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Hindawi Publishing CorporationInternational Journal of Antennas and PropagationVolume 2012, Article ID 712318, 13 pagesdoi:10.1155/2012/712318

Research Article

A Subspace-Based Compensation Method for the MutualCoupling in Concentric Circular Ring Arrays for Near-FieldSource Localisation

Mohammed Jainul Abedin and Ananda Sanagavarapu Mohan

Centre for Health Technologies, School of Electrical, Mechanical and Mechatronic Systems, Faculty of Engineering andInformation Technology, University of Technology, Sydney (UTS), P.O. Box 123, Broadway, NSW 2007, Australia

Correspondence should be addressed to Ananda Sanagavarapu Mohan, [email protected]

Received 14 June 2011; Accepted 5 September 2011

Academic Editor: Hoi Shun Lui

Copyright © 2012 M. J. Abedin and A. S. Mohan. This is an open access article distributed under the Creative CommonsAttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work isproperly cited.

We propose a technique for compensating the effect of mutual coupling on parameter estimation that is suitable with any subspace-based super-resolution algorithms. A Concentric circular ring array (CCRA) formed using thin dipole antennas in the receivingmode is employed to estimate the parameters of electromagnetic sources located in the radiating near field of the array. ACCRA geometry that obtains a lowest Cramer-Rao lower bound (CRLB) in the presence of array mutual coupling is chosen forinvestigation. The mutual coupling among antenna elements of the array would affect the orthogonality of subspaces when MUSICor ESPRIT algorithms are used for parameter estimation. The proposed method obtains a compensation matrix that restores theorthogonality between the subspaces there by improving the accuracy of estimation. To avoid three-dimensional searches, therange parameter is estimated using a cross-correlation-based method. Numerical simulation using a full-wave electromagnetic(EM) solver is employed to demonstrate the effectiveness of the proposed compensation approach.

1. Introduction

A common practical issue in parameter estimation usingarray of antennas is the effect of mutual coupling. Thiseffect has long been recognised as one of the leading causesof degradation of the performance of parameter estimationalgorithms [1]. This degradation occurs due to the presenceof mutual coupling between antenna elements which causesdeviations to the array manifold. Many methods to combatthe effect of mutual coupling on parameter estimationhave been proposed in the literature [1–4]. The conceptof coupling matrix for compensating the effect of mutualcoupling was used in [5]; however, it did not properlyaccount for the platform effects. Full-wave EM solvers werealso employed by many authors. Adve and Sarkar [6] usedthe method of moments for computing the mutual couplingeffect precisely on wire antenna arrays, and Rogier andZutter [7] used full-wave EM techniques for computingthe mutual coupling effects in a planar array. A minimum

norm mutual coupling compensation for the application ofdirection of arrival (DOA) estimation was also reported in[8]. Yuan et al. [9] proposed a method for DOA estimationusing MUSIC algorithm by considering universal steeringvectors which does not require any additional mutualcoupling compensation method. In addition to the above-mentioned methods, calibration techniques have also beenproposed for compensating the effect of mutual couplingin DOA estimation algorithms. For instance, the calibrationprocedures developed by Weiss and Friedlander [10] requiresome perfectly calibrated sensors to compensate for theeffect of mutual coupling as well as perturbations in gainand phase. An iterative least mean square approach wasproposed by Hung [11] to estimate the calibration matrix,which requires some initial calibration. Self-calibration [12]and autocalibration [13] methods were also proposed whichseek to iteratively minimise a function with respect to bothdirection of arrival (DOA) and mutual coupling. All theabove-mentioned techniques mainly aim to compensate for

2 International Journal of Antennas and Propagation

the error due to lack of calibration, but all of them requirea set of known calibration sources at some known locations.In practice, obtaining more than a single calibration sourcecan be challenging for most of the localisation scenarios.Thus, compensation techniques that do not overly dependon multiple calibration sources are desirable. It should alsobe noted that all the above-mentioned works mainly aimed ateither ULA or UCA geometries for compensating the mutualcoupling. Mutual coupling matrix has a banded Toplitzstructure for ULA and symmetric circulant structure withthree bands for UCA [14], both of which can help to simplifythe computation of the compensation matrix. However, suchwell-defined matrix structure may not be available for mostof the complex arbitrary array geometries such as concentriccircular ring arrays (CCRA) for which the effect of mutualcoupling can be more severe. It would, therefore, be ofinterest to develop alternate mutual coupling compensationmethods useful for complex planar arrays, which is the focusof this paper.

The effect of mutual coupling varies in arrays sincethe mutual impedances in a transmitting array can bedifferent from that of a receiving array. Lui et al. [15, 16]have analysed this effect especially for the receiving modeof antenna arrays. They also propose a mutual couplingcompensation technique based on a least square approachfor the receiving mode of antenna arrays [15, 17]. However,their compensation technique can suffer from performancedegradation when the number of emitting sources is less thanthe number of elements in the receiver antenna array. This ismainly because the compensation technique based on leastsquares method does not have a unique solution when eithera single source or a smaller number of sources are present.Further, the investigations reported in [15, 17] on mutualcoupling compensation are carried out only for ULA andUCA geometries for the estimation of azimuth angles of far-field sources. However, in many applications, estimation ofthe parameters of near-field sources when complex arbitraryantenna arrays are employed is of concern and particularlythe techniques to reduce the effects of mutual coupling onthe parameter estimation are of interest.

For near-field parameter estimation, the use of planararrays can be more appropriate to overcome the limitationsof ULAs, namely, the problems in resolving the noncoplanaremitters and the presence of angular ambiguities withinthe azimuthal plane [18]. However, the use of complexplanar array geometries can pose some extra challenges inthe form of the array steering vector imperfections due tomutual coupling which can lead to severe detrimental effects.Further, when parameters need to be estimated, the effectof mutual coupling varies on the type of estimation algo-rithms used. For subspace-based super-resolution estimationalgorithms, the error due to mutual coupling can rotatethe subspaces thereby disrupting the orthogonality betweensignal and noise subspaces. As a result, subspace-basedsuper-resolution algorithms such as MUSIC and ESPRITwould lead to significant degradation of their estimationperformance. These errors must be compensated in order torecover their estimation performance. Hence, there is a needfor mutual coupling compensation methods to improve the

estimation performance of MUSIC and ESPRIT algorithmsfor localising either single or multiple near-field sourceswhen arbitrary planar antenna arrays are employed.

In this paper, we propose a mutual coupling compensa-tion technique applicable for MUSIC and ESPRIT algorithmsand investigate its performance when it is applied on a CCRAfor the near-field parameter estimation. CCRA geometrycan be formed by having rings that have either uniform ornonuniform radii. Similarly, the number of antenna elementsover the circumference of each ring could also be variable. Allthese variations have an impact on the array mutual couplingof CCRA which in turn affects the parameter estimation.Thus, in order to choose an optimised CCRA geometry thatcan lead to lowest estimation bias; we derive CRLB for fourdifferent cases each with different ring radii and number ofantenna elements so as to obtain a particular combinationwhich provides the lowest root mean square error. Theoptimised array geometry is then employed for further pro-cessing, that is, the mutual coupling compensation and near-field parameter estimation using both MUSIC and ESPRITalgorithms. Since the parameter set required to be estimatedfor near-field localisation is larger than for the correspondingfar-field case, it would be useful to reduce the computationalload of conventional MUSIC and ESPRIT algorithms. Toachieve this objective, here the range parameter is estimatedusing a cross-correlation-based method. Further, beamspaceprocessing [19] is applied on CCRA so as to make it suitablefor use with MUSIC and ESPRIT algorithms. We assume thatthe antennas receive correlated signals. Since the beamspaceprocessing maps the array steering manifold of a planararray onto that of a virtual linear array, forward-backwardsmoothing can also be applied for decorrelation [18]. Weemploy a three-ring CCRA with thin wire dipole antennasas elements and obtain the mutual coupling matrix using afull-wave electromagnetic simulator. Simulation results areprovided on the 3D localization of radiating sources placedin the near field of the CCRA by incorporating the proposedmutual coupling compensation method on MUSIC andESPRIT algorithms.

2. Problem Formulation

2.1. Concentric Circular Ring Array (CCRA). A CCRA isdesigned with P concentric rings, and each ring is populatedwith the same number of M omnidirectional dipole antennason its circumference, as shown in Figure 1. The locationof kth element on lth ring is denoted by its Cartesiancoordinate {xkl , ykl} = {ρl cosϕkl , ρl sinϕkl}, where kl =11, 21, . . . ,M1, 12, 22, . . . ,M2, . . . ,MP , with respect to the ori-gin chosen at the centre of the circular ring structure, whereρl is the radius of the lth ring and ϕkl is the angular positionof the kth element of the lth ring. For the uniform placementof array elements on the circumference of each ring, ϕkl =2π(kl − 1)/MP . The radius of the lth ring is given by

ρl = λmin

(4 sin(π/MP)), (1)

International Journal of Antennas and Propagation 3

dth source

θdrd

rdk

k th element

x

y

z

ρ2

ϕkl

ρP

φd

pth ring

Figure 1: Concentric circular ring array (CCRA).

where λ is the wavelength. Now, the elements of the arraysteering vector for the lth ring can be obtained as

akl = e jωτkl (r,θ,φ), (2)

where τkl is the time delay of incoming signal coming from asource located at (r, θ,φ) to kth element of lth ring, and timedelay is determined as

τkl(r, θ,φ

) = 1c

(rkl − r

), (3)

where

rkl = sqrt(r2 +

(x2kl

+ y2kl

)2 − 2 sin θ(xkl cosφ + ykl sinφ

)).

(4)

The array steering vector for the lth ring is given by

al =[e jωτ1l (r,θ,φ), e jωτ2l (r,θ,φ), . . . , e jωτMl

(r,θ,φ)]T

, (5)

where [·]T denotes transpose operation. Considering all therings in the CCRA, the array steering vector can be written as

a(r, θ,φ

) = [a1(r, θ,φ

), a2(r, θ,φ

), . . . , aP

(r, θ,φ

)]T. (6)

A CCRA can be formed either by having same or varyingnumber of elements on each ring [20]. Hence, the arraysteering vector of every ring may not be equal in length andthus can be represented as a group of separate single arraysteering vectors. The dimension of the total array steeringvector is MP × 1, given by

a(r, θ,φ

) =⎡

⎣e jωτ11 (r,θ,φ), e jωτ21 (r,θ,φ), . . . , e jωτM1 (r,θ,φ), e jωτ12 (r,θ,φ),

e jωτ22 (r,θ,φ), . . . , e jωτM2 (r,θ,φ), e jωτ1l (r,θ,φ), e jωτ2l (r,θ,φ), . . . , e jωτMP (r,θ,φ)

⎤

⎦

T

. (7)

With an aim to apply beamspace transformation, we expressthe array steering vector of CCRA into a phase mode as

a f = f Hp a(r, θ,φ

). (8)

This transformation maps the array steering vector of CCRAto that of an equivalent virtual linear array. Defining “B” asthe discrete Fourier transform matrix given by

B =

⎡

⎢⎢⎢⎢⎣

1 e− jhϕ · · · e− jh(M−1)ϕ

......

......

1 e jhϕ · · · e jh(M−1)ϕ

⎤

⎥⎥⎥⎥⎦

B =

⎡

⎢⎢⎢⎢⎣

f H−h...

f Hh

⎤

⎥⎥⎥⎥⎦

,

(9)

where (·)H denotes complex conjugate transpose and theelements of (9) can be expressed as

fp = 1M

[1, e− j pϕ, . . . , e− j p(M−1)ϕ

]T, (10)

where p = −h, −h + 1, . . . ,h.

2.2. Selection of Optimal CCRA Configuration Based on CRLB.Here, our aim is to select an optimal CCRA configurationthat provides lowest estimation bias using Cramer-Rao lowerbound (CRLB) and employ that array for further processingin this paper. CRLB is derived in this section for CCRA in thepresence of mutual coupling. Assuming the received signalvector s to be complex valued and Gaussian distributed, thecovariance matrix can be formed as

R = ARsAH

+ σ2nI, (11)

where A = CA and Rs = E[ssH]. The unknown parameterswhich need to be estimated can be written in a vector form

Θ =[

r, θ,φ, [αk]1:MP ,[βk]

1:MP

]T, (12)

where θ = [θ1, . . . , θd]T , φ = [φ1, . . . ,φd]T , αk = {Crek }, and

βk = {Cimk }. For convenience, we are using notations αk and

βk instead of αkl and βkl , for k = 1, 2, . . . ,MP. ConsideringN snapshots of the received signal, the Fisher InformationMatrix (FIM) can be formed as

Fmn = Ntr{R−1 ∂R

∂ΘmR−1 ∂R

∂Θn

}. (13)


The elements of FIM are derived [21, 22] and presented inthe appendix. Referring to the appendix, the FIM which is a(3D + MP)× (3D + MP) matrix can be written as

FIM � F =

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

Fθθ Fθφ Fθr Fθα Fθβ

FTθφ Fφφ Fφr Fφα Fφβ

FTθr FT

rφ Frr Frα Frβ

FTθα FT

αφ FTrα Fαα Fαβ

FTθβ FT

φβ FTrβ Fβα Fββ

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

, (14)

where each term represents a block matrix, for example,Fθα = [Fθαk ]k=1:MP ,Fαα = [Fαkαk ]k=1:MP , and Fββ =[Fβkβk ]k=1:MP and the other block matrices follow the samenotational convention. CRLB is the inverse of FIM given byJ = F−1, and for all the desired parameters it can be obtainedas

CRLBΘ = sqrt

⎛

⎝ 12D

3D∑

d=1

Jdd

⎞

⎠, (15)

CRLBc = sqrt

⎛

⎝ 1

‖c‖2

3D+MP∑

d=3D+1

Jdd

⎞

⎠× 100%. (16)

In (16), CRLB with mutual coupling is calculated as apercentage as it is a relative value [21]. The CRLB forCCRA is computed by varying either the number of antennaelements or the ring radii, so that we can chose an optimumarray configuration that offers the lowest bias for estimatingthe near-field parameters in the presence of array mutualcoupling. In this paper, we consider four different variationsof CCRA for computing the CRLB and the results areindicated in Figures 2, 3, and 4. It can be observed, fromFigures 2–4, that lowest CRLB results when the number ofantenna elements on each ring is not the same and also whenthe incremental variation of ring radii is not uniform. Thus,the optimal CCRA configuration for performing the near-field parameter estimation is chosen.

2.3. Signal Modelling in the Presence of Mutual Coupling. Thearray response of an M-element CCRA due to dth near-fieldsource located at (r, θ,φ) is given by

x(t) =D∑

d=1

Ca f(rd, θd,φd

)sd(t) + n(t), (17)

where C denotes a matrix that contains the mutual cou-pling among the array elements. The coupling is inverselyproportional to the distance between the elements. Afternormalising the main diagonal elements to unity, the mutualcoupling matrix is expressed as follows:

C = (ZA + ZT)(Z + ZTIN )−1, (18)

where ZA is the impedance of an isolated individual element,ZT is the impedance of each receiver element, and IM is theidentity matrix. The impedance ZT is considered to be the

0 5 10 15 20 25 30 35 4010−3

10−2

10−1

100

SNR (dB)

RM

SEof

φ

CRLB CCRA (a)CRLB CCRA (b)

CRLB CCRA (c)CRLB CCRA (d)

Figure 2: CRLB of estimated azimuth for four different arrange-ments: (a), (b), (c), and (d) of CCRA.

0 5 10 15 20 25 30 35 4010−3

10−2

10−1

100

SNR (dB)



RM

SEof

θ

Figure 3: CRLB of estimated elevation angle for four differentarrangements: (a), (b), (c), and (d) of CCRA.

complex conjugate of ZA ensuring impedance matching formaximum power transfer. The elements of Z matrix are

Z =

∣∣∣∣∣∣∣∣∣∣∣∣∣

ZA + ZT Z1121 · · · Z11MP

Z2111 ZA + ZT · · · Z21MP

......

......

ZMP11 ZMP21 · · · ZA + ZT

∣∣∣∣∣∣∣∣∣∣∣∣∣

. (19)

Now, Z is defined as

Z = [{zklkl}] = [{Rklkl + jXklkl

}], (20)

whereRklkl andXklkl are the resistive and reactive componentsof the impedance Z. Using a number of snapshots of the


0 5 10 15 20 25 30 35 40

RM

SEof

ran

ge



0.22

0.2

0.18

0.16

0.14

0.12

0.1

SNR (dB)

Figure 4: CRLB of estimated range for four different arrangements:(a), (b), (c), and (d) of CCRA.

received signal in (17), the data covariance matrix can beexpressed as

R =D∑

d=1

σ2dCa f

(rd, θd,φd

)aHf(rd, θd,φd

)CH + σ2

nI. (21)

In order to apply subspace-based algorithms such as MUSICor ESPRIT for parameter estimation, an eigenvalue decom-position has to be performed which is given by

R = UsΛsUHs + σ2

nUnUHn , (22)

where Us and Un contain the signal and noise eigenvectors,Λs is the diagonal matrix that contains signal power, and

σ2n denotes noise power. In order to estimate parameters

accurately, the signal and noise subspaces must be mutuallyorthogonal. But, in the presence of mutual coupling C,the orthogonality no longer exists between signal andnoise subspaces. Hence, we need to obtain a matrix Cthat can compensate the effects of mutual coupling andthereby restoring orthogonality between the subspaces. Thus,incorporation of C makes ‖UH

n Ca(rd, θd,φd)‖ = 0 ford = 1, . . . ,D which will lead to proper estimation of theparameters using subspace-based algorithms. However, inpractice, C is not known. Hence, methods are required toaccurately estimate C prior to applying any subspace-basedparameter estimation procedure when the mutual couplingis present among the elements of the array.

2.4. Subspace Rotation due to Array Mutual Coupling. For theCCRA, in an ideal case, that is, when the mutual coupling isignored, the array steering vector due to an incident signalfrom dth source is given by (8). Under the effect of mutualcoupling, the modified array steering vector a f (r, θ,φ) isgiven by

a f(r, θ,φ

) = Ca f(r, θ,φ

). (23)

As noted earlier, the mutual coupling disrupts the orthogo-nality of subspaces since the signal subspace that is spannedby the steering vector is rotated from its true position.Therefore, it can be seen that

span{A f

}= span

{CAf

}, (24)

where Af is the array manifold matrix of size MP ×D that iscreated from the array steering vector (7) in the presence ofD sources. Referring to (17), the mutual coupling matrix Cfor CCRA is given by

C =

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

c1111 c1121 c1131 · · · c11M1 c1112 c1122 · · · c11kl · · · c11MP

c2111 c2121 c2131 · · · c21M1 c2112 c2122 · · · c21kl · · · c21MP

......

......

......

......

c1211

...

c1p11 c1pM1 c1pkl

.... . .

. . .. . .

cMp11 cMp21 · · · cMpM1 · · · ckPkl · · · cMpMp

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

. (25)

Now, the rotated subspace due to C can be expressed as

u(r, θ,φ

) = Cu(r, θ,φ

), (26)

where u are the elements of the rotated signal and noise sub-

spaces due to the presence of mutual coupling. The effect ofmutual coupling C must be compensated so as to make sub-spaces orthogonal in order to accurately estimate parameters.


2.5. Subspace Rotation due to Simplified Mutual CouplingModel in a CCRA. Our aim here is to derive the rotated sub-spaces for CCRA under simplified assumptions to demon-strate our method. The mutual coupling in an array dependson the separation between two adjacent antenna elements,and hence nonadjacent array elements have weaker coupling.Realising this, a single coupling coefficient was consideredfor computing the compensation matrix in the literature[14, 23]. Also, the coupling effect was ignored for an ULAfor interelement separation greater than 0.707λ [21]. Inthis section, in order to derive the subspace rotation dueto mutual coupling, we simplify (25) by assuming largeinterring separation for CCRA so as to make the interringcoupling insignificant and thus can be ignored. Also, inthis case, coupling among elements of outer ring can alsobe ignored due to larger interelement separation since theelements are distributed uniformly on rings with largercircumferences thus leading to larger interelement separa-tion. Under these assumptions, only the mutual couplingamong the adjacent elements of the innermost rings willbe considered to simplify the problem. Thus, the simplifiedmutual coupling matrix C can be obtained from (25) as

C =

⎡

⎢⎢⎢⎢⎢⎢⎢⎣

1 c11 c21 · · · 0

c11 1 c11 · · · 0

.... . .

...

0 0 · · · 1

⎤

⎥⎥⎥⎥⎥⎥⎥⎦

. (27)

Now, simplifying (25) for the above case, the expression forthe rotated subspace can be expressed as

u(r, θ,φ

) =[

1 + c11u, c11 + a + c21u2, . . . , c21u

M−3 + uM−2

+c21uM−1, c21u

M−2 + uM−1]T

.(28)

After further manipulation, (28) can be represented as

u(r, θ,φ

) = (1 + c21u)[

1,u, . . . ,uM−2,uM−1]T

+ c21

[0, 1, . . . ,uM−3,uM−2 − uM

]T

= u1mc

(r, θ,φ

)+ u2

mc

(r, θ,φ

),

(29)

where umc denotes the modified eigenvector in the presenceof mutual coupling. Thus, the expression for the estimatedcovariance matrix modified due to the presence of mutualcoupling becomes

R = U1mc

H(rd, θd,φd

)ΛsU1

mc

(rd, θd,φd

)

+ U2mc

H(rd, θd,φd

)ΛsU2

mc

(rd, θd,φd

)+ σ2

nI.(30)

Comparing (30) and (21), for the simplified case consideredhere, one can observe that the variation in covariancematrix is due to the contribution of the factor 1 + c21uwhich causes subspaces to rotate. As a result, it would leadto performance degradation of the subspace-based super-resolution algorithms.

3. Compensation for the Mutual Coupling

The effect of mutual coupling must be compensated forachieving a desired performance. This can be achieved byformulating a compensation matrix that minimises the errorby which the array manifold gets deviated from that ofactual one [14]. For subspace-based estimation algorithms,the deviation in array manifold due to mutual coupling doesappear as an error that disrupts the orthogonality betweensignal and noise subspaces of the covariance matrix. Similarconcept was also used in [24] where genetic algorithm (GA)was used to minimise the error to obtain a compensationmatrix in order to recover the estimation performance of theMUSIC algorithm. However, the method proposed in [24]requires MUSIC pseudospectrum for a known calibrationsource to correlate with the pseudospectra of unknownsources at each search point in the space which can becomputationally demanding. Alternatively, in this paper,we propose a method that uses an ideal array responsewithout mutual coupling to estimate a compensation matrixby minimising the error between the rotated and idealsubspaces, so that the introduction of the compensationmatrix restores the orthogonality between the subspaces. Theadvantage of the method proposed here as compared to thatused in [24] is that, in the present case, the compensationmatrix can be determined with a lower computational loadas it does not require performing correlation of MUSICpseudospectra at every search point within the parameterspace. As a result, the proposed compensation methodachieves quicker convergence. Also, the proposed methodis applicable for any subspace-based parameter estimationalgorithm, as will be demonstrated below where we alsoprovide a detailed description of the proposed technique.

3.1. Role of the Compensation Matrix. In order to remove theeffect of mutual coupling completely, one needs to estimatethe compensation matrix as C = C−1. However, in actualpractice, the array output model cannot account for all thesources of signal impairments in an antenna array, hence itis difficult to achieve C equal to C−1. Considering (17), evenin a noise-free situation, that is, when n(t) becomes a nullvector, C−1 can only be considered to be proportional to C.Therefore, only an estimated compensation matrix can beobtained. Assuming that ideal array response from referencesources is known, the estimation procedure for C can beexpressed as a minimisation of the function which is givenby

minC

∥∥∥aHkl

(r, θ,φ

)CRxCHakl

(r, θ,φ

)∥∥∥

2

F, (31)

where akl (r, θ,φ) is the orthogonal null steering vector thatspans the equivalent null space of the reference source.Incorporating C obtained from (31) into (26), the subspaceU will approximately be equal to U since CC ≈ I.Also, in (28), all coupling terms will almost vanish oncethe compensation terms are incorporated and thereby thesubspace rotation due to the effect of mutual coupling will


almost be nullified resulting in the effective restoration oforthogonality between subspaces.

Comparing the proposed technique with the other pub-lished compensation techniques in the literature shows thatthe method used in [11, 15] requires at least MP differentnear-field calibration sources around CCRA to achieveakl (r, θ,φ) of (31), where kl = 11, 21, . . . ,M1, 12, 22, . . . ,M2, . . . ,MP . This requires estimation of a compensationmatrix C of the size MP × MP by minimising (31) for allMP calibration sources. For the same case, the least squareapproach would require at least more than MP calibrationsources to achieve a unique solution for the estimationof compensation matrix C. Therefore, methods that donot require multiple calibration sources for estimating thecompensation matrix are desirable.

3.2. Proposed Compensation Method. Here, it is hypothesizedthat the error that destroys the orthogonality between signaland noise subspaces is only due to the mutual couplingand all other possible sources of impairment are ignored.As a result, UH

n A(r, θ,φ) /= 0, which is due to the presenceof mutual coupling among the elements of the receiverantenna array. Here, Un is the noise subspace that is obtainedfrom EVD of covariance matrix formed by the measureddata at the receiver. Since Un is corrupted by the arraymutual coupling, we need to estimate C. For that, instead ofminimising the functional given in (31), we reformulate anobjective function Q as

Q = minC

∥∥∥UH

n A(r, θ,φ

)−UHn CA

(r, θ,φ

)∥∥∥

2

F. (32)

Now, the estimated compensation matrix C can be computedby performing an optimisation method to minimise (32).Here, the true noise subspace Un is obtained from asingle known near-field calibration source for an ideal arraywithout mutual coupling with known array geometry andelement positions and assuming no gain and phase deviationto occur. We employ genetic algorithm to determine Cby minimising Q of (32). Incorporating C restores theorthogonality between subspaces, thereby recovering theperformance of parameter estimation even in the presenceof array mutual coupling.

4. Parameter Estimation after Compensation

In this section, we show as to how the proposed techniqueof compensation is incorporated into both MUSIC andESPRIT algorithms for estimating the parameters of near-field sources. In order to reduce the computational burdenin MUSIC and ESPRIT, we estimate range parameter byusing a method that employs cross-correlation among theeigenvectors of the signal subspace [25]. A brief descriptionof the procedure for parameter estimation of near-fieldsources is given below.

4.1. MUSIC Algorithm for Estimating Azimuth and ElevationAngles. The azimuth and elevation angles of near-field

sources can be obtained from the two-dimensional MUSICpseudospectrum given by

P = 1∥∥∥UnCa f

(θ,φ

)∥∥∥

2 , (33)

where C is the estimated compensation matrix obtained byminimising Q in (32).

4.2. ESPRIT Algorithm for Estimating Azimuth and ElevationAngles. For ESPRIT algorithm, the signal subspace matricescan be written as [26]

ΓU = U−1Ψ + DIU−1Ψ∗, (34)

where Ψ = T−1ΦT , D = diag{(−1)MP−2, . . . , (−1)1, (−1)0,(−1)1, . . . , (−1)MP} and U = UC is the matrix of eigenvectorsthat is obtained after mutual coupling compensation. Thissystem of equations has a unique solution when the numberof sources is less than the number of antenna elements inCCRA. The solution can be obtained as Φ = TΨT−1, andthe eigenvalues of Ψ provide the diagonal elements of Φwhich therefore yield automatically paired source azimuthand elevation angles as

φd = arg(ud + jvd

),

θd = sin−1(

sqrt(u2d + v2

d

)).

(35)

4.3. Cross-Correlation among Signal Eigenvectors for Time-Delay Estimation. Since the eigenvectors contain the timedelay information, the time delay among the signal pathscan be estimated via the cross-correlation of eigenvectors.The compensated signal eigenvectors are ui and u j , where

ui = Cui and u j = Cu j . A normalised cross-correlation ofui and u j is given by

Ri, j(τ) = ri, j(τ)√ri,i(0)r j, j(0)

, (36)

where the ri, j(τ) is the cross-correlation of ui(kl) and u j(kl)and it can be expressed as

ri, j(τ) = 1Ns − τ

MP−τ∑

kl=1

u∗i (kl + τ)uj(kl), (37)

where τ = −MP + 1, . . . ,MP − 1 and ui(n) is the nth elementof u. An estimated time delay can be obtained by maximisingthe Ri, j(τ). Using the time delay, the range parameter isestimated.

5. Simulation Results and Discussion

In order to demonstrate the performance of the proposedcompensation method for near-field source localisation, weconsider a CCRA with 27 half-wave, thin, dipole antennaswhich are distributed uniformly along the circumferences


Table 1: The different CCRA configurations used for obtaining the optimal case.

CCRA configurations Radii of rings No. of elements on each ring

(a) ρ1 = 0.5λ, ρ2 = λ, and ρ3 = 1.5λ M1 = 9, M2 = 9, and M3 = 9

(b) ρ1 = 0.5λ, ρ2 = 1.2λ, and ρ3 = 2λ M1 = 9, M2 = 9, and M3 = 9

(c) ρ1 = 0.5λ, ρ2 = λ, and ρ3 = 1.5λ M1 = 7, M2 = 9, and M3 = 11

(d) ρ1 = 0.5λ, ρ2 = 1.2λ, and ρ3 = 2λ M1 = 7, M2 = 9, and M3 = 11

Table 2: Values of estimated compensation matrix using the proposed method.

c1111 = 1.0000000000000 + 0.0000000000000ic1112 =−16.4480117045500+14.6902890526432i

c1113 =−16.4480117045500+14.6902890526432i

c1121 = −5.96190733331502− 35.7977458691057ic1122 =8.64950808461148− 10.6569065324454i

c1123 =15.3736986943575 + 0.998396037519312i

c1131 = 8.23261772059128 + 15.7095769032906ic1132 =−3.78253875856304+9.10367969640321i

c1133 =−7.48101029224301−9.22054742279721i

c1141 = 46.7641650534059− 30.5168801643121ic1142 =−1.84838342545929+19.3049333382765i

c1143 =−23.2217879987250−18.5353612023957i

c1151 = −22.0857363551641 + 8.42266800899579ic1152 =0.180711829409121− 12.7502896457783i

c1153 =−9.48946897114057+18.3604176420201i

c1161 = 9.43611799993107− 10.1165377535518ic1162 =−0.152420427868246+9.56021137354340i

c1163 =14.9968021182233− 1.56937076599169i

c1171 = −23.2217879987250− 18.5353612023957ic1172 =−1.84838342545925+19.3049333382765i

c1173 =46.7641650534058− 30.5168801643122i

c1181 = 13.1129478953285 + 10.3385379254891ic1182 =4.54841895551983− 12.4182935781109i

c1183 =−18.3079647571629−26.2365863851191i

c1191 = −10.8370296923116− 2.65335182564432ic1192 =−7.02131492970946+7.35188328378729i

c1193 =2.34858705035471 + 18.5759374713535i

of three concentric rings as shown in Figure 1. Antennaelement distribution on each ring and the radii of ringsare given in Table 1. The radius of each half-wave dipole isconsidered to be equal to 5×10−3λ at an operating frequencyof 2.4 GHz and a 50Ω load is assumed to be connectedto the terminal of each of the dipole antenna. The sourcesof EM radiation to be localised are modelled using threevertically polarised thin dipole antennas which are locatedat (2.2λ, 120◦, 40◦), (3λ, 70◦, 70◦), and (4λ, 80◦, 20◦) which liewithin the radiating near field of the receiver antenna array.Each of the source dipole is excited with a voltage source.Induced current at every dipole element of CCRA due tothe incident field emanated from three dipole sources iscomputed by using the full-wave electromagnetic simulationpackage FEKO [27]. The antenna terminal voltage due to thereceived field is calculated as Vm = ZLIm where this terminalvoltage represents (17). Since terminal voltage measured atevery antenna element in the array includes the contributionof the mutual coupling, the proposed compensation methodis applied to compensate the effect of mutual coupling. Asingle (first) row of estimated compensation matrix whichincludes the elements c1111 to c1193 using proposed method istabulated in Table 2.

5.1. Performance of CCRA for Different Arrangements ofAntenna Elements. In order to analyse the effect of mutualcoupling on the near-field parameter estimation using CCRA

and chose optimum array configuration, CRLB for differentarrangements of CCRA are calculated in the presence ofmutual coupling as discussed earlier. Here, we considerfour different arrangements of ring radii and elementdistributions for forming an optimum CCRA as given inTable 1. In every CCRA configuration under consideration,only three concentric rings are considered and the totalnumber of dipole antenna elements in all rings put togetheris equal to 27. The antenna elements are placed uniformlyon the circumference of the each ring. In order to deriveCRLB for all the four cases, a single transmitting half-wavedipole is considered to be positioned at a known locationof (4λ, 80◦, 20◦). The CRLB for estimated azimuth angle,elevation angle, and range for these cases are plotted inFigures 2–4, respectively. It can be observed, from Figures2–4, that the array configuration of case (d) has the lowestCRLB when compared to other cases (a), (b), and (c) andcan be considered as an optimal configuration. Hence, wechoose the optimal CCRA configuration given in case (d) fordemonstrating compensation and parameter estimation.

5.2. Near-Field Parameter Estimation. Here, MUSIC andESPRIT algorithms are applied for parameter estimation ofnear-field sources using the optimal CCRA configuration.Zero mean white Gaussian noise is added to the receivedsignal at every element of CCRA. We apply the beamspacetransformation on the array manifold of CCRA to map


050

100150

200

0

50

1000

10

20

30

40

50

60

Azimuth (deg)

Elevation (deg)

Am

plit

ude

(dB

)

Figure 5: MUSIC pseudospectrum for ideal case where the effect ofmutual coupling is ignored.

050

100150

200

0

50

100

Azimuth (deg)

Elevation (deg)

Am

plit

ude

(dB

)

1.5

1

0.5

0

Figure 6: MUSIC pseudospectrum when mutual coupling isuncompensated.

it to an equivalent virtual linear array. The decorrelationtechnique available in [26] is applied since we considerthe received signals to be correlated. For an idealisedsituation, that is, when the effect of mutual coupling isignored, the estimated azimuth and elevation angles areshown by their pseudospectra in Figure 5. When the effectof mutual coupling is taken into consideration, withoutadopting any compensation, MUSIC pseudospectrum failsto create sharp peaks as shown in Figure 6. After applyingthe proposed compensation method, improved accuracy ofparameter estimation can be observed from the peaks ofMUSIC pseudospectra as shown in Figure 7. Similarly, in thepresence of mutual coupling, that is, when no compensationis applied, ESPRIT algorithm also fails to estimate azimuthand elevation angles properly as shown in histogram plotsof Figures 8 and 9, respectively. However, after applying theproposed compensation method, the estimation accuracyof azimuth and elevation angles using ESPRIT improvessignificantly as shown in histogram plots of Figures 10and 11, respectively. The estimated ranges for near-field

050

100150

200

0

50

100

Azimuth (deg)

Elevation (deg)

Am

plit

ude

(dB

)

40

30

20

10

0

Figure 7: MUSIC pseudospectrum after the proposed compensa-tion.

0 50 100 150 200 250

20

40

60

80

100

120

140

Azimuth (deg)

His

togr

am

Figure 8: Histogram plot of estimated azimuth angles usingESPRIT algorithm when mutual coupling is uncompensated.

sources are calculated from the estimated time delay whichis obtained by using cross-correlation among eigenvectorsof signal subspaces. The coefficients of cross-correlationsbefore compensating the effect of mutual coupling areplotted in Figure 12 whereas Figure 13 shows coefficients ofcross-correlation after applying the proposed compensationmethod. Comparing Figures 12 and 13, it is obvious that,after the compensation, the peaks have become distinctand clearer. The optimal CCRA configuration, as explainedearlier, is chosen for testing the performance of both MUSICand ESPRIT algorithms, and their performance is plotted inFigures 14, 15, and 16 along with CRLB. It can be observedfrom these results that the MUSIC algorithm performsslightly better at lower SNRs when compared to ESPRIT. TheRMSE of cross-correlation-based range estimation method iscloser CRLB at higher SNRs.


10 20 30 40 50 60 70 80 900

5

10

15

Elevation (deg)

His

togr

am

Figure 9: Histogram plot of estimated elevation angles usingESPRIT algorithm when mutual coupling is uncompensated.

60 70 80 90 100 110 120 1300

10

20

30

40

50

60

70

Azimuth (deg)

His

togr

am

Figure 10: Histogram plot of estimated azimuth angles usingESPRIT algorithm after applying proposed compensation.

10 20 30 40 50 60 70 800

10

20

30

40

50

60

70

His

togr

am

Elevation (deg)

Figure 11: Histogram plot of estimated elevation angles usingESPRIT algorithm after applying proposed compensation.

0 1 2 3 4 5 6×10−9

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

(s)

Cro

ss-c

orre

lati

on

Figure 12: The cross-correlation among the eigenvectors of signalsubspace when mutual coupling is uncompensated.

0 1 2 3 4 5 6×10−9

(s)

Cro

ss-c

orre

lati

on

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

Figure 13: The cross-correlation among the eigenvectors of signalsubspace after applying proposed compensation.

0 5 10 15 20 25 30 35 400

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

SNR (dB)

RM

SEof

φ

CRLB

MUSICESPRIT

Figure 14: Performance comparison for the estimation of azimuthangle using MUSIC and ESPRIT algorithms for CCRA design ofcase (d).


0 5 10 15 20 25 30 35 400

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

SNR (dB)

CRLB

RM

SEof

θ

MUSICESPRIT

Figure 15: Performance comparison for the estimation of elevationangle using MUSIC and ESPRIT algorithms using CCRA design ofcase (d).

0 5 10 15 20 25 30 35 400.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

SNR (dB)

Using delayCRLB

RM

SEof

r

Figure 16: Performance of range estimation using cross-correlationbased time-delay estimation method.

6. Conclusions

A mutual coupling compensation technique that is applica-ble with subspace-based super-resolution estimation algo-rithms is proposed in this paper. Concentric circular ringarrays are employed for testing the proposed method ofmutual coupling compensation by estimating the near-field parameters of transmitting dipoles using MUSIC andESPRIT algorithms. In order to achieve a better accuracyon parameter estimation, an optimal CCRA configurationis chosen to obtain lowest CRLB in the presence of mutualcoupling. The proposed compensation method is appliedwith MUSIC and ESPRIT algorithms for estimating azimuth

and elevation angles, and the range parameter is estimatedfrom the time delay obtained from the cross-correlation ofsignal subspace eigenvectors so as to avoid parameter searchover 3D space and reduce the associated computationalburden. The estimation performance after applying theproposed compensation method of mutual coupling iscompared with CRLB for different SNRs. From simulationresults, it can be observed that the proposed compensationmethod can effectively remove the effect of mutual couplingin CCRA and helps to improve the estimation performanceof subspace-based algorithms in the presence of array mutualcoupling.

Appendix

Elements of Fisher information matrix (FIM) (14) can beobtained as

Fθθ = 2N Re{(

RsAHR−1x A′θ

)�(RsA

HR−1x A′θ

)T

+(RsA

HR−1x ARs

)�(AHθ R

−1x A′θ

)T},

Fφφ = 2N Re{(

RsAHR−1x A′φ

)�(RsA

HR−1x A′φ

)T

+(RsA

HR−1x ARs

)�(AHφ R

−1x A′φ

)T},

Frr = 2N Re{(

RsAHR−1x A′r

)�(RsA

HR−1x A′r

)T

+(RsA

HR−1x ARs

)�(AHr R

−1x A′r

)T},

Fθθ = 2N Re{(

RsAHR−1x A′θ

)�(RsA

HR−1x A′θ

)T

+(RsA

HR−1x ARs

)�(AHR−1x A′θ

)T},

Fθφ = 2N Re{(

RsAHR−1

x A′θ)�(RsA

HR−1x A′φ

)T

+(RsA

HR−1x AθRs

)�(A′Hθ R

−1x A′φ

)T},

Fφθ = 2N Re{(

RsAHR−1

x A′φ)�(RsA

HR−1x A′θ

)T

+(RsA

HR−1x AθRs

)�(A′Hφ R

−1x A′θ

)T},

Frφ = 2N Re{(

RsAHR−1

x A′φ)�(RsA

HR−1x A′r

)T

+(RsA

HR−1x ArRs

)�(A′Hφ R

−1x A′r

)T},

Frθ = 2N Re{(

RsAHR−1

x A′θ)�(RsA

HR−1x A′r

)T

+(RsA

HR−1x AθRs

)�(A′Hr R

−1x A′θ

)T},


Fφr = 2N Re{(

RsAHR−1

x A′r)�(RsA

HR−1x A′φ

)T

+(RsA

HR−1x ArRs

)�(A′Hφ R

−1x A′r

)T},

Fθr = 2N Re{(

RsAHR−1

x A′r)�(RsA

HR−1x A′θ

)T

+(RsA

HR−1x AθRs

)�(A′Hr R

−1x A′θ

)T},

Fθαk = 2N Re{(

RsAHR−1

x A′αk)�(RsA

HR−1x A′θ

)T

+(RsA

HR−1x AθRs

)�(A′HαkR

−1x A′θ

)T},

Fθβk = 2N Re{(

RsAHR−1

x A′βk)�(RsA

HR−1x A′θ

)T

+(RsA

HR−1x AθRs

)�(A′HβkR−1x A′θ

)T},

Fφαk = 2N Re{(

RsAHR−1

x A′αk)�(RsA

HR−1x A′φ

)T

+(RsA

HR−1x AφRs

)�(A′HαkR

−1x A′φ

)T},

Fφβk = 2N Re{(

RsAHR−1

x A′βk)�(RsA

HR−1x A′φ

)T

+(RsA

HR−1x AφRs

)�(A′HβkR

−1x A′φ

)T},

Frαk = 2N Re{(

RsAHR−1

x A′αk)�(RsA

HR−1x A′r

)T

+(RsA

HR−1x ArRs

)�(A′HαkR

−1x A′r

)T},

Frβk = 2N Re{(

RsAHR−1

x A′βk)�(RsA

HR−1x A′r

)T

+(RsA

HR−1x ArRs

)�(A′HβkR

−1x A′r

)T}.

(A.1)

Acknowledgment

The work reported in this paper is supported by theAustralian Research Council through a Discovery ProjectGrant DP 0773234.

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