direction of arrival based on the multioutput least squares support vector...

Research ArticleDirection of Arrival Based on the Multioutput Least SquaresSupport Vector Regression Model

Kai Huang 12 Ming-Yi You 12 Yun-Xia Ye12 Bin Jiang12 and An-Nan Lu12

1Science and Technology on Communication Information Security Control Laboratory Jiaxing 314033 Zhejiang China2No 36 Research Institute of CETC Jiaxing 314033 Zhejiang China

Correspondence should be addressed to Ming-Yi You youmingyi126com

Received 14 July 2020 Revised 15 September 2020 Accepted 19 September 2020 Published 30 September 2020

Academic Editor Liangtian Wan

Copyright copy 2020 Kai Huang et al -is 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

-e interferometer is a widely used direction-finding system with high precision When there are comprehensive disturbances inthe direction-finding system some scholars have proposed corresponding correction algorithms but most of them requirehypothesis based on the geometric position of the array -e method of using machine learning that has attracted much attentionrecently is data driven which can be independent of these assumptions We propose a direction-finding method for the in-terferometer by using multioutput least squares support vector regression (MLSSVR) model -e application of this methodincludes the following the construction of MLSSVR model training data training and construction of the MLSSVR model andthe estimation of direction of arrival Finally themethod is verified through numerical simulationWhen there are comprehensivedeviations in the system the direction-finding accuracy can be effectively improved

1 Introduction

Direction of arrival (DOA) estimation is a widely studiedproblem in various fields including wireless communica-tions [1] radar detection [2ndash4] target localization andtracking [5 6] Various methods have been proposed toestimate DOA of emitters such as interferometer [7 8] andarray processing [9ndash11]

-e interferometer estimates the DOA based on thephase difference of different direction-finding baselines -eaccuracy of interferometer is sensitive to the phase differenceof baselines In engineering applications there may bevarious deviations such as phase inconsistency betweenchannels mutual coupling between the antennas and an-tenna location deviations In order to achieve optimal di-rection-finding performance the methods includingcorrelation-coefficient [12] weighted least squares [13] andparameter estimation [14] are always used To facilitatemethod implementation simplified models are establishedto describe the effects of various deviations and autocali-bration processes are proposed to improve DOA estimationprecision [15ndash21] Most of the simplifications on array

deviations are made from mathematical perspectives ap-proximately with various additional assumptions such asuniform linearity or circularity array geometries [15ndash17]constrained antennas location deviations within a particularline or plane [18 19] and intersensor independence of gainand phase errors [20 21]

However the effect of comprehensive disturbanceswhich probably exist in practical systems is much moredifficult to be modeled precisely and calibrated automati-cally -e multiple deviations have a great influence on theamplitude and phase of each receiving channel whichgreatly affects the performance of the interferometer di-rection-finding system -e commonly used method toreduce the effect of multiple deviations is the external fieldcalibration method ie for the different directions of in-coming signal with a large signal-to-noise ratio the mea-sured values of the phase difference from baselines aredirectly recorded and saved together with the known cali-bration directions [22] In the application of the direction-finding system the DOA is calculated by least squares be-tween the measured value of each phase difference with thesaved values from the external field -e external field

HindawiMathematical Problems in EngineeringVolume 2020 Article ID 8601376 8 pageshttpsdoiorg10115520208601376

calibration method is simple in principle and easy to operateand has been widely used in engineering Due to the in-fluence of multiple deviations even if there are differentgains of each antenna the external field calibration methodalways uses the equal-weight least squares method -einfluence of the inconsistency between the antennas on thedirection-finding accuracy has not been fully considered iethe external field calibration method fails to maximize theeffectiveness of the direction-finding system

Recently some scholars have used machine learning tosolve the DOA estimation with comprehensive distur-bances [23ndash31] Machine learning have significant ad-vantages over traditional methods based on arraygeometries and least square in solving direction-findingproblems in complicated scenarios with multiple devia-tions such as radial basis function (RBF) [23] least squaressupport vector classification (LSSVC) [24] support vectorregression (SVR) [25ndash29] and deep neural networks(DNN) [30 31] -ese methods are data driven and do notrely on preassumptions about array geometries andwhether they are calibrated or not Despite its potentialusefulness the standard formulation of the LS-SVR cannotcope with the multioutput case [32] -erefore with theangles around 0deg and 360deg the accuracy has not been ef-fectively improved -e DNN for direction finding requireslarge samples to perform well resulting in too long modeltraining time and if the samples size is insufficient thismethod cannot effectively perform -is paper proposes adirection-finding method for the interferometer based onthe multioutput least squares support vector regression(MLSSVR) model -e multioutput mode can avoid largeerrors in the single-output mode around 0deg and 360deg and itis expected to achieve higher direction-finding accuracywithin the angle range Due to the MLSSVR model themodel training time can be greatly shortened -e appli-cation of this method includes the following (1) theconstruction of MLSSVR model training data (2) trainingand construction of the MLSSVR model and (3) the es-timation of direction of arrival

-e organization of the rest of the paper is as followsSection 2 formulates the array output model with compli-cated deviations Section 3 presents the process of theMLSSVR model for DOA estimation Section 4 carries outsimulations to verify the validity of the method Section 5concludes this work

-e main notation used in this paper is listed in Table 1

2 Problem Formulation

Assume that the waveform of signal is s(t) and direction ofsignal is α -en the antenna output is

x(t) a(α)s(t) + v(t) (1)

When the comprehensive disturbances exist in the di-rection-finding system such as phase inconsistency betweenchannels mutual coupling between the antennas and an-tenna location deviations these disturbances cause devia-tions to a(α) and the actual antenna output is

1113957x(t) a(α e)s(t) + v(t) (2)

a(α e) is the direction vector of array with compre-hensive disturbances

-e phase difference between the antennas may be faraway from the theoretical value Obviously at this timehigh-precision direction finding cannot be performedaccording to the theoretical geometry of the antennas In thiscase the external field calibration method can be usedgenerally We propose the MLSSVR model to achievehigher-precision direction finding and this method is datadriven and do not rely on preassumptions about array ge-ometries and whether they are calibrated or not

3 MLSSVR Model for DOA Estimation

31 Construction Training Data of the MLSSVR ModelFor the MLSSVR model the training data are constructed asa matrix comprising phase differences of baselines and theircorresponding vectors related to the direction of incomingsignal which is

φt ηlabel1113858 1113859 φt1 φt2 φtL

ηlabel1 ηlabel2 ηlabelL1113890 1113891

T

(3)

where φti φ12ti φ(Nminus 1)N

ti1113960 1113961

T (i 1 L) is the phase

difference vector of the i-th training sample where T is thetransposed symbol and ηlabeli is the i-th vector which isrelated to the DOA of the signal In order to avoid the largedirection-finding error around 0deg or 360deg caused by theunreasonable loss function of the single-output SVR modelwe use the dual output form namely

ηlabeli cos αlabeli( 1113857 sin αlabeli( 11138571113858 1113859T (4)

Here αlabeli is the DOA corresponding to the i-thtraining sample

It should be pointed out that the MLSSVR model is notacquired through only one experiment and the acquisitionof its training data should be carried out under the conditionof similar signal-to-noise ratio in the application scenarioOn the other hand in training data of the MLSSVR modelfor a given DOA it is usually necessary to obtain severalsamples (generally more than 10 [32]) In addition the DOAof the signal should cover the angle range in the applicationas much as possible namely

αl le αlabeli le αu (5)

Here αu and αl represent the upper and lower limits ofthe possible direction of arrival respectively

If the directions of adjacent known incoming signal inthe training samples are not the same they will differ by afixed step Δαt ie αlabeli+1 αlabeli or αlabeli+1 αlabeli + Δαt-e value should generally be less than the requisite accuracyof direction finding in the application Between αu and αlthe number of sample categories is M

32 Train and Build a MLSSVR Model Given a data set(xi yi)

Li1 xi isin Rd and yi isin Rm-e purpose of multioutput

2 Mathematical Problems in Engineering

regression is to give a set of input vectors x isin Rd and predicta set of output vectors y isin Rm -e MLSSVR model solvesthis problem by finding W (w1 wm) isin Rnhtimesm and b

(b1 bm)T isin Rm that minimize the following objectivefunction with constraints

minτ(WΞ) 12trace WTW1113872 1113873 + c

12trace ΞTΞ1113872 1113873

st Y ZTW + repmat bT L 11113872 1113873 + Ξ

(6)

Here Y [yi] isin RLtimesm Ζ (δ(x1) δ(xL)) isin RnhtimesLδ(middot) is the mapping function (kernel function) ofRd⟶ Rnh whose purpose is to transform xi into a moredistinguishable high-dimensional feature space with nh di-mensions Ξ (ξ1 ξm) isin RLtimesm

+ ξi (ξi1 ξim)T andξij is a slack variable trace(A) 1113936

mi Aii where A is a m times m

matrix and c isin R+ is a regularization parameterIn order to achieve the solution of (6) the objective

function can be constructed as a heuristic Bayesian archi-tecture Let wi w0 + vi where w0 isin Rnh contains commonparameter information and vi isin Rnh carries the individualinformation of each sample To obtain a solution w0V (v1 vm) and b the following objective functionwith constraints can be constructed

minτ w0V Γ( 1113857 12wT

0w0 +12λmtrace VTV1113872 1113873 + c

12trace ΓTΓ1113872 1113873

st Y ZTW + repmat bT L 11113872 1113873 + Γ

(7)

Among them λ isin R+ is another regularizationparameter

Equation (7) can be transformed into Lagrangersquosequation and then the optimal solutions w lowast0 V

lowast and blowastcan be achieved using the KarushndashKuhnndashTucker (KKT)optimization conditions After that the corresponding de-cision function is [32]

f(x) δ(x)TWlowast + blowast

T

δ(x)Trepmat w lowast0 1 m( 1113857

+ δ(x)TVlowast + blowast

T

(8)

33 DOA Estimate Given a set of phase difference data φbased on equation (8) we have

η cos(α) sin(α)1113858 1113859T

δ(φ)Trepmat w lowast0 1 m( 1113857

+ δ(φ)TVlowast + blowast

T

(9)

Based on the result of equation (9) the final DOA isestimated as

α atan2(sin(α) cos(α)) atan2(η(2 1) η(1 1))

(10)

Here atan2 is the four-quadrant inverse tangentfunction

4 Simulations and Analysis

-is section gives a numerical simulation in a typical sce-nario to demonstrate the effectiveness of theMLSSVRmodelapplied to direction finding Consider a 5-element uniformcircular array with a radius-to-wavelength ratio of 04 -eradial basis kernel function is adopted to solve formula (8)κ(x z) exp(minuspx minus z2) pgt 0 After multiple trainingsdetermine the parameters c 05 and λ 4 in formula (7)and the radial basis function parameter p 1

41 Phase Inconsistency between Channels Assume that theRMSE of phase difference under 10 dB signal-to-noise ratiois 25deg 5deg 5deg 10deg and 15deg in 5 channels respectively -etraining data sets consist of phase difference and cosine andsine functions of each angle from 0deg to 360deg with a step of 1degand 10 groups of samples are collected per angle ie thenumber of training data sets L 3600 Under the samesignal-to-noise ratio a total of 3600 testing samples of phase

Table 1 Symbol and notation

Symbol Explanations(t) Waveform of signala(α) Direction vector of arrayx(t) -e theoretical output of antennasL -e number of training samplesRa a-Dimensional real number spaceR+ One-dimensional positive real number spaceC Mutual coupling matrix

φkjti

-emeasured value of phase difference from baselines formed between the k-th antenna and the j-th antenna in the i-th trainingsample

α Direction of signalv(t) Zero-mean Gaussian noise1113957x(t) -e actually output of antennasN -e number of antennasRatimesb a times b-Dimensional real number spaceRatimesb

+ a times b-Dimensional positive real number spaceφt Phase differences matrix in the training dataηlabel -e matrix formed by cosine and sine function of DOA in the training data

Mathematical Problems in Engineering 3

difference are generated from 0deg to 360deg As a comparisonthe DOA of testing samples is also calculated by using leastsquares (LS) with training data sets Figure 1 shows thedirection-finding error of testing samples

42 Mutual Coupling between the Antennas When there ismutual coupling between antennas assume the mutualcoupling matrix as follows

C

1 07821 + 02583j 04576 + 02469j 04576 + 02469j 07821 + 02583j

07821 + 02583j 1 07821 + 02583j 04576 + 02469j 04576 + 02469j

04576 + 02469j 07821 + 02583j 1 07821 + 02583j 04576 + 02469j

04576 + 02469j 04576 + 02469j 07821 + 02583j 1 07821 + 02583j

07821 + 02583j 04576 + 02469j 04576 + 02469j 07821 + 02583j 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(11)

Selecting 10 samples for each angle from 0deg to 360deg andthe training data sets consist of phase difference and cosineand sine functions of each angle Under the same signal-to-noise ratio and mutual coupling 3600 groups of phasedifference from different DOA are generated as the testingsamples Figure 2 shows the direction-finding error oftesting samples

43 Both Phase Inconsistency and Mutual Coupling In thepresence of phase inconsistency and mutual coupling si-multaneously 3600 groups of phase difference and cosineand sine functions of DOA are generated as the training datasets for 0degsim360deg with a step of 1deg and 10 samples are selectedfor each angle Under the same signal-to-noise ratio with thecomprehensive disturbances which consist of phase in-consistency and mutual coupling the testing samples arecomposed of 3600 groups of phase difference from differentDOA Figure 3 shows the direction-finding error of testingsamples

44 Different SNRs In the presence of phase inconsistencyand mutual coupling simultaneously the training data setsare generated in the same way as mentioned in Section 43under 10 dB SNR while testing samples are generated underthe 5 dB SNR -e number of training data sets and testingsamples are both 3600 Figure 4 shows the direction-findingerror of testing samples

Table 2 shows the RMSE of DOA of the testing samplesby using LS and MLSSVR respectively under differentdisturbance scenarios From the results in Table 2 it can beseen that the MLSSVR model can significantly reduce thedirection-finding error and obtain high-precision direction-finding results in the full range of 360deg

45 Comparison of MLSSVR and Neural Networks -etraining data sets and testing samples are generated in thesame way as mentioned in Section 43 -e number oftraining data sets and the number of testing samples areboth 3600 In additional both phase inconsistency and

0ndash30

ndash20

ndash10

0

10

20

30

Dire

ctio

n-fin

ding

erro

r (deg)

300 35025020015010050

DOA of testing samples (deg)

In the presence of phase inconsistency

LSMLSSVR

Figure 1 Direction-finding error of testing samples with phase inconsistency


mutual coupling are still in existence For the sametraining data sets and testing samples the two-layerconvolutional neural network is used to estimate theDOA-eMaxEpochs is set to 15 (For the neural networkmodel it was obvious that the value of MaxEpochs inneural network could be larger to reduce the direction-finding error of the model Here a small value was de-liberately selected to reduce the training time of neural

network to approximate the training time of MLSSVRmodel) so that the training time of the neural networkmodel is close to the MLSSVR model Figure 5 shows thedirection-finding error of testing samples

Table 3 shows the MLSSVR model is suitable for smalltraining data sets and compared with neural networkmodelthe MLSSVR model ensures direction-finding accuracywhile shortening training time

0ndash100

ndash50

0

50

100

150

200

Dire

ctio

n-fin

ding

erro

r (deg)

300 35025020015010050


In the presence of mutual coupling

LSMLSSVR

Figure 2 Direction-finding error of testing samples with mutual coupling

0ndash200

ndash150

Dire

ctio

n-fin

ding

erro

r (deg)

300 35025020015010050


In the presence of phase inconsistency and mutual coupling

LSMLSSVR

ndash100

ndash50

0

50

100

150

200

Figure 3 Direction-finding error of testing samples with phase inconsistency and mutual coupling


0

ndash100

ndash50

ndash200

ndash150

0

50

100

150

200

Dire

ctio

n-fin

ding

erro

r (deg)

300 35025020015010050


Different SNRs

LSMLSSVR

Figure 4 Direction-finding error of testing samples with different SNRs

Table 2 RMSE of DOA of the testing samples

-e number of testing samples Phase inconsistency Mutual coupling Phase inconsistency mutual coupling Different SNRsRMSE of DOA (deg)

LS 3600 40870 102529 159103 201085MLSSVR 29695 52498 79172 113850

0

ndash100

ndash150

ndash200

ndash50

0

50

100

150

200

Dire

ctio

n-fin

ding

erro

r (deg)

300 35025020015010050


Neural network and MLSSVR

Neural networkMLSSVR

Figure 5 Direction-finding error of testing samples


RMSE of DOA (deg) Training times (s)Neural network 147228 2531350MLSSVR 77232 2548054


5 Conclusion

-e standard formulation of support vector regression canonly deal with the single-output case and when it is appliedto radio direction-finding there may be a problem that thedirection-finding results have large errors around 0deg or 360deg-is paper applies the MLSSVR model to the field of radiodirection-finding the training data sets consist of phasedifferences of each baseline and the cosine and sine functionsof each angle from 0deg to 360deg -e DOA is calculated by thesine function and cosine function of the incident angle thusavoiding a larger case finding the error results in the vicinityof 0deg or 360deg In the case of comprehensive disturbances inthe direction-finding system the effectiveness of theMLSSVR model is verified by numerical simulation Andwith small training data sets it can still effectively improvethe direction-finding accuracy compared to the LS method

Data Availability

All training data sets and testing samples used to support thefindings of this study are available from the correspondingauthor upon request

Conflicts of Interest

-e authors declare that they have no conflicts of interest

Acknowledgments

-is work was supported by the No 36 Research Institute ofCETC under the project no CX05

References

[1] L Wan L Sun X Kong Y Yuan K Sun and F Xia ldquoTask-driven resource assignment in mobile edge computingexploiting evolutionary computationrdquo IEEE Wireless Com-munications vol 26 no 6 pp 94ndash101 2019

[2] F Wen Z Zhang K Wang G Sheng and G Zhang ldquoAngleestimation and mutual coupling self-calibration for ULA-based bistatic MIMO radarrdquo Signal Processing vol 144pp 61ndash67 2018

[3] F Wen and J Shi ldquoFast direction finding for bistatic EMVS-MIMO radar without pairingrdquo Signal Process vol 173 2020

[4] X Wang L Wang X Li and G Bi ldquoNuclear norm mini-mization framework for DOA estimation in MIMO radarrdquoSignal Processing vol 135 pp 147ndash152 2017

[5] L Wan X Kong and F Xia ldquoJoint range-Doppler-angleestimation for intelligent tracking of moving aerial targetsrdquoIEEE Internet of =ings Journal vol 5 no 3 pp 1625ndash16362018

[6] K C Ho and Y T Chan ldquoAn asymptotically unbiased es-timator for bearings-only and Doppler-bearing target motionanalysisrdquo IEEE Transactions on Signal Processing vol 54no 3 pp 809ndash822 2006

[7] J-H Lee J-H Lee and J-M Woo ldquoMethod for obtainingthree- and four-element array spacing for interferometerdirection-finding systemrdquo IEEE Antennas and WirelessPropagation Letters vol 15 pp 897ndash900 2016

[8] J H Lee J K Kim H K Ryu and Y J Park ldquoMultiple arrayspacings for an interferometer direction finder with high

direction-finding accuracy in a wide range of frequenciesrdquoIEEE Antennas andWireless Propagation Letters vol 17 no 4pp 563ndash566 2018

[9] D Meng XWang M Huang L Wan and B Zhang ldquoRobustweighted subspace fitting for DOA estimation via block sparserecoveryrdquo IEEE Communications Letters vol 24 no 3pp 563ndash567 2020

[10] J Zheng T Yang H Liu and T Su ldquoEfficient data trans-mission strategy for IIoTs with arbitrary geometrical arrayrdquoIEEE Transactions on Industrial Informatics vol 99 Article ID2993586 2020

[11] J Zheng T Yang H Liu T Su and L Wan ldquoAccuratedetection and localization of UAV swarms-enabled MECsystemrdquo IEEE Transactions on Industrial Informatics vol 99Article ID 3015730 2020

[12] C S Park and D Y Kim ldquo-e fast correlative interferometerdirection finder using IQ demodulatorrdquo in Proceedings of theAsia-Pacific Conference on Information Processing pp 1ndash5Busan South Korea 2006

[13] G Zhu Y Wang and S Mi ldquoResearch on direction findingtechnique based on the combination of weighted least squaresmethod and MUSIC algorithmrdquo Rader Science and Tech-nology vol 17 no 3 pp 319ndash323 2019

[14] J Zheng H Liu and Q H Liu ldquoParameterized centroidfrequency-chirp rate distribution for LFM signal analysis andmechanisms of constant delay introductionrdquo IEEE Transac-tions on Signal Processing vol 65 no 24 pp 6435ndash6447 2017

[15] B Friedlander and A J Weiss ldquoDirection finding in thepresence of mutual couplingrdquo IEEE Transactions on Antennasand Propagation vol 39 no 3 pp 273ndash284 1991

[16] T Svantesson ldquoModeling and estimation of mutual couplingin a uniform linear array of dipolerdquo in Proceedings of the IEEEInternational Conference on Acoustics Speech and SignalProcessing IEEE Phoenix AZ USA pp 2961ndash2964 March1999

[17] M Lin and L Yang ldquoBlind calibration and DOA estimationwith uniform circular arrays in the presence of mutualcouplingrdquo IEEE Antennas and Wireless Propagation Lettersvol 5 no 1 pp 315ndash318 2006

[18] A J Weiss and B Friedlander ldquoArray shape calibration usingeigen-structure methodsrdquo Signal Processing vol 22 no 3pp 251ndash258 1991

[19] B P Flanagan and K L Bell ldquoArray self-calibration with largesensor position errorsrdquo Signal Processing vol 81 no 10pp 2201ndash2214 2001

[20] A Paulraj and T Kailath ldquoDirection of arrival estimation byeigen-structure methods with unknown sensor gain andphaserdquo Acoustics Speech and Signal Processingrdquo in Pro-ceedings of the ICASSPrsquo85 IEEE International Conference onAcoustics Speech and Signal Processing pp 640ndash643 IEEETampa FL USA April 1985

[21] Y Li and M Er ldquo-eoretical analyses of gain and phase errorcalibration with optimal implementation for linear equi-spaced arrayrdquo IEEE Transactions on Signal Processing vol 54no 2 pp 712ndash723 2006

[22] Q Zhai ldquoResearch on least square direction-finding methodrdquoRadio Engineering vol 38 no 3 pp 55ndash57 2008

[23] C S Shieh and C T Lin ldquoDirection of arrival estimationbased on phase differences using neural fuzzy networkrdquo IEEETransactions on Antennas and Propagation vol 48 no 7pp 1115ndash1124 2000

[24] C G Christodoulou J A Rohwer and C T Abdallah ldquo-euse of machine learning in smart antennasrdquo in Proceedings of


the IEEE Antennas and Propagation Society Symposiumpp 321ndash324 IEEE Monterey CA USA June 2004

[25] J A Rohwer C T Abdallah and C G Christodoulou ldquoLeastsquares support vector machines for direction of arrival es-timation with error control and validationrdquo in Proceedings ofthe GLOBECOMrsquo03 IEEE Global Telecommunications Con-ference pp 2172ndash2176 IEEE San Francisco CA USA De-cember 2003

[26] C A M Lima C Junqueira R Suyama F J V Zuben andJ M T Romano ldquoLeast-square support vector machines forDOA estimation a step-by-step description and sensitivityanalysisrdquo in Proceedings of the International Joint Conferenceon Neural Networks pp 3226ndash3231 Montreal Canada July2005

[27] S Vigneshwaran N Sundararajan and P SaratchandranldquoDirection of arrival (DOA) estimation under array sensorfailures using a minimal resource allocation neural networkrdquoIEEE Transactions on Antennas and Propagation vol 55no 2 pp 334ndash343 2007

[28] M Dehghanpour V T T Vakili and A Farrokhi ldquoDOAestimation using multiple kernel learning SVM consideringmutual couplingrdquo in Proceedings of the 2012 Fourth Inter-national Conference on Intelligent Networking and Collabo-rative Systems pp 55ndash61 Bucharest Romania September2012

[29] R Wang B Wen and W Huang ldquoA support vector re-gression-based method for target direction of arrival esti-mation from HF radar datardquo IEEE Geoscience and RemoteSensing Letters vol 15 no 5 pp 674ndash678 2018

[30] Z-M Liu C Zhang and P S Yu ldquoDirection-of-arrival es-timation based on deep neural networks with robustness toarray imperfectionsrdquo IEEE Transactions on Antennas andPropagation vol 66 no 12 pp 7315ndash7327 2018

[31] S Abeywickrama L Jayasinghe H Fu S Nissanka andC Yuen ldquoRF-based direction finding of UAVs uing DNNrdquo inProceedings of the IEEE International Conference on Com-munication Systems (ICCS) December 2018

[32] S Xu X An X Qiao L Zhu and L Li ldquoMulti-output least-squares support vector regression machinesrdquo Pattern Rec-ognition Letters vol 34 no 9 pp 1078ndash1084 2013


calibration method is simple in principle and easy to operateand has been widely used in engineering Due to the in-fluence of multiple deviations even if there are differentgains of each antenna the external field calibration methodalways uses the equal-weight least squares method -einfluence of the inconsistency between the antennas on thedirection-finding accuracy has not been fully considered iethe external field calibration method fails to maximize theeffectiveness of the direction-finding system

Recently some scholars have used machine learning tosolve the DOA estimation with comprehensive distur-bances [23ndash31] Machine learning have significant ad-vantages over traditional methods based on arraygeometries and least square in solving direction-findingproblems in complicated scenarios with multiple devia-tions such as radial basis function (RBF) [23] least squaressupport vector classification (LSSVC) [24] support vectorregression (SVR) [25ndash29] and deep neural networks(DNN) [30 31] -ese methods are data driven and do notrely on preassumptions about array geometries andwhether they are calibrated or not Despite its potentialusefulness the standard formulation of the LS-SVR cannotcope with the multioutput case [32] -erefore with theangles around 0deg and 360deg the accuracy has not been ef-fectively improved -e DNN for direction finding requireslarge samples to perform well resulting in too long modeltraining time and if the samples size is insufficient thismethod cannot effectively perform -is paper proposes adirection-finding method for the interferometer based onthe multioutput least squares support vector regression(MLSSVR) model -e multioutput mode can avoid largeerrors in the single-output mode around 0deg and 360deg and itis expected to achieve higher direction-finding accuracywithin the angle range Due to the MLSSVR model themodel training time can be greatly shortened -e appli-cation of this method includes the following (1) theconstruction of MLSSVR model training data (2) trainingand construction of the MLSSVR model and (3) the es-timation of direction of arrival

-e organization of the rest of the paper is as followsSection 2 formulates the array output model with compli-cated deviations Section 3 presents the process of theMLSSVR model for DOA estimation Section 4 carries outsimulations to verify the validity of the method Section 5concludes this work

-e main notation used in this paper is listed in Table 1

2 Problem Formulation

Assume that the waveform of signal is s(t) and direction ofsignal is α -en the antenna output is

x(t) a(α)s(t) + v(t) (1)

When the comprehensive disturbances exist in the di-rection-finding system such as phase inconsistency betweenchannels mutual coupling between the antennas and an-tenna location deviations these disturbances cause devia-tions to a(α) and the actual antenna output is

1113957x(t) a(α e)s(t) + v(t) (2)

a(α e) is the direction vector of array with compre-hensive disturbances

-e phase difference between the antennas may be faraway from the theoretical value Obviously at this timehigh-precision direction finding cannot be performedaccording to the theoretical geometry of the antennas In thiscase the external field calibration method can be usedgenerally We propose the MLSSVR model to achievehigher-precision direction finding and this method is datadriven and do not rely on preassumptions about array ge-ometries and whether they are calibrated or not

3 MLSSVR Model for DOA Estimation

31 Construction Training Data of the MLSSVR ModelFor the MLSSVR model the training data are constructed asa matrix comprising phase differences of baselines and theircorresponding vectors related to the direction of incomingsignal which is

φt ηlabel1113858 1113859 φt1 φt2 φtL

ηlabel1 ηlabel2 ηlabelL1113890 1113891

T

(3)

where φti φ12ti φ(Nminus 1)N

ti1113960 1113961

T (i 1 L) is the phase

difference vector of the i-th training sample where T is thetransposed symbol and ηlabeli is the i-th vector which isrelated to the DOA of the signal In order to avoid the largedirection-finding error around 0deg or 360deg caused by theunreasonable loss function of the single-output SVR modelwe use the dual output form namely

ηlabeli cos αlabeli( 1113857 sin αlabeli( 11138571113858 1113859T (4)

Here αlabeli is the DOA corresponding to the i-thtraining sample

It should be pointed out that the MLSSVR model is notacquired through only one experiment and the acquisitionof its training data should be carried out under the conditionof similar signal-to-noise ratio in the application scenarioOn the other hand in training data of the MLSSVR modelfor a given DOA it is usually necessary to obtain severalsamples (generally more than 10 [32]) In addition the DOAof the signal should cover the angle range in the applicationas much as possible namely

αl le αlabeli le αu (5)

Here αu and αl represent the upper and lower limits ofthe possible direction of arrival respectively

If the directions of adjacent known incoming signal inthe training samples are not the same they will differ by afixed step Δαt ie αlabeli+1 αlabeli or αlabeli+1 αlabeli + Δαt-e value should generally be less than the requisite accuracyof direction finding in the application Between αu and αlthe number of sample categories is M

32 Train and Build a MLSSVR Model Given a data set(xi yi)

Li1 xi isin Rd and yi isin Rm-e purpose of multioutput





12trace ΞTΞ1113872 1113873


(6)






minτ w0V Γ( 1113857 12wT

0w0 +12λmtrace VTV1113872 1113873 + c

12trace ΓTΓ1113872 1113873


(7)





T



T

(8)


η cos(α) sin(α)1113858 1113859T



T

(9)



(10)







φkjti







C

1 07821 + 02583j 04576 + 02469j 04576 + 02469j 07821 + 02583j

07821 + 02583j 1 07821 + 02583j 04576 + 02469j 04576 + 02469j

04576 + 02469j 07821 + 02583j 1 07821 + 02583j 04576 + 02469j

04576 + 02469j 04576 + 02469j 07821 + 02583j 1 07821 + 02583j

07821 + 02583j 04576 + 02469j 04576 + 02469j 07821 + 02583j 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(11)






0ndash30

ndash20

ndash10

0

10

20

30

Dire

ctio

n-fin

ding

erro

r (deg)

300 35025020015010050



LSMLSSVR






0ndash100

ndash50

0

50

100

150

200

Dire

ctio

n-fin

ding

erro

r (deg)

300 35025020015010050



LSMLSSVR


0ndash200

ndash150

Dire

ctio

n-fin

ding

erro

r (deg)

300 35025020015010050



LSMLSSVR

ndash100

ndash50

0

50

100

150

200



0

ndash100

ndash50

ndash200

ndash150

0

50

100

150

200

Dire

ctio

n-fin

ding

erro

r (deg)

300 35025020015010050


Different SNRs

LSMLSSVR




LS 3600 40870 102529 159103 201085MLSSVR 29695 52498 79172 113850

0

ndash100

ndash150

ndash200

ndash50

0

50

100

150

200

Dire

ctio

n-fin

ding

erro

r (deg)

300 35025020015010050








5 Conclusion


Data Availability




Acknowledgments


References








































12trace ΞTΞ1113872 1113873


(6)






minτ w0V Γ( 1113857 12wT

0w0 +12λmtrace VTV1113872 1113873 + c

12trace ΓTΓ1113872 1113873


(7)





T



T

(8)


η cos(α) sin(α)1113858 1113859T



T

(9)



(10)







φkjti







C

1 07821 + 02583j 04576 + 02469j 04576 + 02469j 07821 + 02583j

07821 + 02583j 1 07821 + 02583j 04576 + 02469j 04576 + 02469j

04576 + 02469j 07821 + 02583j 1 07821 + 02583j 04576 + 02469j

04576 + 02469j 04576 + 02469j 07821 + 02583j 1 07821 + 02583j

07821 + 02583j 04576 + 02469j 04576 + 02469j 07821 + 02583j 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(11)






0ndash30

ndash20

ndash10

0

10

20

30

Dire

ctio

n-fin

ding

erro

r (deg)

300 35025020015010050



LSMLSSVR






0ndash100

ndash50

0

50

100

150

200

Dire

ctio

n-fin

ding

erro

r (deg)

300 35025020015010050



LSMLSSVR


0ndash200

ndash150

Dire

ctio

n-fin

ding

erro

r (deg)

300 35025020015010050



LSMLSSVR

ndash100

ndash50

0

50

100

150

200



0

ndash100

ndash50

ndash200

ndash150

0

50

100

150

200

Dire

ctio

n-fin

ding

erro

r (deg)

300 35025020015010050


Different SNRs

LSMLSSVR




LS 3600 40870 102529 159103 201085MLSSVR 29695 52498 79172 113850

0

ndash100

ndash150

ndash200

ndash50

0

50

100

150

200

Dire

ctio

n-fin

ding

erro

r (deg)

300 35025020015010050








5 Conclusion


Data Availability




Acknowledgments


References







































C

1 07821 + 02583j 04576 + 02469j 04576 + 02469j 07821 + 02583j

07821 + 02583j 1 07821 + 02583j 04576 + 02469j 04576 + 02469j

04576 + 02469j 07821 + 02583j 1 07821 + 02583j 04576 + 02469j

04576 + 02469j 04576 + 02469j 07821 + 02583j 1 07821 + 02583j

07821 + 02583j 04576 + 02469j 04576 + 02469j 07821 + 02583j 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(11)






0ndash30

ndash20

ndash10

0

10

20

30

Dire

ctio

n-fin

ding

erro

r (deg)

300 35025020015010050



LSMLSSVR






0ndash100

ndash50

0

50

100

150

200

Dire

ctio

n-fin

ding

erro

r (deg)

300 35025020015010050



LSMLSSVR


0ndash200

ndash150

Dire

ctio

n-fin

ding

erro

r (deg)

300 35025020015010050



LSMLSSVR

ndash100

ndash50

0

50

100

150

200



0

ndash100

ndash50

ndash200

ndash150

0

50

100

150

200

Dire

ctio

n-fin

ding

erro

r (deg)

300 35025020015010050


Different SNRs

LSMLSSVR




LS 3600 40870 102529 159103 201085MLSSVR 29695 52498 79172 113850

0

ndash100

ndash150

ndash200

ndash50

0

50

100

150

200

Dire

ctio

n-fin

ding

erro

r (deg)

300 35025020015010050








5 Conclusion


Data Availability




Acknowledgments


References








































0ndash100

ndash50

0

50

100

150

200

Dire

ctio

n-fin

ding

erro

r (deg)

300 35025020015010050



LSMLSSVR


0ndash200

ndash150

Dire

ctio

n-fin

ding

erro

r (deg)

300 35025020015010050



LSMLSSVR

ndash100

ndash50

0

50

100

150

200



0

ndash100

ndash50

ndash200

ndash150

0

50

100

150

200

Dire

ctio

n-fin

ding

erro

r (deg)

300 35025020015010050


Different SNRs

LSMLSSVR




LS 3600 40870 102529 159103 201085MLSSVR 29695 52498 79172 113850

0

ndash100

ndash150

ndash200

ndash50

0

50

100

150

200

Dire

ctio

n-fin

ding

erro

r (deg)

300 35025020015010050








5 Conclusion


Data Availability




Acknowledgments


References





































0

ndash100

ndash50

ndash200

ndash150

0

50

100

150

200

Dire

ctio

n-fin

ding

erro

r (deg)

300 35025020015010050


Different SNRs

LSMLSSVR




LS 3600 40870 102529 159103 201085MLSSVR 29695 52498 79172 113850

0

ndash100

ndash150

ndash200

ndash50

0

50

100

150

200

Dire

ctio

n-fin

ding

erro

r (deg)

300 35025020015010050








5 Conclusion


Data Availability




Acknowledgments


References





































5 Conclusion


Data Availability




Acknowledgments


References





































direction of arrival based on the multioutput least squares support vector...

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