research article a new wideband mutual coupling...

Research ArticleA New Wideband Mutual Coupling Compensation Method forAdaptive Arrays Based on Element Pattern Reconstruction

Qiulin Huang Feng Wei Lihua Yuan Hongxing Zhou and Xiaowei Shi

Science and Technology on Antenna and Microwave Laboratory Xidian University Xirsquoan 710071 China

Correspondence should be addressed to Qiulin Huang qiulhuangmailxidianeducn

Received 19 November 2013 Revised 10 January 2014 Accepted 13 January 2014 Published 20 February 2014

Academic Editor Sembiam R Rengarajan

Copyright copy 2014 Qiulin Huang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A new mutual coupling compensation method for wideband adaptive arrays is proposed The new method is developed bycombining the element pattern reconstruction method and the system identification method The element pattern reconstructionmethod is valid and effective in the mutual coupling compensation for adaptive arrays such as dipole arrays and microstrip arraysEach entry of the wideband compensation matrix is represented as an analytical expression against frequency The polynomialcoefficients and orders of all entries are obtained via the system identification method The new wideband compensation methodis characterized by the good adaptability of element structures and polarizations owing to the advantages of element patternreconstructionmethod Awidebandmicrostrip array is designed to test the validity and effectiveness of thewideband compensationmethod

1 Introduction

At present wideband antenna arrays are widely used in theadaptive signal processing systems and microstrip antennasare often employed in these systems However the per-formance of adaptive array signal processing is seriouslyinfluenced by the mutual coupling effect of arrays especiallyin the lower frequency band It can be explained via therelationship between the frequency and electric length Overthe past 30 years various mutual coupling compensationmethods have been proposed such as the open circuitvoltage method the receiving mutual impedance methodthe minimum norm method the method proposed by Suet al and the method introduced by Yuan et al [1ndash8] Inmost of themethods the compensationmatrices are obtainedthrough the analysis of current distributions or induced ter-minal voltages on antenna arrays The open circuit scatteringof arrays was not taken into account in the open circuitvoltage method which degrades the compensation effect ofthis method [1] The receiving mutual impedance methodand minimum norm method can provide more accuratemutual coupling compensations owing to the improvementsin processing the scattering effect [2ndash4] The pattern of idealpoint source model was used in the method proposed by

Su et al [5ndash7] For the antenna array composed of dipolesor monopoles H-plane pattern of the element is isotropicwhich is the samewith the ideal point source In this case Sursquosmethod can provide accurate mutual coupling compensationfor the incident signals in the H-plane However methodsproposed in [2ndash7] are more suitable for antenna arrayscomposed of wire elements such as dipoles and monopolesFor arrays composed of planar elements such as microstripantennas it is difficult to find a proper compensation matrixto match the incident signals coming from various directionsby using the compensation methods mentioned above Inthe method proposed by Yuan et al the universal steeringvector was used in DOA estimations so that the receivedvoltages of the array can be used directly to find directionswithout calibration [8] The universal steering vector iscomposed of the embedded element patterns at given direc-tion However the compensation matrix was given throughthe relationship between the universal steering vector andthe conventional steering vector due to the ideal sourcemodel and the universal steering vector was related with theincident angles and polarization of the incident signals whichlimits the applications of this method in adaptive antennaarrays Although the compensation matrix independent ofincident angles was obtained in [9ndash11] the compensation

Hindawi Publishing CorporationInternational Journal of Antennas and PropagationVolume 2014 Article ID 386920 9 pageshttpdxdoiorg1011552014386920

2 International Journal of Antennas and Propagation

matrix also described the relationship between the universalsteering vector and the conventional steering vector It isexpected that applications of ideal source model in the abovecompensation methods cannot guarantee the decoupling ofreceived signals Actually residual mutual coupling effectsstill remain when the ideal source model is employed inthe compensation method since the received signal dueto the isolated antennas is uninfluenced by the mutualcoupling

The joint estimation of direction-of-arrivals (DOAs) andcalibration matrix provided a feasible approach for the mut-ual coupling calibration of microstrip arrays [12ndash14] How-ever joint estimation methods are usually used for uniformlinear arrays (ULAs) and uniform circular arrays (UCAs)The number of mutual coupling parameters can be decreasedfor regular array structures since the calibration matrix ischaracterized by the complex Toeplitz structure which isimportant for solving the joint estimation problems Moreimportantly the time consumption of joint estimations isdifficult to satisfy the real-time requirement inmany practicalapplications

In the literature a wideband mutual coupling compen-sation method for a dipole array was investigated and thesystem identification method was employed to obtain thewideband calibration matrix [15ndash17] It was verified that thesystem identification method was effective in calculating thewideband compensation matrix However the system identi-fication method proposed above was based on the receivingmutual impedancemethod that ismore suitable for the arrayscomposed of wire elements thus limiting the applications ofthis wideband compensation method

Recently the element pattern reconstructionmethod wasproposed by the authors to compensate for the mutual cou-pling effect of adaptive arrays [18 19]Thismethodwas devel-oped on the basis of the relationship between the embeddedelement patterns and isolated element patterns of an array Itcan be used not only in arrays composed of wire elementsbut also in arrays composed of planar elements such asmicrostrip antennas Additionally this method is suitable forlinearly polarized arrays as well as circularly polarized arraysIt is important that the compensation matrix is indepe-ndent of the incident angle and polarization of signals Thegood performance of the method was verified with a dipolearray linearly polarized microstrip array circularly polarizedmicrostrip array and conformal microstrip array Basedon this consideration the element pattern reconstructionmethod has the potential to develop more effective compen-sation methods for the wideband adaptive arrays

In this paper the combination of the element patternreconstruction method and system identification method isproposed for the wideband mutual coupling compensationThe new method has a good adaptability for the elementstructures and polarizations owing to the characteristics ofthe element pattern reconstructionmethodThe newmethodcan be employed in the wideband frequency-hopping modeand for proper wideband signals A wideband microstriparray with five elements is designed to verify the effectivenessof the new wideband compensation method

2 Theory

21 Element Pattern Reconstruction Method It is known thatan antenna pattern corresponds with the magnitude of thereceived signals in the given directions In addition thepattern of an element in the isolated state with other elementsremoved from the array is not influenced by the mutualcoupling while the pattern of an embedded element willbe influenced by mutual coupling The above considerationsprovide us with an approach to compensate for the mutualcoupling of adaptive arrays via the transformation of elementpatternsThe element pattern reconstructionmethod is basedon the fact that when all embedded element patterns aretransformed to coincide with the corresponding isolatedelement patterns in a certain direction the received signalsowing to this direction would be decoupled after beingtransformed in the same way The compensation matrix isobtained through the transformation of the element patternsin the proposed method which is different with existingmethods

Generally the main polarization component and thecross polarization component are seen by an antenna simul-taneously For the linearly polarized antenna the mainpolarization component is often the 120579 or 120593 component ofthe electric field For circularly polarized antennas the mainpolarization component is the electric field of the left handcircular polarization (LHCP) or the right hand circular polar-ization (RHCP) components In a well-designed microstripantenna the main polarization component contributes thedominant part in the total electric field Consequently themain polarization component plays a dominant role in themutual coupling calibrations The main polarization compo-nent is thus considered for the element pattern reconstructionmethod

Consider an antenna arraywith119873 elementswith each ele-ment terminated in a load119885

119871 Two kinds of element patterns

are involved in the element pattern reconstruction methodOne is the embedded element pattern that is calculatedwith other elements terminated with a load Another is theisolated element pattern that is calculatedwith other elementsremoved from the array A relationship between two kindsof patterns is established so as to transform the embeddedelement patterns into the isolated element patterns that is

[[[[

[

119864119894

1(120579 120593)

119864119894

2(120579 120593)

119864119894

119873(120579 120593)

]]]]

]

= C[[[[

[

1198641(120579 120593)

1198642(120579 120593)

119864119873(120579 120593)

]]]]

]

(1)

where 119864119894119899(120579 120593) and 119864

119899(120579 120593) with 119899 = 1 2 119873 represent

the element pattern in the isolated state and in the embeddedstate respectively Matrix C represents the compensationmatrix which can be used to compensate for the mutualcoupling effect of the array It is to be noted that bothmagnitude and phase of electric fields are taken into accountin (1) In order to accurately calculate the compensationmatrix C that can adapt to wide angle range multiple

International Journal of Antennas and Propagation 3

directions should be sampled from the element patternsAssume that119872 directions are chosen one obtains

[[[[[

[

119864119894

1(1205791 1205931) sdot sdot sdot 119864

119894

1(120579119872 120593119872)

119864119894

2(1205791 1205931) sdot sdot sdot 119864

119894

2(120579119872 120593119872)

119864119894

119873(1205791 1205931) sdot sdot sdot 119864

119894

119873(120579119872 120593119872)

]]]]]

]

= C[[[[

[

1198641(1205791 1205931) sdot sdot sdot 119864

1(120579119872 120593119872)

1198642(1205791 1205931) sdot sdot sdot 119864

2(120579119872 120593119872)

119864119873(1205791 1205931) sdot sdot sdot 119864

119873(120579119872 120593119872)

]]]]

]

(2)

Consequently the least square solution that satisfies

min 10038171003817100381710038171003817CEminusE11989410038171003817100381710038171003817

(3)

can be obtained that is

C = E119894E119867 (EE119867)minus1

(4)

where E119894 represents the electric field matrix owing to theisolated elements and E the electric field matrix owing to theembedded elements Superscript 119867 denotes the Hermitiantranspose operation and operator sdot denotes the Frobeniusnorm of amatrix For matrixCwith the dimension of119873times119872the Frobenius norm is written as

C = 119873

sum

119899=1

119872

sum

119898=1

100381610038161003816100381611988811989911989810038161003816100381610038162

12

(5)

where 119888119899119898

is an entry of matrix CA compensation matrix can be calculated via (4) This

solution is valid for the sampled directions It is knownthat slow continuous change of element pattern exists in themain lobe and especially in the 3 dB beamwidth It is thuspossible to obtain a compensation matrix that is valid forthe 3 dB beamwidth or a bit larger angle range In naturethe compensationmatrix is characterized by the symmetricalstructure and will be of complex Toeplitz structure when thearray is a ULA or UCA However limited degrees of freedomand direction samples bring about the nonsymmetry andnon-Toeplitz characters of the compensation matrix

Once the compensation matrix is obtained it can beemployed to compensate for themutual coupling of the arrayAssume that the sample matrix for received signals is X1015840which includes the noise the calibrated sample matrix canbe calculated by the following formula

X = CX1015840 (6)

After being calibrated by the compensation matrix thereceived signals can be imported to adaptive array processingalgorithms In order to verify the performance of the ele-ment pattern reconstruction method many DOA estimationalgorithms can be employed such as the multiple signalclassification (MUSIC) the estimations of signal parameters

via rotational invariance techniques (ESPRIT) themaximumlikelihood (ML) and the subspace fitting (SF) [20ndash23]

For an actual antenna array the received signals influ-enced by the mutual coupling effect can be generated viathe electric field matrix E which acts as the actual directionmatrix Another approach to obtain the actual directionmatrix is to calculate the induced terminal voltage vectorof the array due to the incident plane waves in the samedirections with the incident signals Actually two approachesare equivalent

22 Wideband Mutual Coupling Compensation In somewideband adaptive systems such as the frequency-hoppingsystem mutual coupling compensation is carried out fornarrowband frequencies although the entire operating fre-quency band of the system is of wideband In this case thecompensation matrix at any frequency is needed Howeverit is difficult and unnecessary to store the compensationmatrices for all frequencies owing to the limited memoryspace of the hardware For the wideband system eachentry of the wideband compensation matrix would varyagainst the frequency However it is difficult to obtain ananalytical expression for the mathematical relation betweenthe entries of wideband compensation matrix at differentfrequencies from a pure electromagnetic theory considera-tion [15] From the point of system identification methodapproximate expressions can be utilized to represent theentries of wideband compensation matrix [16] Assume that119888119899119898(119895119896) is an approximate expression of the entry 119888

119899119898(119895119896) one

obtains

119888119899119898(119895119896) =

119860119899119898

0+ 119860119899119898

1(119895119896) + 119860

119899119898

2(119895119896)2+ 119860119899119898

3(119895119896)3+ sdot sdot sdot

1198611198991198980+ 1198611198991198981(119895119896) + 119861119899119898

2(119895119896)2+ 1198611198991198983(119895119896)3+ sdot sdot sdot

(7)

where 119896 = 120596radic12058301205760 denotes the wave number or the pro-pagation constant and the polynomial coefficients in (7) willdetermine the curve corresponding to 119888

119899119898in the frequency

band In order to calculate the polynomial coefficientscomplex-curve fitting method proposed by Levy is an effec-tivemethod that can be employed [17] Different fromwhat isproposed byLevy the variable involved in (7) is 119895119896 rather than119895120596 which can avoid too small polynomial coefficients andprovide us with better computational accuracy of the entriesFor convenience (7) can be written by

119888119899119898(119895119896) = ((119860

119899119898

0minus 119860119899119898

21198962+ 119860119899119898

41198964minus sdot sdot sdot )

+119895119896 (119860119899119898

1minus 119860119899119898

31198962+ 119860119899119898

5) 1198964minus sdot sdot sdot )

times ((119861119899119898

0minus 119861119899119898

21198962+ 119861119899119898


+119895119896 (119861119899119898

1minus 119861119899119898

31198962+ 119861119899119898

51198964minus sdot sdot sdot ))

minus12

(8)

Except for the calculation of polynomial coefficients thepolynomial orders for each entry are also to be determined


Assume that the polynomial orders of the numerator anddenominator are denoted as 119901 and 119902 respectively The calcu-lation of polynomial coefficients

Θ = [119860119899119898

0 119860119899119898

1 119860

119899119898

119901 119861119899119898

0 119861119899119898

1 119861

119899119898

119902] (9)

can be expressed by the following optimization problem [16]

Θ = argminΘ

119863 (Θ)

119863 (Θ) =

119873

sum

119894=1

1003816100381610038161003816119888119899119898 (119895120596119894radic12058301205760) minus 119888119899119898 (119895120596119894radic12058301205760)10038161003816100381610038162

(10)

where 119863(Θ) is calculated in all sampled frequencies ofthe bandwidth In order to obtain an accurate widebandcompensation matrix the compensation matrices at suffi-cient numbers of frequencies need to be calculated by theelement pattern reconstruction method The complex-curvefitting method is thus employed to fit each entry of thewideband compensation matrix Entries of the widebandcompensation matrix vary slowly with the frequency There-fore it is convenient to try the polynomial orders with thebeginning of the lowest order and then gradually increasethe orders until the fitting error gets negligibly small Forsimplicity the polynomial orders of the numerator anddenominator are set equal In this case the above optimiza-tion problem can be carried out by the procedure shown inFigure 1

As mentioned above the compensation matrix obtainedby the element pattern reconstruction method is not char-acterized rigorously by the symmetry and complex Toeplitzstructure Therefore all entries of the compensation matrixare involved in the complex-curve fitting in order to getmore accurate compensation effect in the wideband systemsOnce the above process is completed the compensationmatrix at any frequency included in the frequency bandcan be calculated via (8) with a little effort and only a fewpolynomial coefficients that describe the curves need to bestored For the wideband frequency-hopping adaptive arrayit is convenient to obtain the compensation matrix for therequired frequencies via (8)

The proposed wideband compensation method can bealso employed in the DOA estimation of wideband signals[20 24] In this case DOA estimation method in thefrequency domain can be used for wideband signals Thewideband signals are decomposed into multiple narrowbandcomponents which are calibrated at the corresponding sub-bands using the compensation matrices pertaining to thesesubbands The DOA estimation is thus carried out by usingthe MUSIC algorithm at each subband [20] Finally thespatial spectrum for the wideband signals is obtained byaveraging all the subband spatial spectra over the wholebandwidth In general the noise involved in MUSIC algo-rithms is white Gaussian noise (AWGN) and would becomecolored after being multiplied by the compensation matrixHowever the compensation matrix is of diagonal dominanceand full rank which would result in limited influence on theperformance of MUSIC algorithms

No

Yes

Yes

Establish the array model and set the frequency band

Calculate the compensation matrix at all sampled frequencies in the frequency band

Select an entry to be fitted

Set the polynomial order

Carry out the complex-curve fitting for the selected entry

Fitting error smaller than a given threshold

All entries are involved

End

Increase the order

No

Figure 1 Flowchart for the calculation of wideband compensationmatrix

It should be noted that the new wideband compensationmethod is characterized by the good adaptability of elementstructures and polarizations owing to the performance of theelement pattern reconstruction method It can be used notonly for the dipole or monopole arrays but also for properwideband adaptive arrayswithwidebandmicrostrip elementsor other elements with wideband characteristics

3 Numerical Examples

In this section a wideband microstrip array is designedto verify the wideband compensation method proposed inthis paper As is shown in Figure 2 the array is composedof five wideband microstrip elements The VSWR curve ofthe element is shown in Figure 3 The frequency band ofthe isolated element is larger than 500MHz with VSWR le

15 For the microstrip array however the frequency bandunder consideration is 300MHz with the center frequencyof 265GHz The 3 dB beamwidth at 265GHz of the elementis about 90∘ The element spacing of the array is 54mmthat is 0477120582

0 where 120582

0is the wavelength at 265GHz EM

simulation tool HFSS Version 13 is utilized to calculate the


Coaxial probe

y

Hh

L

d

a

x

Air

Patch

Ground

Lg

(a)

x

y

z

120579

1 2 3 4 5

(b)

Figure 2 Microstrip element profile (a) and array structures (b)119871119892= 90mm 119889 = 8mm 119886 = 12mm 119871 = 50mm119867 = 15mm and

ℎ = 12mm

5

4

3

2

1

22 24 26 28 30

Frequency (GHz)

VSW

R

VSWR

Figure 3 VSWR curve of the element in the isolated state

electric fields of the elements and the array In the followingsimulations of DOA estimations all incident signals areuncorrelated and impinge upon the array from 119909119900119911 planeMUSIC algorithm is employed SNR for all incident signalsis 20dB and the data sample is 3000

20

15

10

5

0minus100 minus50 0 50 100

Angle (deg)

Mag

nitu

de

Isolated stateEmbedded stateReconstructed

(a)

minus100 minus50 0 50 100

Angle (deg)


100

0

minus100

Phas

e (de

g)

(b)

Figure 4 Pattern reconstruction of 119864120593for element number 3 at

265GHz (a) magnitude and (b) phase

31 Mutual Coupling Compensation Using Element PatternReconstruction Method The element patterns are recon-structed by the compensation matrix obtained via the ele-ment pattern reconstruction method For the linearly polar-ized microstrip array the main polarization component isthe 120593 component of the electric field in 119909119900119911 plane which isplane of interest Therefore the 120593 component of the electricfield is employed to obtain the compensation matrix Inorder to reconstruct the element patterns in a larger anglerange 121 directions in the angle range of [minus60∘ 60∘] in 119909119900119911plane are sampled to calculate the compensation matrix Asan example the pattern reconstruction for element number3 at 265GHz is given in Figure 4 It can be seen that



Angle (deg)

40

35

30

25

20

15

10

5

0

Mag

nitu

de (d

B)

EPRMOpen circuitSu998400s method

Figure 5 Spatial spectrum of the MUSIC algorithm for threeincident signals

the magnitude and phase of the reconstructed pattern areconsistent with those of the isolated element pattern in theangle range of [minus50∘ 50∘] The angle range of consistence isbeyond the 3 dBbeamwidth Itmeans that the incident signalscoming from this angle rangewill be suitably calibrated by thecompensation matrix

DOA estimations for three incident signals are employedto further verify the effectiveness of the element patternreconstruction method The array operates at 265GHzand the compensation matrix obtained above is utilized tocalibrate the received signals Three uncorrelated signals areincident from 120579 = minus50

∘ minus20∘ and 10∘ in 119909119900119911 plane respe-

ctively As shown in Figure 5 the element pattern reconstruc-tion method can provide more accurate DOA estimationsin contrast with the open circuit voltage method and Sursquosmethod and DOA estimations for all signals are resolvedusing the element pattern reconstruction method (EPRM) inFigure 5

32 Calculation of the Wideband Compensation Matrix Inorder to implement the optimization procedure to obtain thewideband compensation matrix 31 compensation matriceswith the frequency interval of 10MHz are calculated usingthe element pattern reconstruction method A total of 121directions in the angle range of [minus60∘ 60∘] 119909119900119911 plane aresampled to calculate each compensation matrix Complex-curve fitting under given polynomial orders is carried out foreach entry of the calibration matrix Following the process asshown in Figure 1 proper polynomial orders and coefficientsare obtained The polynomial orders for all entries are listedin Table 1 Analytical expressions for all entries are given inthe Appendix of this paperThemagnitudes and phases of theentries including 119888

11 11988821 11988831 11988841 and 119888

51are shown in Figures

6(a) and 6(b) respectively It can be seen that the magnitudeand phase of each entry obtained through complex-curve

10

09

08

02

01

0025 26 27 28

Frequency (GHz)

Mag

nitu

de

c11

c21

c51c41c31

FittedDirectly calculated

(a)

150

100

50

0

minus50

minus100

minus150

Phas

e (de

g)

25 26 27 28

Frequency (GHz)


c11

c21

c51

c41

c31

(b)

Figure 6 Complex-curve fitting for 11988811 11988821 11988831 11988841 and 119888

51 (a)

magnitude and (b) phase

Table 1 Polynomial orders for all entries of the compensationmatrix

Polynomial order Entries of calibration matrix3 119888

11 11988823 11988831 11988832 11988834 11988841 11988843 11988844 11988855

4 11988812 11988813 11988824 11988833 11988835 11988853 11988854

5 11988842 11988851

6 11988821 11988822 11988852

7 11988814 11988815 11988825 11988845

fitting agree well in the frequency band with those dueto direct calculation via the element pattern reconstructionmethod respectively


45

40

35

30

25

20

15

10

5

0

Mag

nitu

de (d

B)


Angle (deg)


(a)

40

35

30

25

20

15

10

5

0

Mag

nitu

de (d

B)


Angle (deg)


(b)

40

35

30

25

20

15

10

5

0

Mag

nitu

de (d

B)


Angle (deg)


(c)

Figure 7 Comparison of DOA estimations due to the directly calculated compensation matrices via element pattern reconstruction methodand the fitted compensation matrices via complex-curve fitting at (a) 25 GHz (b) 265GHz and (c) 28 GHz

For verifying the impact of the fitting error on the mutualcoupling compensation three examples of DOA estimationsat 25 GHz 265GHz and 28GHz obtained from the directlycalculated compensation matrices via the element patternreconstruction method and the fitted compensation matricesvia the complex-curve fitting are shown in Figures 7(a) 7(b)and 7(c) In the DOA estimations three uncorrelated signalsare incident from 120579 = minus10

∘ 20∘ and 50∘ respectively

From the spatial spectra of MUSIC algorithm at threefrequencies it can be seen that the accuracy of DOA esti-mations is virtually unattacked by the fitting error It canbe inferred that the complex-curve fitting is robust in thewhole frequency band which would guarantee the real-timecalculation of the compensation matrix at any frequency inthe frequency band

33 DOAEstimations of theWideband Signals Thewidebandsignal under consideration has a bandwidth of 100MHz cove-ring 25sim26GHz DOA estimations of wideband signals arecarried out in the narrowband way as described in Section 2Through the discrete Fourier transform (DFT) the receivedsignal on each element is decomposed into 10 subbandsignals The subband signal vectors are calibrated by thecompensationmatrices owing to the center frequencies of thecorresponding subbands The DOA estimations at each sub-band are then implemented via MUSIC algorithm Finallythe average spatial spectrum on the bandwidth provides theDOA estimations of wideband signals

Spatial spectra of MUSIC algorithm for three widebandsignals at all subbands are given in Figure 8 where the com-pensation matrices via the complex-curve fitting are


262

266

270Freq

uenc

y (G

Hz)

806040200minus20minus40minus60minus80Angle (deg)

4035302520151050

Mag

nitu

de (d

B)

Figure 8 Spatial spectra of the MUSIC algorithm for three wide-band signals at all subbands The fitted compensation matrices inthe frequency band are employed

40

35

30

25

20

15

10

5

0

Mag

nitu

de (d

B)


Angle (deg)

New method

Open circuitSu998400s method

Figure 9 Average spatial spectra due to the new method the opencircuit voltage method and Sursquos method

employed Three uncorrelated wideband signals are incidentfrom 120579 = 0

∘ minus20∘ and minus50

∘ respectively Consistenceof DOA estimations at various subbands can be seen inFigure 8 which is further indicated by the average spatialspectra as shown in Figure 9 The incident angles of threewideband signals can be found accurately in the averagespatial spectra using the proposed wideband compensationmethod For comparison the open circuit voltage methodand Sursquos method are employed All compensation matricesobtained with the open circuit voltage method and Sursquosmethod at sampled frequencies are calculated and directlyused in the DOA estimations respectively It can be seen thatlarge errors of DOA estimations exist for the open circuitvoltage method and Sursquos method Estimations bias due tothree methods can be found in Table 2

Table 2 Absolute DOA bias for the wideband signals in three cases

Incident signal 120579 (∘) Absolute DOA bias (∘)New method Open circuit Sursquos method

Signal 1 minus50 07 78 30Signal 2 minus20 03 72 27Signal 3 0 01 34 16

4 Conclusion

In this paper a new wideband compensation method foradaptive arrays is proposed The element pattern reconstruc-tion method and the system identification method are com-bined to obtain themutual coupling compensationmatrix forthe wideband adaptive array Owing to the performance ofthe element pattern reconstruction method the new wide-band compensation method is characterized by the goodadaptability of element structures and polarizations Thecompensation matrix at any frequency in the frequency bandcan be obtained conveniently through the analytical expre-ssions of all entries which benefitswidebandmutual couplingcompensation The effectiveness of the proposed widebandmutual coupling compensation method is verified by DOAestimations using a wideband microstrip array

Conflict of Interests

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

Acknowledgments

This work was supported in part by the FundamentalResearch Funds for the Central Universities (K5051202012)and the National Natural Science Foundation of China(61201019)

References

[1] 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

[2] 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

[3] H T Hui ldquoA new definition of mutual impedance for appli-cation in dipole receiving antenna arraysrdquo IEEE Antennas andWireless Propagation Letters vol 3 no 1 pp 364ndash367 2004

[4] C K Edwin Lau R S Adve and T K Sarkar ldquoMinimum normmutual coupling compensation with applications in directionof arrival estimationrdquo IEEE Transactions on Antennas andPropagation vol 52 no 8 pp 2034ndash2041 2004

[5] T Su KDandekar andH Ling ldquoSimulation ofmutual couplingeffect in circular arrays for directionmdashfinding applicationrdquoMicrowave andOptical Technology Letters vol 26 no 5 pp 331ndash336 2000

[6] K R Dandekar H Ling and G Xu ldquoExperimental study ofmutual coupling compensation in smart antenna applicationsrdquo


IEEE Transactions onWireless Communications vol 1 no 3 pp480ndash487 2002

[7] R J Weber and Y Huang ldquoAn automatic calibration system forsmart antenna arraysrdquo in Proceedings of the IEEE Antennas andPropagation Society International Symposium (APSURSI rsquo09)pp 1ndash4 Charleston SC USA June 2009

[8] 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

[9] S Henault S K Podilchak S M Mikki and Y M M Antar ldquoAmethodolgy for mutual coupling estimation and compensationin antennasrdquo IEEE Transactions on Antennas and Propagationvol 61 no 3 pp 1119ndash1131 2013

[10] S Henault and Y M M Antar ldquoLimitations of online calibra-tion methods in antenna arraysrdquo in Proceedings of the IEEEAntennas and Propagation Society International Symposium(APSURSI rsquo10) pp 1ndash4 Toronto Canada July 2010

[11] S Henault and Y M M Antar ldquoAccurate evaluation of mutualcoupling for array calibrationrdquo in 2009 Computational Electro-magnetics International Workshop (CEM rsquo09) pp 34ndash37 IzmirTurkey July 2009

[12] B Friedlander and A J Weiss ldquoDirection finding in the pre-sence of mutual couplingrdquo IEEE Transactions on Antennas andPropagation pp 273ndash284 1991

[13] M Lin andLYang ldquoBlind calibration andDOAestimationwithuniform circular arrays in the presence of mutual couplingrdquoIEEE Antennas and Wireless Propagation Letters vol 5 no 1pp 315ndash318 2006

[14] F Sellone and A Serra ldquoAn iterative algorithm for the compen-sation of toeplitzmutual coupling inUniform and linear arraysrdquoin Proceedings of the 12th Digital Signal ProcessingWorkshop pp438ndash443 Teton National Park Wyo USA September 2006

[15] B H Wang and H T Hui ldquoWideband mutual coupling com-pensation for receiving antenna arrays using the system iden-tification methodrdquo IET Microwaves Antennas and Propagationvol 5 no 2 pp 184ndash191 2011

[16] R Pintelon P Guillaume Y Rolain J Schoukens and H Vanhamme ldquoParametric identification of transfer functions in thefrequency domainmdasha surveyrdquo IEEE Transactions on AutomaticControl vol 39 no 11 pp 2245ndash2260 1994

[17] E C Levy ldquoComplex-curve fittingrdquo IRE Transactions on Auto-matic Control pp 37ndash43 1959

[18] Q Huang H Zhou J Bao and X Shi ldquoAccurate DOA esti-mations using microstrip adaptive arrays in the presence ofmutual coupling effectrdquo International Journal of Antennas andPropagation vol 2013 Article ID 919545 8 pages 2013

[19] Q Huang H Zhou J Bao and X Shi ldquoAccurate calibrationof mutual coupling for conformal antenna arraysrdquo ElectronicLetters vol 49 no 23 pp 1418ndash1420 2013

[20] R O Schmidt ldquoMultiple emitter location and signal parameterestimationrdquo IEEE Transactions on Antennas and Propagationvol AP-34 no 3 pp 276ndash280 1986

[21] R Roy and T Kailath ldquoESPRIT-estimation of signal parametersvia rotational invariance techniquesrdquo IEEE Transactions onAcoustics Speech and Signal Processing vol 37 no 7 pp 984ndash995 1989

[22] A G Jaffer ldquoMaximum likelihood direction finding of stochas-tic sources a separable solutionrdquo in Proceedings of the Interna-tional Conference on Acoustics Speech and Signal Processingvol 5 pp 2893ndash2896 New York NY USA 1988

[23] J A Cadzow ldquoA high resolution direction-of-arrival algorithmfor narrow-band coherent and incoherent sourcesrdquo IEEE Trans-actions on Acoustics Speech and Signal Processing vol 36 no 7pp 965ndash979 1988

[24] MWax T Shan and T Kailath ldquoSpatio-temporal spectral ana-lysis by eigenstructure methodsrdquo IEEE Transactions on Acous-tics Speech and Signal Processing vol 32 no 4 pp 817ndash8271984

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



matrix also described the relationship between the universalsteering vector and the conventional steering vector It isexpected that applications of ideal source model in the abovecompensation methods cannot guarantee the decoupling ofreceived signals Actually residual mutual coupling effectsstill remain when the ideal source model is employed inthe compensation method since the received signal dueto the isolated antennas is uninfluenced by the mutualcoupling

The joint estimation of direction-of-arrivals (DOAs) andcalibration matrix provided a feasible approach for the mut-ual coupling calibration of microstrip arrays [12ndash14] How-ever joint estimation methods are usually used for uniformlinear arrays (ULAs) and uniform circular arrays (UCAs)The number of mutual coupling parameters can be decreasedfor regular array structures since the calibration matrix ischaracterized by the complex Toeplitz structure which isimportant for solving the joint estimation problems Moreimportantly the time consumption of joint estimations isdifficult to satisfy the real-time requirement inmany practicalapplications

In the literature a wideband mutual coupling compen-sation method for a dipole array was investigated and thesystem identification method was employed to obtain thewideband calibration matrix [15ndash17] It was verified that thesystem identification method was effective in calculating thewideband compensation matrix However the system identi-fication method proposed above was based on the receivingmutual impedancemethod that ismore suitable for the arrayscomposed of wire elements thus limiting the applications ofthis wideband compensation method

Recently the element pattern reconstructionmethod wasproposed by the authors to compensate for the mutual cou-pling effect of adaptive arrays [18 19]Thismethodwas devel-oped on the basis of the relationship between the embeddedelement patterns and isolated element patterns of an array Itcan be used not only in arrays composed of wire elementsbut also in arrays composed of planar elements such asmicrostrip antennas Additionally this method is suitable forlinearly polarized arrays as well as circularly polarized arraysIt is important that the compensation matrix is indepe-ndent of the incident angle and polarization of signals Thegood performance of the method was verified with a dipolearray linearly polarized microstrip array circularly polarizedmicrostrip array and conformal microstrip array Basedon this consideration the element pattern reconstructionmethod has the potential to develop more effective compen-sation methods for the wideband adaptive arrays

In this paper the combination of the element patternreconstruction method and system identification method isproposed for the wideband mutual coupling compensationThe new method has a good adaptability for the elementstructures and polarizations owing to the characteristics ofthe element pattern reconstructionmethodThe newmethodcan be employed in the wideband frequency-hopping modeand for proper wideband signals A wideband microstriparray with five elements is designed to verify the effectivenessof the new wideband compensation method

2 Theory

21 Element Pattern Reconstruction Method It is known thatan antenna pattern corresponds with the magnitude of thereceived signals in the given directions In addition thepattern of an element in the isolated state with other elementsremoved from the array is not influenced by the mutualcoupling while the pattern of an embedded element willbe influenced by mutual coupling The above considerationsprovide us with an approach to compensate for the mutualcoupling of adaptive arrays via the transformation of elementpatternsThe element pattern reconstructionmethod is basedon the fact that when all embedded element patterns aretransformed to coincide with the corresponding isolatedelement patterns in a certain direction the received signalsowing to this direction would be decoupled after beingtransformed in the same way The compensation matrix isobtained through the transformation of the element patternsin the proposed method which is different with existingmethods

Generally the main polarization component and thecross polarization component are seen by an antenna simul-taneously For the linearly polarized antenna the mainpolarization component is often the 120579 or 120593 component ofthe electric field For circularly polarized antennas the mainpolarization component is the electric field of the left handcircular polarization (LHCP) or the right hand circular polar-ization (RHCP) components In a well-designed microstripantenna the main polarization component contributes thedominant part in the total electric field Consequently themain polarization component plays a dominant role in themutual coupling calibrations The main polarization compo-nent is thus considered for the element pattern reconstructionmethod

Consider an antenna arraywith119873 elementswith each ele-ment terminated in a load119885

119871 Two kinds of element patterns

are involved in the element pattern reconstruction methodOne is the embedded element pattern that is calculatedwith other elements terminated with a load Another is theisolated element pattern that is calculatedwith other elementsremoved from the array A relationship between two kindsof patterns is established so as to transform the embeddedelement patterns into the isolated element patterns that is

[[[[

[

119864119894

1(120579 120593)

119864119894

2(120579 120593)

119864119894

119873(120579 120593)

]]]]

]

= C[[[[

[

1198641(120579 120593)

1198642(120579 120593)

119864119873(120579 120593)

]]]]

]

(1)

where 119864119894119899(120579 120593) and 119864

119899(120579 120593) with 119899 = 1 2 119873 represent

the element pattern in the isolated state and in the embeddedstate respectively Matrix C represents the compensationmatrix which can be used to compensate for the mutualcoupling effect of the array It is to be noted that bothmagnitude and phase of electric fields are taken into accountin (1) In order to accurately calculate the compensationmatrix C that can adapt to wide angle range multiple



[[[[[

[

119864119894

1(1205791 1205931) sdot sdot sdot 119864

119894

1(120579119872 120593119872)

119864119894

2(1205791 1205931) sdot sdot sdot 119864

119894

2(120579119872 120593119872)

119864119894

119873(1205791 1205931) sdot sdot sdot 119864

119894

119873(120579119872 120593119872)

]]]]]

]

= C[[[[

[

1198641(1205791 1205931) sdot sdot sdot 119864

1(120579119872 120593119872)

1198642(1205791 1205931) sdot sdot sdot 119864

2(120579119872 120593119872)

119864119873(1205791 1205931) sdot sdot sdot 119864

119873(120579119872 120593119872)

]]]]

]

(2)


min 10038171003817100381710038171003817CEminusE11989410038171003817100381710038171003817

(3)


C = E119894E119867 (EE119867)minus1

(4)


C = 119873

sum

119899=1

119872

sum

119898=1

100381610038161003816100381611988811989911989810038161003816100381610038162

12

(5)

where 119888119899119898




X = CX1015840 (6)





119899119898(119895119896) one

obtains

119888119899119898(119895119896) =

119860119899119898

0+ 119860119899119898

1(119895119896) + 119860

119899119898

2(119895119896)2+ 119860119899119898

3(119895119896)3+ sdot sdot sdot

1198611198991198980+ 1198611198991198981(119895119896) + 119861119899119898

2(119895119896)2+ 1198611198991198983(119895119896)3+ sdot sdot sdot

(7)




119888119899119898(119895119896) = ((119860

119899119898

0minus 119860119899119898

21198962+ 119860119899119898


+119895119896 (119860119899119898

1minus 119860119899119898

31198962+ 119860119899119898


times ((119861119899119898

0minus 119861119899119898

21198962+ 119861119899119898


+119895119896 (119861119899119898

1minus 119861119899119898

31198962+ 119861119899119898


minus12

(8)




Θ = [119860119899119898

0 119860119899119898

1 119860

119899119898

119901 119861119899119898

0 119861119899119898

1 119861

119899119898

119902] (9)


Θ = argminΘ

119863 (Θ)

119863 (Θ) =

119873

sum

119894=1


(10)




No

Yes

Yes








End

Increase the order

No






0 where 120582




Coaxial probe

y

Hh

L

d

a

x

Air

Patch

Ground

Lg

(a)

x

y

z

120579

1 2 3 4 5

(b)


ℎ = 12mm

5

4

3

2

1

22 24 26 28 30

Frequency (GHz)

VSW

R

VSWR



20

15

10

5

0minus100 minus50 0 50 100

Angle (deg)

Mag

nitu

de


(a)


Angle (deg)


100

0

minus100

Phas

e (de

g)

(b)






Angle (deg)

40

35

30

25

20

15

10

5

0

Mag

nitu

de (d

B)








11 11988821 11988831 11988841 and 119888



10

09

08

02

01

0025 26 27 28

Frequency (GHz)

Mag

nitu

de

c11

c21

c51c41c31


(a)

150

100

50

0

minus50

minus100

minus150

Phas

e (de

g)

25 26 27 28

Frequency (GHz)


c11

c21

c51

c41

c31

(b)


51 (a)




11 11988823 11988831 11988832 11988834 11988841 11988843 11988844 11988855

4 11988812 11988813 11988824 11988833 11988835 11988853 11988854

5 11988842 11988851

6 11988821 11988822 11988852

7 11988814 11988815 11988825 11988845



45

40

35

30

25

20

15

10

5

0

Mag

nitu

de (d

B)


Angle (deg)


(a)

40

35

30

25

20

15

10

5

0

Mag

nitu

de (d

B)


Angle (deg)


(b)

40

35

30

25

20

15

10

5

0

Mag

nitu

de (d

B)


Angle (deg)


(c)








262

266

270Freq

uenc

y (G

Hz)


4035302520151050

Mag

nitu

de (d

B)


40

35

30

25

20

15

10

5

0

Mag

nitu

de (d

B)


Angle (deg)

New method









4 Conclusion




Acknowledgments


References





























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











[[[[[

[

119864119894

1(1205791 1205931) sdot sdot sdot 119864

119894

1(120579119872 120593119872)

119864119894

2(1205791 1205931) sdot sdot sdot 119864

119894

2(120579119872 120593119872)

119864119894

119873(1205791 1205931) sdot sdot sdot 119864

119894

119873(120579119872 120593119872)

]]]]]

]

= C[[[[

[

1198641(1205791 1205931) sdot sdot sdot 119864

1(120579119872 120593119872)

1198642(1205791 1205931) sdot sdot sdot 119864

2(120579119872 120593119872)

119864119873(1205791 1205931) sdot sdot sdot 119864

119873(120579119872 120593119872)

]]]]

]

(2)


min 10038171003817100381710038171003817CEminusE11989410038171003817100381710038171003817

(3)


C = E119894E119867 (EE119867)minus1

(4)


C = 119873

sum

119899=1

119872

sum

119898=1

100381610038161003816100381611988811989911989810038161003816100381610038162

12

(5)

where 119888119899119898




X = CX1015840 (6)





119899119898(119895119896) one

obtains

119888119899119898(119895119896) =

119860119899119898

0+ 119860119899119898

1(119895119896) + 119860

119899119898

2(119895119896)2+ 119860119899119898

3(119895119896)3+ sdot sdot sdot

1198611198991198980+ 1198611198991198981(119895119896) + 119861119899119898

2(119895119896)2+ 1198611198991198983(119895119896)3+ sdot sdot sdot

(7)




119888119899119898(119895119896) = ((119860

119899119898

0minus 119860119899119898

21198962+ 119860119899119898


+119895119896 (119860119899119898

1minus 119860119899119898

31198962+ 119860119899119898


times ((119861119899119898

0minus 119861119899119898

21198962+ 119861119899119898


+119895119896 (119861119899119898

1minus 119861119899119898

31198962+ 119861119899119898


minus12

(8)




Θ = [119860119899119898

0 119860119899119898

1 119860

119899119898

119901 119861119899119898

0 119861119899119898

1 119861

119899119898

119902] (9)


Θ = argminΘ

119863 (Θ)

119863 (Θ) =

119873

sum

119894=1


(10)




No

Yes

Yes








End

Increase the order

No






0 where 120582




Coaxial probe

y

Hh

L

d

a

x

Air

Patch

Ground

Lg

(a)

x

y

z

120579

1 2 3 4 5

(b)


ℎ = 12mm

5

4

3

2

1

22 24 26 28 30

Frequency (GHz)

VSW

R

VSWR



20

15

10

5

0minus100 minus50 0 50 100

Angle (deg)

Mag

nitu

de


(a)


Angle (deg)


100

0

minus100

Phas

e (de

g)

(b)






Angle (deg)

40

35

30

25

20

15

10

5

0

Mag

nitu

de (d

B)








11 11988821 11988831 11988841 and 119888



10

09

08

02

01

0025 26 27 28

Frequency (GHz)

Mag

nitu

de

c11

c21

c51c41c31


(a)

150

100

50

0

minus50

minus100

minus150

Phas

e (de

g)

25 26 27 28

Frequency (GHz)


c11

c21

c51

c41

c31

(b)


51 (a)




11 11988823 11988831 11988832 11988834 11988841 11988843 11988844 11988855

4 11988812 11988813 11988824 11988833 11988835 11988853 11988854

5 11988842 11988851

6 11988821 11988822 11988852

7 11988814 11988815 11988825 11988845



45

40

35

30

25

20

15

10

5

0

Mag

nitu

de (d

B)


Angle (deg)


(a)

40

35

30

25

20

15

10

5

0

Mag

nitu

de (d

B)


Angle (deg)


(b)

40

35

30

25

20

15

10

5

0

Mag

nitu

de (d

B)


Angle (deg)


(c)








262

266

270Freq

uenc

y (G

Hz)


4035302520151050

Mag

nitu

de (d

B)


40

35

30

25

20

15

10

5

0

Mag

nitu

de (d

B)


Angle (deg)

New method









4 Conclusion




Acknowledgments


References





























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











Θ = [119860119899119898

0 119860119899119898

1 119860

119899119898

119901 119861119899119898

0 119861119899119898

1 119861

119899119898

119902] (9)


Θ = argminΘ

119863 (Θ)

119863 (Θ) =

119873

sum

119894=1


(10)




No

Yes

Yes








End

Increase the order

No






0 where 120582




Coaxial probe

y

Hh

L

d

a

x

Air

Patch

Ground

Lg

(a)

x

y

z

120579

1 2 3 4 5

(b)


ℎ = 12mm

5

4

3

2

1

22 24 26 28 30

Frequency (GHz)

VSW

R

VSWR



20

15

10

5

0minus100 minus50 0 50 100

Angle (deg)

Mag

nitu

de


(a)


Angle (deg)


100

0

minus100

Phas

e (de

g)

(b)






Angle (deg)

40

35

30

25

20

15

10

5

0

Mag

nitu

de (d

B)








11 11988821 11988831 11988841 and 119888



10

09

08

02

01

0025 26 27 28

Frequency (GHz)

Mag

nitu

de

c11

c21

c51c41c31


(a)

150

100

50

0

minus50

minus100

minus150

Phas

e (de

g)

25 26 27 28

Frequency (GHz)


c11

c21

c51

c41

c31

(b)


51 (a)




11 11988823 11988831 11988832 11988834 11988841 11988843 11988844 11988855

4 11988812 11988813 11988824 11988833 11988835 11988853 11988854

5 11988842 11988851

6 11988821 11988822 11988852

7 11988814 11988815 11988825 11988845



45

40

35

30

25

20

15

10

5

0

Mag

nitu

de (d

B)


Angle (deg)


(a)

40

35

30

25

20

15

10

5

0

Mag

nitu

de (d

B)


Angle (deg)


(b)

40

35

30

25

20

15

10

5

0

Mag

nitu

de (d

B)


Angle (deg)


(c)








262

266

270Freq

uenc

y (G

Hz)


4035302520151050

Mag

nitu

de (d

B)


40

35

30

25

20

15

10

5

0

Mag

nitu

de (d

B)


Angle (deg)

New method









4 Conclusion




Acknowledgments


References





























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Coaxial probe

y

Hh

L

d

a

x

Air

Patch

Ground

Lg

(a)

x

y

z

120579

1 2 3 4 5

(b)


ℎ = 12mm

5

4

3

2

1

22 24 26 28 30

Frequency (GHz)

VSW

R

VSWR



20

15

10

5

0minus100 minus50 0 50 100

Angle (deg)

Mag

nitu

de


(a)


Angle (deg)


100

0

minus100

Phas

e (de

g)

(b)






Angle (deg)

40

35

30

25

20

15

10

5

0

Mag

nitu

de (d

B)








11 11988821 11988831 11988841 and 119888



10

09

08

02

01

0025 26 27 28

Frequency (GHz)

Mag

nitu

de

c11

c21

c51c41c31


(a)

150

100

50

0

minus50

minus100

minus150

Phas

e (de

g)

25 26 27 28

Frequency (GHz)


c11

c21

c51

c41

c31

(b)


51 (a)




11 11988823 11988831 11988832 11988834 11988841 11988843 11988844 11988855

4 11988812 11988813 11988824 11988833 11988835 11988853 11988854

5 11988842 11988851

6 11988821 11988822 11988852

7 11988814 11988815 11988825 11988845



45

40

35

30

25

20

15

10

5

0

Mag

nitu

de (d

B)


Angle (deg)


(a)

40

35

30

25

20

15

10

5

0

Mag

nitu

de (d

B)


Angle (deg)


(b)

40

35

30

25

20

15

10

5

0

Mag

nitu

de (d

B)


Angle (deg)


(c)








262

266

270Freq

uenc

y (G

Hz)


4035302520151050

Mag

nitu

de (d

B)


40

35

30

25

20

15

10

5

0

Mag

nitu

de (d

B)


Angle (deg)

New method









4 Conclusion




Acknowledgments


References





























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











Angle (deg)

40

35

30

25

20

15

10

5

0

Mag

nitu

de (d

B)








11 11988821 11988831 11988841 and 119888



10

09

08

02

01

0025 26 27 28

Frequency (GHz)

Mag

nitu

de

c11

c21

c51c41c31


(a)

150

100

50

0

minus50

minus100

minus150

Phas

e (de

g)

25 26 27 28

Frequency (GHz)


c11

c21

c51

c41

c31

(b)


51 (a)




11 11988823 11988831 11988832 11988834 11988841 11988843 11988844 11988855

4 11988812 11988813 11988824 11988833 11988835 11988853 11988854

5 11988842 11988851

6 11988821 11988822 11988852

7 11988814 11988815 11988825 11988845



45

40

35

30

25

20

15

10

5

0

Mag

nitu

de (d

B)


Angle (deg)


(a)

40

35

30

25

20

15

10

5

0

Mag

nitu

de (d

B)


Angle (deg)


(b)

40

35

30

25

20

15

10

5

0

Mag

nitu

de (d

B)


Angle (deg)


(c)








262

266

270Freq

uenc

y (G

Hz)


4035302520151050

Mag

nitu

de (d

B)


40

35

30

25

20

15

10

5

0

Mag

nitu

de (d

B)


Angle (deg)

New method









4 Conclusion




Acknowledgments


References





























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










45

40

35

30

25

20

15

10

5

0

Mag

nitu

de (d

B)


Angle (deg)


(a)

40

35

30

25

20

15

10

5

0

Mag

nitu

de (d

B)


Angle (deg)


(b)

40

35

30

25

20

15

10

5

0

Mag

nitu

de (d

B)


Angle (deg)


(c)








262

266

270Freq

uenc

y (G

Hz)


4035302520151050

Mag

nitu

de (d

B)


40

35

30

25

20

15

10

5

0

Mag

nitu

de (d

B)


Angle (deg)

New method









4 Conclusion




Acknowledgments


References





























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










262

266

270Freq

uenc

y (G

Hz)


4035302520151050

Mag

nitu

de (d

B)


40

35

30

25

20

15

10

5

0

Mag

nitu

de (d

B)


Angle (deg)

New method









4 Conclusion




Acknowledgments


References





























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









research article a new wideband mutual coupling...

Documents