Joint calibration algorithm for gain-phase and mutual coupling errors in uniform linear array

Li Weixing,Lin Jianzhi,Zhang Yue,Chen Zengping

Science and Technology on Automatic Target Recognition Laboratory,National University of Defense Technology,Changsha 410073,China

Joint calibration algorithm for gain-phase and mutual coupling errors in uniform linear array

Li Weixing,Lin Jianzhi,Zhang Yue,Chen Zengping*

Science and Technology on Automatic Target Recognition Laboratory,National University of Defense Technology,Changsha 410073,China

The effect of gain-phase perturbations and mutual coupling significantly degrades the performance of digital array radar(DAR).This paper investigates array calibration problems in the scenario where the true locations of auxiliary sources deviate from nominal values but the angle intervals are known.A practical algorithm is proposed to jointly calibrate gain-phase errors and mutual coupling errors.Firstly,a simplified model of the distortion matrix is developed based on its special structure in uniform linear array(ULA).Then the model is employed to derive the precise locations of the auxiliary sources by one-dimension search.Finally,the least-squares estimation of the distortion matrix is obtained.The algorithm has the potential of achieving considerable improvement in calibration accuracy due to the reduction of unknown parameters.In addition,the algorithm is feasible for practical applications,since it requires only one auxiliary source with the help of rotation plat forms.Simulation results demonstrate the validity,robustness and high performance of the proposed algorithm.Experiments were carried out using an S-band DAR test-bed.The results of measured data show that the proposed algorithm is practical and effective in application.

1.Introduction

Digital array radar(DAR)employs a full digital beam forming(DBF)architecture in the receiving and transmitting system.It has the potential of forming multiple simultaneous beams while providing high anti-interference capabilities.In the last decade,DAR has attracted considerable attention and has been widely used in space surveillance.1,2Most array signal processing algorithms,such as DBF and direction of arrival(DOA),rely crucially on the assumption that the array manifold is perfectly known.However,in actual systems,the array manifold is inevitably affected by gain-phase perturbations and mutual coupling effects.As a result,the performance of DAR may be seriously degraded.3

Traditional algorithm for array calibration is to carry out measurements using computational electromagnetic solvers,which has been applied in some actual radar systems.4,5The algorithm has the problem of time consuming and high demands for testing environments.It may be impractical once the array systems are in operation due to the complex electromagnetic environments.

In order to address the problem,a number of selfcalibration algorithms that make use of signal processing technology have been developed.Ref.6estimates DOA parameters and mutual coupling coefficients using the space alternating generalized expectation maximization algorithm.In Ref.7,the mutual coupling effects in the uniform linear array(ULA)are inherently eliminated without any calibration sources,but the algorithm requires some extended elements.Refs.8–10present a category of algorithms that can iteratively estimate the array manifold errors and the DOAs of impinging signals based on the subspace principle.These calibration algorithms usually suffer from low accuracy,high computational complexity and serious ambiguous problems.

Compared with self-calibration algorithms,active calibration algorithms use auxiliary sources to overcome suboptimal convergence problems and have the potential to achieve better calibration accuracy.The algorithm in Ref.11calibrates mutual coupling errors in an arbitrary array using several ideal instrumental elements.But it is difficult to find the ideal elements in practice.A maximum likelihood approach is presented in Refs.12,13to estimate the unknown gain-phase,mutual coupling as well as sensor positions.It has the drawbacks of high computation,and the iterations may not be convergent under some conditions.Refs.14–16have proposed a category of eigenstructure algorithm that treats gain-phase and mutual coupling errors as a whole.The closed- form of the distortion matrix is derived with the help of some time-disjoint auxiliary sources.The algorithms have been implemented to improve the performance of actual systems.17,18But they do not consider about the special structure of distortion matrix and have a strict requirement pertaining to the number of auxiliary sources.The algorithm in Ref.19eliminates the repeated entries in the distortion matrix of ULA to reduce the unknown parameters and reaches a better accuracy than the algorithms in Refs.14,16.However,the algorithm requires the precise knowledge of locations of auxiliary sources,which may not be available in some actual applications.

In practical situations,it may be difficult to access the precise directions of auxiliary sources.However,it is possible to determine the angle intervals between them using additional equipment,such as rotating plat forms.This paper focuses on the problems of joint calibration of gain-phase and mutual coupling errors in the above scenario.The proposed algorithm firstly develops a simplified form of the distortion matrix according to its special structure.Then the relationship between the distortion matrix and the DOAs of calibration sources is derived.Finally one-dimension searching is employed to obtain the angles and the least-square estimation of distortion matrix is also provided.The proposed algorithm achieves high accuracy and behaves robustly when the incident angles of auxiliary sources are not known precisely.

The paper is organized as follows.In Section 2,the signal model of ULA is demonstrated and the problem of array calibration is illustrated.In Section 3,the proposed algorithm for array calibration in the presence of gain-phase errors and mutual coupling errors is developed.Computer simulations and experimental results of measured data are presented and analyzed in Section 4,followed by conclusions in Section 5.

2.Signal model and problem formulation

Consider a ULA consisting of N omnidirectional antenna elements with the space d between neighboring elements.There are M narrowband signals s1（t），s2（t），...，sM（t）located in thefar-field region.The signals imping on the ULA from different directions of φ1，φ2，...，φM,with respect to the normal line of the ULA.The signals are incoherent with each other with a wavelength of λ.The additive noise is zero-mean,random process with a variance of σ2.The outputs of the array can be written as

wherex（t）= ［x1（t），x2（t），...，xN（t）］T,s（t）= ［s1（t），s2（t），...，sM（t）］T,n（t）= ［n1（t），n2（t），...，nN（t）］Tare the output vector,signal vector and noise vector, respectively. A=［a（φ1），a（φ2），...，a（φM）］is the ideal array manifold matrix,where a（φi）= ［1，e-j2πdsinφi/λ，...，e-j2π（N-1）dsinφi/λ］Tdenotes the ideal steering vector of the i th signal.

Taking gain-phase perturbations and mutual coupling effects into consideration,the outputs can be modified as

where Γ =diag（τ1，τ2，...，τN）is a diagonal matrix and τidenotes gain-phase errors of the i th element.C ∈ CN×Nis the mutual coupling matrix(MCM).

Since the structure of C is highly dependent on the physical structure of the array,it can be considered as a banded symmetric Toeplitz matrix in the case of ULA.7Indeed,mutual coupling effects tend to be reciprocal to the distance between elements and may be negligible for the elements separated by a few wavelengths.There fore,C may be expressed as

where ciis the mutual coupling coefficient between the first and the ith element.P is the number of non-zero complex coefficients in the first row of the MCM.

The covariance matrix of array output vector is defined as

where Rs（t）=E｛s（t）sH（t）｝ is the covariance matrix of signals,which is nonsingular when the signals are incoherent.INis the N×N identity matrix.

Per forming eigen-decomposition on the output covariance matrix,it can be written as

In Eq.(5),?1≥ ?2≥ ···≥ ?Mare the M large eigenvalues of Rx（t）,and ?M+1= ?M+2= ···= ?N= σ2aresmalleigenvalues.Σs=diag（?1，?2，...，?M）andΣno=diag（?M+1，?M+2，...，?N）are diagonal matrices.Us= ［e1，e2，...，eM］∈ CN×Mis composed of the eigenvectors corresponding to the M large eigenvalues,while Uno= ［eM+1，eM+2，...，eN］∈ CN×（N-M）contains the rest N-M eigenvectors.

According to the subspace theory,Usspans the same space with the distortion array manifold matrix CΓA,which is orthogonal to the space spanned by Uno.Thus thefollowing relation holds true:

where span｛X｝denotes the subspace spanned by X.

In Eq.(6),Uscan be derived from the receiving data of the array.There fore,the optimal estimation of unknown distortion parameters may be obtained based on the relationships between CΓA and Us.

3.Proposed algorithm for joint calibration of gain-phase and mutual coupling errors

3.1.Least-squares solutions of distortion matrix

Suppose there are M far- field auxiliary sources impinging on the array at different times.The incident angles are θ，θ+Δθ1，...，θ+ΔθM-1.Here we concern about the situation where θ is unknown,but the angles’intervals Δθ1，Δθ2，...，can be measured precisely.As for the mth auxiliary source,the covariance matrix of receiving data is denoted asand the eigenvector corresponding to the maximum eigenvalue ofdenoted as em.It can be determined from Eq.(6)that emis related to the distortion steering vector by a complex scaling constant.The following equation is obtained ignoring noise and finite length of snapshots:

where ηmis an unknown constant.There fore,a least-squares problem may be formulated with the cost function as

where Λ = ［η1，η2，...，ηM］Tis denoted as the scaling vector,‖ ·‖F(xiàn)is the Frobenius norm.

Here we define Z=CΓ as the array distortion matrix,which contains gain-phase errors and mutual coupling errors.It is considered that there are N2unknown parameters in Z.However,making use of the special structure of Γ and C,the unknown parameters can be decreased.

Taking advantage of the diagonal structure of gain-phase matrix,Γ can be expressed by the diagonal elements:

As noted from Eq.(3),the MCM of ULA is a banded symmetric Toeplitz matrix and may be trans formed to a mutual coupling vector as

where j=1，2，...，L and p=1，2，...，P.

It is evident that the simplified form of the distortion matrix in Eq.(12)only contains L unknown parameters.Due to the reduction of unknown parameters,the estimation accuracy will be improved and the requirements for auxiliary sources will be relaxed as well.

By substituting Eq.(12)into Eq.(8),the cost function can be rewritten as

Optimization of Eq.(14)with respect to g whilst keeping Λ and θ unchanged provides a solution for g:

Eq.(16)specifies the least-squares estimation of the distortion vector g with unknown parameters Λ and θ,the solutions of which will be discussed in the following section.

3.2.Determination of incident angles

Inserting Eq.(16)into Eq.(14)yields the optimization problem:

In Eq.(17),ηm（m=1，2，...，M）can be expressed concisely by the vector Λ. Defining an L×M matrix as T′=and the m th principle eigenvectorwe have the forms

where 0p×qdenotes p × q null matrix.Um= ［0N×（m-1），em，0N×（M-m）］is an N × M matrix,with the mth column of emand other columns of 0N×1.

Substituting Eqs.(18)and(19)into Eq.(17),the cost function becomes

where Vm=TmT0T′-Um∈ CN×M.

The solution of the optimization problem in Eq.(20)requires a constraint to avoid the trivial solution.Without loss of generality,we impose the line constraint ΛHε=1,where ε= ［1，0，...，0］T∈ CM×1.Using the method of Lagrange multipliers,the optimization solution of Λ is derived as

Substituting Eq.(21)into Eq.(20),the cost function may be expressed as

There fore,the incident angles of calibration sources can be obtained by one-dimensional searching:

With the knowledge of θ,the scaling vector Λ is derived by Eq.(21).Finally,the distortion vector g is calculated by Eq.(16)and the distortion matrix Z is reconstructed according to Eq.(12).

3.3.Discussion

In the above sections,the proposed algorithm has been described in detail.Referring to Ref.20,a necessary but not sufficient condition for the uniqueness of solutions has been presented:the number of independent equations in the cost function is more than the number of unknown parameters.

In Eq.(14),there are 2PN real parameters containing in g,2M real parameters in Λ and the unknown incident angle θ.Thus,it follows that the number of real parameters is 2PN+2M+1.The number of independent equations,however,is 2MN.There fore,the uniqueness condition requires that

From Eq.(25),we know that the number of auxiliary sources for the proposed algorithm is mainly dependent on the number of non-zero elements in MCM,instead of the number of antenna elements.As a result,it requires only a few time-disjoint auxiliary sources even for large arrays.It is noted that the algorithm in Ref.21also makes use of the symmetric Toeplitz property of the MCM.However,the simplified form is established in a different way from Eq.(12).Consequently,the derivations of the corresponding solutions are different either.

The proposed algorithm will be simplified significantly when only mutual coupling errors are considered.In the absence of gain-phase errors,there are only P unknown parameters in g.Thus,only two auxiliary sources are demanded and the accuracy will be improved as a result of the reduction of unknown parameters.

In most DAR systems,the antenna arrays are mounted on rotation plat forms.The directions of auxiliary sources with respect to the array can be changed by varying the orientations of rotation plat forms.In this way,calibration signals from different directions are captured at different times.Since the proposed algorithm demands that calibration sources from different directions are time-disjoint,it requires only one auxiliary source to generate these calibration signals with the help of rotation plat forms.Though the incident angles are difficult to measure,the angle intervals may be determined according to the records of rotation plat forms.There fore,the requirements for time-disjoint auxiliary sources are easily satisfied with the help of rotation plat forms.As a result,it is convenient to apply the proposed algorithm in practical situations.

4.Simulations and experimental results

4.1.Computer simulations

In this section,some representative simulations are carried out to demonstrate the performance of the proposed algorithm in comparison with previous algorithms.Consider an eightelement omnidirectional ULA with the space between neighboring elements equal to 0.05 m.Each channel is perturbed by gain-phase errors with coefficients of τ= ［1，0.96-0.08j，0.96+0.09j，1.13+0.12j，0.85+0.11j，1.05-0.14j，1.06+0.04j，1.05-0.003j］T.As far as the mutual coupling is concerned,we assume that the number of nonzero mutual coupling coefficients is P=3 and the mutual coupling vector is c= ［1，0.31+0.18j，0.12-0.09j］T.The relative root mean square error(RMSE)of the distortion matrix defined in Eq.(26)is chosen as a measurement of the estimation accu-racy.In the following simulations,200 independent Monte Carlo trials were carried out.

where Mcis the number of Monte Carlo trials.

In Simulation 1,the accuracy of the distortion matrix estimation is investigated.Suppose there are eleven auxiliary sources with a frequency of 2.6 GHz located in the far-field region.The direction of the first auxiliary sourcesis θ=-50°and the angle interval between adjacent sources is Δθ =10°.

Firstly,we assume that the incident angle of the first auxiliary source is known precisely.The solid lines in Figs.1 and 2 depict the relative RMSE of the distortion matrix versus SNR and snapshots.The Cramer-Rao lower bound(CRLB)9,22is also given for comparison.The results show that the proposed algorithm and the algorithm in Ref.21achieve approximately the same performance and the relative RMSE is close to the CRLB.Both algorithms reach better estimation accuracy than the algorithms in Refs.14,16due to the utilization of the unique structure of distortion matrix.Secondly,suppose the preknowledge of θ deviates from the true value with a bias of 0.6°,the relative RMSE are illustrated by the dashed lines in Figs.1 and 2.When the DOAs of auxiliary sources are not known precisely,the proposed algorithm behaves slightly worse.However,the performance of the other algorithms degrades seriously.Thirdly,the robustness of the algorithms is investigated,as is shown in Fig.3.With the increase of the bias of θ,the proposed algorithm can still provide a good estimation of the distortion matrix,but other algorithms lose efficacy quickly.

Fig.1 Relative RMSE of distortion matrix versus SNR(snapshots=512).

Fig.2 Relative RMSE of distortion matrix versus snapshots(SNR=5 dB).

Simulation 2 concentrates on the performance of DOA estimation in the presence of array errors.Suppose there are two incoherent signals impinging on the ULA from-6°and 3°,with a SNR of 6 dB and the snapshots of 1024.The DOA estimation accuracy is measured by the RMSE defined in Eq.(27),where L is the number of signals.Keeping the settings of auxiliary sources the same as simulation 1,the RMSE of the DOA estimation versus the SNR and snapshots of auxiliary sources are shown in Figs.4–6.With the increase of the SNR and snapshots of auxiliary sources,the distortion matrix may be estimated more precisely,resulting in higher DOA estimation accuracy.It can be seen that the proposed algorithm per forms better than others,especially for the situation where the first incident angle is not known precisely.The conclusions are coincident with Simulation 1.

Fig.3 Relative RMSE of distortion matrix versus angle bias of auxiliary sources(SNR=5 dB,snapshots=512).

Fig.4 RMSE of DOA estimation versus SNR of auxiliary sources(snapshots=512).

Fig.5 RMSE of DOA estimation versus snapshots of auxiliary sources(SNR=5 dB).

Fig.6 RMSE of DOA estimation versus angle bias of auxiliary sources(SNR=5 dB,snapshots=512).

In certain situations,gain-phase errors may be well compensated by some auxiliary instruments such as interior calibration networks.There fore,only mutual coupling effects are concerned.In simulation 3,the performance of the calibration algorithms for mutual coupling errors is studied.The array parameters are the same as simulation 1 except that there are no gain-phase errors.Two auxiliary sources are located in thefar-field region with an angle interval of 35°.The relative RMSE of MCM by the proposed algorithm are shown in Figs.7 and 8,compared with the rank reduction(RARE)algorithm in Ref.9,the recursive RARE algorithm in Ref.10,the resiliency algorithm in Ref.7and the CRLB.It can be seen that the proposed algorithm achieves a considerable accuracy and outper forms other algorithms,particularly at low values of SNR and snapshots.Moreover,the proposed algorithm rarely suffers from ambiguous problems and does not require any additional antenna elements.

Fig.7 Relative RMSE of MCM versus SNR(snapshots=512).

Fig. 8 Relative RMSE of MCM versus snapshots(SNR=5 dB).

4.2.Experimental results

The algorithms in the previous literature were mainly based on simulations and few are verified by measured data.In this section,experiments were carried out by an S-band DAR test-bed to evaluate the performance of the proposed algorithm.The test-bed has 8 tapered slot antennas arranged in a line with an adjacent space of 0.05 m.Digital receivers are designed to realize a high level of digitalization.Radio frequency(RF)signals up to 3 GHz are directly sampled by the 10 bits ADCs with a sampling rate of 1.2 Gsps utilizing the band-pass sampling theory.Signal processors based on the Open-VPX standard employ powerful Xilinx Virtex-6 FPGAs and TMS320C6678 DSPs to support system calibration,DBF,DOA and other real-time algorithms.

The experiments were conducted in an anechoic chamber(see Fig.9).The antenna array is placed on the rotation plat form.Sinusoidal signals at a frequency of 2.6 GHz are trans formed into parallel waves through a reflector.Incident angles were measured according to the records of the rotation plat form with a precision of 0.05°.After power on,eleven calibration signals were received,uniformly distributed in［-50°，50°］.Then the plat form was rotated from-50°to 49°at the step of 1°,with measured data acquired at each direction for further processing.It is noted that in the anechoic chamber,directions of calibration signals could be measured with the help of the reflector and rotation plat form.However,it is difficult to determine the directions in the outside environments where the conditions are not so ideal.In order to investigate the robustness of the proposed algorithm,we impose a bias on the incident angles of calibration signals in some experiments.The proposed algorithm is compared with the algorithm in Ref.14,16,and the gain-phase errors calibration(GPC)algorithm in Ref.21,which only calibrates gain-phase errors but ignores mutual coupling errors.

Fig.9 Experimental setup in anechoic chamber.

Thefirst experiment concerns about the DOA estimation in the presence of array manifold errors.Fig.10 shows the results of DOA estimation of signals from-27°.In Fig.10(a),the records of orientations of the rotation plat form are considered as the directions of auxiliary sources.It can be seen that the algorithms in Refs.14,16and the proposed algorithm can successfully estimate the DOA.But the curve of the gain-phase calibration algorithm is not sharp and has a poor accuracy due to mutual coupling effects.Imposing a bias of 2°on the directions of thefirst auxiliary source while maintaining angle intervals of the calibration sources,the proposed algorithm is still able to obtain the correct values.However,the curves of other algorithms skew from true positions in Fig.10(b).

The second experiment employs the array patterns3to evaluate the performance of the calibration algorithms.The array patterns,shown in Fig.11,are measured by applying the steering vectors with hanning window on signals from directions of -50°to 49°.Similar to the first experiment a bias of 2°is also imposed on the angles of auxiliary sources to investigate the performance of the calibration algorithms.The array patterns steer at 0°in Fig.11(a)and 10°in Fig.11(b),respectively.The gain-phase calibration algorithm results in high sidelobes and serious distortion of the pattern,since it ignores mutual coupling effects.Although the algorithm in Refs.14,16can provide low levels of sidelobes,the patterns deviate from the ideal case.It is worth mentioning that the pattern of the proposed algorithm is closer to the ideal pattern and not affected by the angle bias of auxiliary sources.

Fig.10 DOA estimation results of signals from-27°.

Fig.11 Array patterns of measured data with hanning window.

Fig.12 Measured element patterns after calibration by different algorithms.

In the third experiment,the consistence of element patterns5is investigated.Since all the transmitting signals are the same,the element patterns can be measured by the power of receiving signals after calibration.Fig.12 shows the measured patterns of all the antenna elements,taking thefirst element as reference and normalized to unity at θ =0°.Different lines in each subfigure represent the patterns of different elements.Affected by mutual coupling,the element patterns obtained by the gain-phase calibration algorithm differ from each other,which will lead to a deterioration of array performance.Taking mutual coupling errors into consideration,the consistence of element patterns is greatly improved.It is shown that the proposed algorithm outper forms other algorithms again.When incident angles are biased,the proposed algorithm behaves robustly,while the performance of other algorithms degrades.

5.Conclusions

This paper has presented a practical algorithm for array calibration in the presence of gain-phase and mutual coupling errors.The investigations are summarized as follows:

(1)The algorithm behaves robustly against angle bias of auxiliary sources.Since the calibration signals are time-disjoint,the algorithm requires only one auxiliary source with the help of rotation plat forms.

(2)The algorithm achieves high estimation accuracy and rarely suffers from ambiguous problems.

(3)Experimental results of measured data show that the algorithm is practical and useful to improve the performance of the DAR test-bed.

(4)The authors will devote to extend the algorithm to the calibration of other array manifold imperfections,such as sensor array errors.

(5)The authors will try to extend the proposed algorithm to the case when both the incident angles and angle intervals are unknown.

Acknowledgement

This work was supported by the National Natural Science Foundation of China(No.61571449).

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Li Weixing received the B.S.degree in 2009 in electronic engineering from the National University of Defense Technology,and he is a Ph.D.student there.His current research interests include array signal processing and radar system design.

Lin Jianzhi received the B.S.degree in 2009 in electronic engineering from the National University of Defense Technology,and he is a Ph.D.student there.His current research interests include radar signal processing and imaging.

Zhang Yue received the B.S.and Ph.D.degrees from the National University of Defense Technology in 2004 and 2010,respectively.He is currently a lecturer with research interests include radar signal processing and system design.

Chen Zengping is a prof essor and a Ph.D.supervisor at the National University of Defense Technology and his research interests include signal processing,radar system,and automatic target recognition.

16 October 2015;revised 24 January 2016;accepted 7 March 2016

Available online 22 June 2016

Calibration accuracy;

Digital array;

Gain and phase errors;

Least-squares;

Mutual coupling;

Uniform linear array

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E-mail address:atrchen@sina.com(Z.Chen).

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This is an open access article under the CC BY-NC-ND license(http://creativecommons.org/licenses/by-nc-nd/4.0/).