119898 = 119899

119862119898119899119881

app119899

(25)

where 119881app119899







0 20 40 60 80

0

10

20

minus60

minus50

minus40

minus30

minus20

minus10



1205901205822

(dB)


(a)

0 20 40 60 80

0

10

20

minus60

minus50

minus40

minus30

minus20

minus10



1205901205822

(dB)


(b)


minus5120582 1198850= 125Ω and 119885


119904= 0∘ (b) 120579

119904= 85∘

0 20 40 60 80

0

10

20

minus60

minus50

minus40

minus30

minus20

minus10



1205901205822

(dB)


minus80

(a)

0 20 40 60 80

0

10

20

minus60

minus50

minus40

minus30

minus20

minus10



1205901205822

(dB)


minus80

(b)


1198850= 150Ω and 119885


119904= 0∘ (b) 120579

119904= 85∘



















4 Conformal Array




















1198671015840= 119862119877119867119862119879 (26)


119877 and 119862





6 Summary





References













































































































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










where [119881119899119888] and [119881





119900 with the


[[[[

[

1198811199001

1198811199002

119881119900119873

]]]]

]

=

[[[[[[[[[[[

[

1 +11988511

119885119871

11988512

119885119871


119885119871

11988521

119885119871

1 +11988522

119885119871


119885119871

d

1198851198731

119885119871

1198851198732

119885119871

sdot sdot sdot 1 +119885119873119873

119885119871

]]]]]]]]]]]

]

[[[[

[

1198811198881

1198811198882

119881119888119873

]]]]

]

(17)

where 119885119898119899





119860ULA (120601) = [119886ULA (1206011) 119886ULA (1206012) 119886ULA (120601119889)](18a)

with


[[[[

[

1

119890119895119896119889 sin120601

119890119895119896119889(119873minus1) sin120601

]]]]

]

(18b)


05 1

SIN

R (d

B)

0

8

16

24

32

minus1 minus05 0sin(120601119889)

120585119889 = 5dB120585119889 = 10dB





119886 (120601) = cos119899 (120601) 119886ULA (120601) (19)




119889



SINR = 120585119889119880119879

119889119877minus1

119899119880lowast

119889 (20)






1 15

Dipole


2

1205822 1205822 1205822 1205822

119889 119889

119873 = 16

(a)

0

6

12

18


05 1

SIN

R (d

B)

minus1 minus05 0sin(120601119889)


119889 = 05120582

119889 = 120582

(b)


119889= 90∘




as [108]

119877119899= 119868 +

119898

sum

119896=1

120585119894119896119880lowast

119894119896119880119879

119894119896

119882 = 119870119877minus1

119899119880lowast

119889

(21)





119877119899= 119885lowast

0119885119879

0+

119898

sum

119896=1

120585119894119896119880lowast

119894119896119880119879

119894119896

119882 = 119870119885119879

0119877minus1

119899119880lowast

119889

(22)

where 1198850






NR

(dB)

0

12

24

36

48


119873 = 6

119873 = 16

119873 = 32

119873 = 64

119873 = 256

119873 = 512

119873 = 1024

119863(120582)


0 2 4 6 8 10 12

SIN

R (d

B)

0

4

8

12

16

Number of elements



119889= 5 dB 119885

119871=

119885lowast

119894119894 (120579119889 120601119889) = (90



119877119909119909= 1205902[119868 +

119898

sum

119896=1

120585119894119896(119885minus1

119900119880119894119896) lowast (119885

minus1

119900119880119894119896)119879

+ 120585119889(119885minus1

119900119880119889) lowast (119885

minus1

119900119880119889)119879

]

(23)









119862119898119899

=119884119898119899

119884119899119899+ 119884119892

(24)

where 119884119898119899





21

Receiveport

Terminating load

Dipoles

Couplers Matched

termination

λ2

Phaseshifters

119889 = 01120582

1205822 1205822 1205822

119873 minus 1 119873 = 30119889 = 01120582



0 20 40 60 80

0

10

20

minus60

minus50

minus40

minus30

minus20

minus10



1205901205822

(dB)


(a)

0 20 40 60 80

0

10

20

minus60

minus50

minus40

minus30

minus20

minus10



1205901205822

(dB)


(b)


1198850= 75Ω and 119885


119904= 0∘ (b) 120579

119904= 85∘


119881119899= 119881

app119899

minus

119873

sum

119898=1

119898 = 119899

119862119898119899119881

app119899

(25)

where 119881app119899







0 20 40 60 80

0

10

20

minus60

minus50

minus40

minus30

minus20

minus10



1205901205822

(dB)


(a)

0 20 40 60 80

0

10

20

minus60

minus50

minus40

minus30

minus20

minus10



1205901205822

(dB)


(b)


minus5120582 1198850= 125Ω and 119885


119904= 0∘ (b) 120579

119904= 85∘

0 20 40 60 80

0

10

20

minus60

minus50

minus40

minus30

minus20

minus10



1205901205822

(dB)


minus80

(a)

0 20 40 60 80

0

10

20

minus60

minus50

minus40

minus30

minus20

minus10



1205901205822

(dB)


minus80

(b)


1198850= 150Ω and 119885


119904= 0∘ (b) 120579

119904= 85∘



















4 Conformal Array




















1198671015840= 119862119877119867119862119879 (26)


119877 and 119862





6 Summary





References













































































































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










1 15

Dipole


2

1205822 1205822 1205822 1205822

119889 119889

119873 = 16

(a)

0

6

12

18


05 1

SIN

R (d

B)

minus1 minus05 0sin(120601119889)


119889 = 05120582

119889 = 120582

(b)


119889= 90∘




as [108]

119877119899= 119868 +

119898

sum

119896=1

120585119894119896119880lowast

119894119896119880119879

119894119896

119882 = 119870119877minus1

119899119880lowast

119889

(21)





119877119899= 119885lowast

0119885119879

0+

119898

sum

119896=1

120585119894119896119880lowast

119894119896119880119879

119894119896

119882 = 119870119885119879

0119877minus1

119899119880lowast

119889

(22)

where 1198850






NR

(dB)

0

12

24

36

48


119873 = 6

119873 = 16

119873 = 32

119873 = 64

119873 = 256

119873 = 512

119873 = 1024

119863(120582)


0 2 4 6 8 10 12

SIN

R (d

B)

0

4

8

12

16

Number of elements



119889= 5 dB 119885

119871=

119885lowast

119894119894 (120579119889 120601119889) = (90



119877119909119909= 1205902[119868 +

119898

sum

119896=1

120585119894119896(119885minus1

119900119880119894119896) lowast (119885

minus1

119900119880119894119896)119879

+ 120585119889(119885minus1

119900119880119889) lowast (119885

minus1

119900119880119889)119879

]

(23)









119862119898119899

=119884119898119899

119884119899119899+ 119884119892

(24)

where 119884119898119899





21

Receiveport

Terminating load

Dipoles

Couplers Matched

termination

λ2

Phaseshifters

119889 = 01120582

1205822 1205822 1205822

119873 minus 1 119873 = 30119889 = 01120582



0 20 40 60 80

0

10

20

minus60

minus50

minus40

minus30

minus20

minus10



1205901205822

(dB)


(a)

0 20 40 60 80

0

10

20

minus60

minus50

minus40

minus30

minus20

minus10



1205901205822

(dB)


(b)


1198850= 75Ω and 119885


119904= 0∘ (b) 120579

119904= 85∘


119881119899= 119881

app119899

minus

119873

sum

119898=1

119898 = 119899

119862119898119899119881

app119899

(25)

where 119881app119899







0 20 40 60 80

0

10

20

minus60

minus50

minus40

minus30

minus20

minus10



1205901205822

(dB)


(a)

0 20 40 60 80

0

10

20

minus60

minus50

minus40

minus30

minus20

minus10



1205901205822

(dB)


(b)


minus5120582 1198850= 125Ω and 119885


119904= 0∘ (b) 120579

119904= 85∘

0 20 40 60 80

0

10

20

minus60

minus50

minus40

minus30

minus20

minus10



1205901205822

(dB)


minus80

(a)

0 20 40 60 80

0

10

20

minus60

minus50

minus40

minus30

minus20

minus10



1205901205822

(dB)


minus80

(b)


1198850= 150Ω and 119885


119904= 0∘ (b) 120579

119904= 85∘



















4 Conformal Array




















1198671015840= 119862119877119867119862119879 (26)


119877 and 119862





6 Summary





References













































































































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










NR

(dB)

0

12

24

36

48


119873 = 6

119873 = 16

119873 = 32

119873 = 64

119873 = 256

119873 = 512

119873 = 1024

119863(120582)


0 2 4 6 8 10 12

SIN

R (d

B)

0

4

8

12

16

Number of elements



119889= 5 dB 119885

119871=

119885lowast

119894119894 (120579119889 120601119889) = (90



119877119909119909= 1205902[119868 +

119898

sum

119896=1

120585119894119896(119885minus1

119900119880119894119896) lowast (119885

minus1

119900119880119894119896)119879

+ 120585119889(119885minus1

119900119880119889) lowast (119885

minus1

119900119880119889)119879

]

(23)









119862119898119899

=119884119898119899

119884119899119899+ 119884119892

(24)

where 119884119898119899





21

Receiveport

Terminating load

Dipoles

Couplers Matched

termination

λ2

Phaseshifters

119889 = 01120582

1205822 1205822 1205822

119873 minus 1 119873 = 30119889 = 01120582



0 20 40 60 80

0

10

20

minus60

minus50

minus40

minus30

minus20

minus10



1205901205822

(dB)


(a)

0 20 40 60 80

0

10

20

minus60

minus50

minus40

minus30

minus20

minus10



1205901205822

(dB)


(b)


1198850= 75Ω and 119885


119904= 0∘ (b) 120579

119904= 85∘


119881119899= 119881

app119899

minus

119873

sum

119898=1

119898 = 119899

119862119898119899119881

app119899

(25)

where 119881app119899







0 20 40 60 80

0

10

20

minus60

minus50

minus40

minus30

minus20

minus10



1205901205822

(dB)


(a)

0 20 40 60 80

0

10

20

minus60

minus50

minus40

minus30

minus20

minus10



1205901205822

(dB)


(b)


minus5120582 1198850= 125Ω and 119885


119904= 0∘ (b) 120579

119904= 85∘

0 20 40 60 80

0

10

20

minus60

minus50

minus40

minus30

minus20

minus10



1205901205822

(dB)


minus80

(a)

0 20 40 60 80

0

10

20

minus60

minus50

minus40

minus30

minus20

minus10



1205901205822

(dB)


minus80

(b)


1198850= 150Ω and 119885


119904= 0∘ (b) 120579

119904= 85∘



















4 Conformal Array




















1198671015840= 119862119877119867119862119879 (26)


119877 and 119862





6 Summary





References













































































































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










21

Receiveport

Terminating load

Dipoles

Couplers Matched

termination

λ2

Phaseshifters

119889 = 01120582

1205822 1205822 1205822

119873 minus 1 119873 = 30119889 = 01120582



0 20 40 60 80

0

10

20

minus60

minus50

minus40

minus30

minus20

minus10



1205901205822

(dB)


(a)

0 20 40 60 80

0

10

20

minus60

minus50

minus40

minus30

minus20

minus10



1205901205822

(dB)


(b)


1198850= 75Ω and 119885


119904= 0∘ (b) 120579

119904= 85∘


119881119899= 119881

app119899

minus

119873

sum

119898=1

119898 = 119899

119862119898119899119881

app119899

(25)

where 119881app119899







0 20 40 60 80

0

10

20

minus60

minus50

minus40

minus30

minus20

minus10



1205901205822

(dB)


(a)

0 20 40 60 80

0

10

20

minus60

minus50

minus40

minus30

minus20

minus10



1205901205822

(dB)


(b)


minus5120582 1198850= 125Ω and 119885


119904= 0∘ (b) 120579

119904= 85∘

0 20 40 60 80

0

10

20

minus60

minus50

minus40

minus30

minus20

minus10



1205901205822

(dB)


minus80

(a)

0 20 40 60 80

0

10

20

minus60

minus50

minus40

minus30

minus20

minus10



1205901205822

(dB)


minus80

(b)


1198850= 150Ω and 119885


119904= 0∘ (b) 120579

119904= 85∘



















4 Conformal Array




















1198671015840= 119862119877119867119862119879 (26)


119877 and 119862





6 Summary





References













































































































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 20 40 60 80

0

10

20

minus60

minus50

minus40

minus30

minus20

minus10



1205901205822

(dB)


(a)

0 20 40 60 80

0

10

20

minus60

minus50

minus40

minus30

minus20

minus10



1205901205822

(dB)


(b)


minus5120582 1198850= 125Ω and 119885


119904= 0∘ (b) 120579

119904= 85∘

0 20 40 60 80

0

10

20

minus60

minus50

minus40

minus30

minus20

minus10



1205901205822

(dB)


minus80

(a)

0 20 40 60 80

0

10

20

minus60

minus50

minus40

minus30

minus20

minus10



1205901205822

(dB)


minus80

(b)


1198850= 150Ω and 119885


119904= 0∘ (b) 120579

119904= 85∘



















4 Conformal Array




















1198671015840= 119862119877119867119862119879 (26)


119877 and 119862





6 Summary





References













































































































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation























4 Conformal Array




















1198671015840= 119862119877119867119862119879 (26)


119877 and 119862





6 Summary





References













































































































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













4 Conformal Array




















1198671015840= 119862119877119867119862119879 (26)


119877 and 119862





6 Summary





References













































































































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation




















1198671015840= 119862119877119867119862119879 (26)


119877 and 119862





6 Summary





References













































































































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










6 Summary





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
























































































































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





















































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











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Documents

Review Article Mutual Coupling in Phased Arrays: A Reviewdownloads.hindawi.com/journals/ijap/2013/348123.pdf · Mutual Coupling in Phased Arrays: A Review HemaSingh,H.L.Sneha,andR.M.Jha