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Signal Processing 88 (2008) 1152–1164

Experimental antenna array calibrationwith artificial neural networks

Hugo Bertrand�, Dominic Grenier, Sebastien Roy

Department of Electrical and Computer Engineering, Universite Laval, Que., Canada G1K 7P4

Received 23 August 2006; received in revised form 12 October 2007; accepted 7 November 2007

Available online 28 November 2007

Abstract

It is well known that to perform accurate Direction of Arrivals (DOA) estimation using algorithms like MUSIC

(MUltiple SIgnals Classification), antenna array data must be calibrated to match the theoretical model upon which DOA

algorithms are based. This paper presents experimental measurements from independent sources obtained with a linear

antenna array and proposes a novel calibration technique based on artificial neural networks trained with experimental

and theoretical steering vectors. In this context, the performance of 3 types of neural networks—ADAptive LInear Neuron

(ADALINE) network, Multilayer Perceptrons (MLP) network and Radial Basis Functions (RBF) network—is assessed.

This is then compared with other calibration techniques, thus demonstrating that the proposed technique works well while

being very simple to implement. The presented results cover operation with a single signal source and with two

uncorrelated sources. The proposed method is applicable to arbitrary array topologies, but is presented herein in

conjunction with a uniform linear array (ULA).

r 2007 Elsevier B.V. All rights reserved.

Keywords: Antenna arrays; Calibration; Artificial neural networks; Array processing; Beamforming; DOA; MUSIC

1. Introduction

Array antenna usage in various communication,radar and instrumentation systems has been grow-ing dramatically over the last few years for severalreasons. The emergence of new antenna shapessimplified their production, given their simplicity.Several such antennas can be integrated, togetherwith some circuitry, into a monolithic small-scaledevice, effectively resulting in low-cost, diminutive

antenna arrays. The advent of increasingly cheapersignal processing chips (DSP) makes possible thesophisticated processing of the plurality of signalsfrom an antenna array at a low-cost, low-powerconsumption, and in a small form factor. Given thedigital processing power available to analog todigital interface can be moved closer to the antenna,thus reducing the complexity and cost of the RFsection. This core signal processing itself can takevarious forms, can include parameter estimation,adaptive filtering and detection.

In the case of this present work, two well-knownclasses of space-time processing algorithms are ofinterest: beamforming and Direction of Arrivals(DOA) estimation.

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www.elsevier.com/locate/sigpro

0165-1684/$ - see front matter r 2007 Elsevier B.V. All rights reserved.

doi:10.1016/j.sigpro.2007.11.006

�Corresponding author

E-mail addresses: [email protected],

[email protected] (H. Bertrand), [email protected]

(D. Grenier), [email protected] (S. Roy).


However, in a real-world, there are some reasonswhich limit the general use of antenna arrays. One ofthem is the precise calibration required such arrayswhen they are used for specific tasks, such asbeamforming and DOA estimation which requirethe acquisition of the precise amplitude and phaserelationships of the signals collected at each element.These relations are unfortunately sensitive to manypotential error sources, leading to severe performanceconstraints. First, phase and gain imbalances betweenthe in-phase (I) and quadrature (Q) branches of eachelement receiver/transmitter subsystem and the I/Qbias errors due to electronic DC offsets cause adivergence from the statistical model upon whichsignal processing algorithms depend; more details onthis source of error are available in [1,2]. Indeed, thesaid algorithms are constructed based on variousassumptions, such as the ideal properties of complexGaussian variates originally postulated by Goodman[3]. Another deviation between theory and practice isdue to the co-channel gain and phase errors, i.e. thevariations between the multiple element receiver/transmitter subsystems themselves. The third errorsource is the mutual coupling effects between theelements comprising the array. The fourth errorsource in an experimental antenna array setup is theelement location errors or uncertainties. Finally,scattering by the antenna mounting structure or othernearby structures constitutes a fifth source. Thesecauses of errors can be considered statics during asufficient period of time.

All of these perturbations consequently affect thespecific structure of the data covariance matrix.Furthermore, the theoretical steering vectors canfind themselves outside the experimental signalsubspace. These effects imply a poor or erroneousperformance by signal processing algorithms, as canbe seen in Section 5 and in [4,5] which analyze theperformance of these algorithms in the presence ofmodel errors.

The solution to all of these problems is inevitablysome form of data calibration to fit the theoreticalmodel or, equivalently, the experimental arraymanifold calibration with respect to the presumedsteering vectors set. Fig. 1 shows a schematicrepresentation of a linear uniformly spaced antennaarray used to perform DOA estimation for M

sources. The RF front-end is detailed in Section 5while the digital processing apparatus is discussed inSections 2 and 4.

A lot of contributions can be found in the litera-ture on the topic of antenna array calibration. Most

are based only on simulation, which by itselfprobably yields an incomplete picture of thisessentially experimental issue. These works cancollectively be slated into three main approaches.First, some authors [6] attempt to bypass thecalibration problem by constructing a new datacovariance matrix which has a Toeplitz structure.The second class is referred to as auto calibration oronline calibration, because it does not require knowl-edge of the directions of the calibrating sources.Finally, the last approach is designated trained

calibration or offline calibration and it impliesknowledge of the calibrating sources’ (pilots) exactbearings.

The first approach is only applicable in the casewhere sensor location errors alone are present. Thesecond one is generally iterative and is not reallyself-sufficient, because such algorithms, globallybased on some form of least-squares fit, requireinitial calibration [7–12], some a priori informationabout a received signal (such as a CDMA code [7,8])and/or it is assumed that part of the response isknown (like in [13] where the sensor gains areassumed known). Also, these iterative algorithmscan potentially converge to a local minimum ratherthan the global one if the initial conditions are notsufficiently close to the solution [7,10,14–16].Furthermore, the majority of these pseudo-self-calibration algorithms have been studied by focus-ing on one or more error sources, but rarely on all.For example, [12,17–20] focus only on the ampli-tude and/or phase errors, [21] on mutual couplingand scattering, [22–25] on I/Q imbalances and[15,26] on sensor position errors or uncertainties.Also, some techniques [27,28] necessitate a differentcalibration step for each error source.

On the other hand, training-based calibrationrequires knowledge of the calibrating sources’angular positions [4,12,14,29–32]. As explained

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Fig. 1. A schematic representation of a linear equispaced array

receiver performing DOA estimation.

H. Bertrand et al. / Signal Processing 88 (2008) 1152–1164 1153


previously, however, these self-calibration algo-rithms are also heavily dependent on variousidealizations.

So, the concept of supervised calibration online isinvestigated and it is found that artificial neuralnetworks (ANNs) constitute a powerful algorithmictool for this application. In the literature, the jointuse of neural networks and antenna arrays servesalmost exclusively to directly estimate the directionsof arrivals of signals impinging on the array [33,34].The principal advantage of neural networks overclassical DOA algorithms is that they can learnfrom non-calibrated data (calibration is implicit tothe learning process), but the training procedure formore than one source is in our view too demanding(i.e. the number of training samples grows in acombinatorial fashion with the number of sources)for a practical implementation [35] and this is whyin previous efforts such as [36] experiments arelimited to the one source case. To the best of theauthors’ knowledge, there is only two papers whichperforms explicit antenna array calibration vianeural networks [37,39], but it provides no detailson the data used to train the network and no explicitresults of the calibration or comparisons with otheralgorithms. Furthermore, the neural network se-lected comprises many neurons and the trainingalgorithm, a variation of the backpropagationalgorithm, seems heavy for this problem, leadingto significant implementation complexity.

The above observation is based on the fact thatour implementation gives excellent experimentalresults even though it is based on a single-layerlinear neural network comprising far fewer neuronsthan the solution presented in [37]. Furthermore, itis trained with the experimental and theoreticalsteering vectors by a low-complexity algorithmbased on least-squares optimization. This simplicitymakes the proposed algorithm an ideal candidatefor real-time low-cost implementation. It shouldalso be noted that, although the experimental arrayused herein is of the uniform linear (ULA) variety,this is not restrictive and the proposed method canbe applied to any topology.

The paper is organized as follows: Section 2describes the antenna array model; Section 3introduces the ANN model; Section 4 describesthe proposed algorithms as well as two otheralgorithms often seen in the literature for compar-ison and Section 5 presents the experimental setupand results. Finally, Section 6 concludes the paperby discussing and summarizing the findings.

2. Antenna array model

We consider here an N-sensor uniform lineararray (ULA) with I/Q receivers at each sensor asseen in Fig. 2, but any other N-sensor arraygeometry can be used. The output of the nth sensoris formed by the in-phase In and the out-of-phaseQn components of the received signal. Fromsampling all receiver outputs, we build the matrixX composed by K independent snapshot vectors xk.We can assume that a snapshot comprises N

Gaussian random components of the Goodmanclass [3] with zero-mean and covariance matrix R.This distribution originates in part from the factthat the signals emitted by the M sources areindependent (and hence uncorrelated). It is alsobased on the assumption of independent Gaussiansamples which is widely used in signal processing[38, Appendix J] and is known to work well whenapplied to linear systems.

Each of these snapshot vectors can be expressed as

xk ¼ ½ðI1 þ jQ1ÞðI2 þ jQ2Þ � � � ðIN þ jQNÞ�T

¼XMi¼1

aiðkÞaðyiÞ þ gk, ð1Þ

where aiðkÞ are complex random Gaussian variableswith zero-mean and variance s2i related to the powerreceived from the ith source, aðyiÞ are the steeringvectors associated with the ith source and gk arevectors of zero-mean additive Gaussian noise withvariance s2Z.

For a ULA, the steering vector of the ith source istheoretically given by

aðyiÞ ¼1ffiffiffiffiffiNp ½1 ejji ej2ji � � � ejðN�1Þji �T, (2)

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Fig. 2. A linear equispaced array.

H. Bertrand et al. / Signal Processing 88 (2008) 1152–11641154


where ji is the electrical delay between eachantenna’s signal due to the inter-antenna distanced, the ith source’s position yi and his wavenumber li

and is expressed

ji ¼2pd

li

sinðyiÞ. (3)

The data matrix can now be expressed as

X ¼ ½x1 x2 � � � xK �

¼ ASþN, ð4Þ

with

A ¼ ½aðy1Þ aðy2Þ � � � aðyMÞ�, (5)

N ¼ ½g1 g2 � � � gK �, (6)

S ¼ ½s1 s2 � � � sK �, (7)

sk ¼ ½a1ðkÞ a2ðkÞ � � � aMðkÞ�T. (8)

In this ideal situation, the data covariance matrix R

is Hermitian Toeplitz and can be approximated by

R � XXH, (9)

where h�iH denotes the complex conjugate transposeoperation. We can perform its eigendecomposition:

R ¼XMi¼1

livivHi þ

XN

i¼Mþ1

livivHi

¼ VsKsVHs þ s2ZI, ð10Þ

where flig are the N eigenvalues in decreasingmagnitude order, associated with the eigenvectorsfvig. The Hermitian symmetry of this matrix ensuresthat the eigenvalues are real and the eigenvectors areorthogonal. The first M eigenvectors constitute thesignal subspace and the others form the noisesubspace.

We know that the M first eigenvectors span thesame subspace as the M assumed steering vectorsaðy1Þ to aðyMÞ and the other eigenvectors define asubspace orthogonal to the steering vectors.

This structure is exploited by classic DOAalgorithms. For example, the MUSIC (MUltipleSIgnals Classification) algorithm [40] is based on theordinary Euclidian distance between all the possiblesteering vectors and the signal subspace and can beexpressed as

PmusicðyÞ ¼ 20 log101

aHðyÞPnaðyÞ

� �, (11)

where aðyÞ is built like (2) using y as a scanningangle variable and Pn ¼ I� VsV

Hs is the projector to

the noise subspace which is perpendicular to thesignal subspace spanned by the eigenvectors Vs.Eq. (11) is called the MUSIC pseudospectrum.

Unfortunately, when experimental data from anuncalibrated array are used with this algorithm, ityields erroneous results (as can be seen in Fig. 13).This is due to the fact that the theoretical steeringvectors do not lie exactly in the experimental signalsubspace. It follows that we have to modify theunderlying theoretical model to take into accountthe effects of the non-ideal behavior of the experi-mental antenna array.

Based on the simplifying assumption that theproblem is linear, the experimental data matrixcollected from an experimental uncalibrated anten-na array is now given by

X ¼ CASþN, (12)

where C is a complex distortion square matrix whichembodies all the non-ideal effects due to imperfec-tions in the experimental array. If the array presentsonly gain and phase errors between the I and Qbranches of each sensor, C is diagonal. In thecases of branch-to-branch gain and phase mis-matches and mutual coupling, C includes off-diagonal elements. Sensor location errors can alsobe taken into account, since they constitute anothersource of branch-to-branch gain and phase errors.It follows that all non-ideal effects of the experi-mental setup can be encapsulated in this distortionmatrix.

Hence, the goal of the calibration procedure is tocompensate for the distortion matrix—which em-bodies all linear imperfections—in addition to anynon-linear distortions which might be present.

3. ANN model

ANNs are used herein as a distributed parallelcomputing mechanism which establishes a mappingbetween the input and output spaces of the network,like a lookup table with intrinsic interpolationbetween samples. This mapping is constructedimplicitly via supervised learning, i.e. the presenta-tion to the network of training samples comprisingan input signal and a corresponding desiredresponse. During training, the values of weightsand biases are modified to minimize the errorbetween the output of the network and the desiredone until the network converges to a steady state.

Fig. 3 presents the model of an ANN with twolayers in matrix notation. The network output y2 is

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linked to the input p via

y2 ¼ F2ðn2Þ ¼ F2ðW2½F1ðW1pþ b1Þ� þ b2Þ, (13)

where fWig are the weight matrices, fbig the biasvectors, fyig the output vectors and Fi are theactivation functions of the ith layer.

The proposed calibration technique has beenimplemented with different types of ANNs, i.e.ADAptive LInear NEuron (ADALINE), Multi-layer Perceptrons (MLP) and Radial Basis Func-tions (RBF). The simplest type of neural networkused is ADALINE [41], which consists of a singlelayer and a linear activation function as illustratedin Fig. 4.

The supervised training of this network is basedon an approximation of the minimization of thesum of the squared errors between the outputs ofthe network and the desired responses, as describedin Section 4.2.

The output y of this network is obtained with

y ¼Wpþ b. (14)

The second type of neural network used is a form ofMLP which consists of two neuron layers, the firstone having a logsigmoid activation function

F1 ¼ logsigðnÞ ¼1

1þ e�n, (15)

and the second one having the linear activationfunction illustrated in Fig. 5. The weights and biasesof the network are iteratively adjusted to minimizethe network performance function, i.e. the averagesquared error between the network outputs and thetarget outputs. This iterative training procedure is

performed using the backpropagation algorithm[42] which uses the negative of the gradient of theperformance function to update the values of biasesand weights at each iteration.

The output y2 of this network is obtained with

y2 ¼W2y1 þ b2 ¼W2 logsigðW1pþ b1Þ þ b2. (16)

The third neural network used to perform thecalibration procedure is an RBF comprising twolayers of neurons. The first one is composed ofradial basis functions

F1 ¼ radbasðnÞ ¼ e�n2 , (17)

and the second one of linear activation functions, asillustrated in Fig. 6. The training procedure of thisnetwork is quite different from the others: theweights of the first layer are fixed to the values of thetraining patterns, the biases are all equal to aconstant value determined by

b1 ¼0:8326

SPREAD, (18)

where SPREAD is a numerical value foundempirically which determines the width of an areain the input space to which each neuron responds.The second layer is trained in a manner identical tothe ADALINE network. The output y2 of thisnetwork is obtained with

y2 ¼W2y1 þ b2 ¼W2 radbasðkW1 � pk � b1Þ þ b2,

(19)

where kW1 � pk is the Euclidean distance betweeneach rows of W1 and p, and � is the Hadamardproduct.

ARTICLE IN PRESS

Fig. 3. ANN model.

Fig. 4. ADALINE neural network mdel.

Fig. 5. MLP neural network model.

Fig. 6. RBF neural network model.



The choice of the ADALINE neural network ismotivated by its simplicity and relies on the linearassumption discussed earlier. The MLP structure isdrawn from [37] where it is used for calibrationpurposes and serves as a performance benchmark.Indeed, it constitutes a rare example of a publishedapplication of neural networks to the calibrationproblem. Finally, the RBF neural network isincluded because it can be quickly and easilydesigned and constitutes a natural solution in amapping application between the input and outputspaces of the network. Both the MLP and RBFnetworks perform non-linear processing (whichimplies higher complexity) and can therefore helpto determine whether the linear assumption isjustified.

For further details on the neural network modelsand associated learning algorithms, see [41–43].

4. Calibration algorithms

In what follows, we review existing algorithmsbefore presenting the proposed algorithm in relationwith the state of the art.

4.1. Known algorithms used for comparison

As stated previously, the theoretical steeringvectors aðyiÞ do not lie completely in the experi-mental signal subspace without calibration. Mostprevious works (see for example [14]) propose tofind a calibration matrix G, it being the inverse of Cin (12) in the least-squares sense. Mathematically,we have

G ¼ minGkA�GAek, (20)

where Ae denotes an estimate of the experimentalsteering vectors obtained via eigenanalysis of thespatial covariance data matrix. The matrix A is thetheoretical one obtained from (2) given the exactposition of sources.

The minimization (20) can be solved to yield [42]:

G ¼ AðAH

e AeÞ�1A

H

e . (21)

The calibration procedure is then simply

Y ¼ GX, (22)

where Y is the calibrated data matrix. The majorproblem with this method is the possibility ofconvergence to a local minimum rather than theglobal one.

Another calibration algorithm [22], which cor-rects only I/Q imbalances, is based on the principleof the Gram–Schmidt orthogonalization. Considerthe in-line I i and quadrature Qi signals at eachantenna

I iðtÞ ¼ ð1þ �ÞA cosðotÞ þ �I ,

QiðtÞ ¼ A sinðotþ fÞ þ �Q, ð23Þ

where � represents the amplitude difference betweenthese components, �I and �Q the signal offsets and fthe phase imbalance between them. First of all, theDC offsets �I and �Q must be removed to obtain newsignals I 0i and Q0i

I 0iðtÞ ¼ ð1þ �ÞA cosðotÞ,

Q0iðtÞ ¼ A sinðotþ fÞ. ð24Þ

By virtue of the Gram–Schmidt procedure, a scalarE must be found to normalize I 0i and a part P of I 0imust be subtracted from Q0i to complete theorthogonalization and finally obtain signals I 00i andQ00i . In matrix notation, we have

I 00i ðtÞ

Q00i ðtÞ

" #¼

E 0

P 1

� �I 0iðtÞ

Q0iðtÞ

" #.

The desired result being

kI 00i ðtÞk ¼ kQ00i ðtÞk,

I 00i ðtÞ

Q00i ðtÞ¼ e�j

p2, ð25Þ

we find after some algebraic manipulations

E ¼cosðfÞ1þ �

,

P ¼� sinðfÞ1þ �

. ð26Þ

The major advantage of this technique is itsautonomy, i.e. it does not rely on a set of knownsources for training purposes. Hence, it can bereferred to as absolute calibration; its performance,however, is poor compared to our expectation asillustrated in Section 5.

4.2. Proposed algorithm

The originality of the proposed approach centerson the use of ANNs to perform calibration. ANNscan easily provide real-time solutions thanks to theirrelatively low computational complexity and theirmostly parallel architecture which makes themnatural candidates for implementation in VLSI[44–46]. Experiments have shown that they also



alleviate the problem of convergence to a localminimum since they typically always converge tothe global one when training and structure areadequate. We find that a linear neural networktrained with L input/target vector pairs formed bythe real and imaginary parts of the experimental andassumed steering vectors gives excellent results evenif the learning algorithm is very simple and thenumber of neurons is small.

Fig. 7 shows a block diagram overview of theDOA processing chain with the proposed calibra-tion approach. It consists of the standard sequenceof operations detailed in Section 2 which lead to theMUSIC pseudospectrum, with the inclusion at thefront end of the neural network which compensatesfor all distortions by minimizing a single globalquality measure, namely the mean-square error(MSE). At first, we begin by the learning process

which adjusts the values of the weights and biases byminimizing an approximation of MSE between theoutput vectors yðkÞ (acalðkÞ, the calibrated steeringvectors) and the target vectors tðkÞ (aðkÞ, the exactsteering vectors) for the input vectors pðkÞ (aeðkÞ, theexperimental steering vectors), i.e.

MSE ¼ F ðeÞ �1

L

XL

k¼1

X2N

i¼1

ðeiðkÞÞ2¼

1

L

XL

k¼1

eðkÞTeðkÞ,

(27)

with eðkÞ ¼ tðkÞ � yðkÞ. This performance index F isa quadratic form which ensures the convergence ofthe training process to a global minimum (thequadratic error surface is monotonic), as confirmedby the experimental results. Using the LMS algo-rithm [43], which approximates the MSE-basedsteepest descent procedure by using the instanta-neous squared error as an estimate of the MSE ateach iteration, the rules to update the network

parameters at iteration k þ 1 are

Wðk þ 1Þ ¼WðkÞ þ 2aeðkÞpTðkÞ,

bðk þ 1Þ ¼ bðkÞ þ 2aeðkÞ, ð28Þ

where a is the step size which is empirically chosento ensure convergence.

One source located at various known angles ofarrival is used at this step. Then, only theeigenvector associated to the highest eigenvalue atthe output of the eigendecomposition box pðkÞ (seeFig. 7) is taken for the training. This eigenvector isequivalent to the experimental steering vector ofthe source aeðkÞ for each angle of arrival. TheANN adjusts its parameters to give a calibratedsteering vector acalðkÞ very close to the exact steeringvector aðkÞ.

The calibration based on the two other consid-ered types of neural networks gives similar resultswhile requiring longer training (MLP) or compris-ing more neurons (RBF), as detailed below.

Once the training is complete, the DOA proces-sing can now be applied with unknown sources. Inthis step, all eigenvectors are putted in the input ofthe ANN to obtain calibrated eigenvectors. The M

eigenvectors associated to the highest eigenvaluesspan the same signal space that one spanned by thecorresponding calibrated steering vectors. So, weget Pncal which is taken to compute the ordinaryEuclidian distance as in (11).

5. Experimental setup and results

Fig. 8 shows the experimental antenna arraycomprising 8 horn antennas operating in the X band(8–12GHz). While the design of this setup was asrigorous as possible, it presents all the imperfectionsdiscussed previously, i.e. phase and gain imbalances

ARTICLE IN PRESS...

Computeeigendecomposition

ComputenetworkNeural

Trainingset

minimizationLMS–based MSE

+

Computepseudospectrum

MUSICRF

andADCs

p [k]

R = XX H

y [k]

t [k]

Pn

Fig. 7. Block diagram of DOA processing chain yielding MUSIC pseudospectrum (11) with neural network-based array calibration.



between the in-phase (I) and quadrature (Q)branches of each antenna receiver subsystem, I/Qbias errors, branch-to-branch gain and phase errorsdue to the variations between the antenna receiversubsystems, mutual coupling effects between theantennas on the array and sensor location errors oruncertainties and potentially scattering by the anten-na mounting structure or other nearby structures.

The numbers in Fig. 8 refer to Fig. 9 whichpresents a block diagram of the experimental setupfor a one-antenna receiver subsystem. The differentelements are: 1. horn antenna, 2. waveguide WR-90to SMA adaptor, 3. low noise RF amplifier andmixer, 4. RF generator, 5. RF amplifier, 6. powerdivider 1–8 (for each subsystem), 7. low frequencyamplifier, 8. power divider 1–2 (for I/Q branches), 9.low frequency signal generator, 10. quadraturepower divider, 11. power divider 1–8 (for eachsubsystem) and 12. low frequency mixer.

The effects of branch-to-branch gain and phaseerrors and sensor location errors are illustrated inFig. 10.

Fig. 11 shows the calibration impact on signalswith the proposed algorithm: branch-to-branch gainand phase errors are corrected, but it was observedthat the calibration procedures do not alleviatecompletely the I/Q imbalances: the Lissajou I/Qplot [47] is still an ellipse as shown in Fig. 12 for atypical branch.

Finally, the compound effects of these perturba-tions on the performance of DOA algorithms isobvious when looking at Fig. 13, which presents thepseudospectra obtained before and after calibrationfor one source at 0�.

Given the effects of the imperfections of theexperimental setup, details of the calibration procedurefollow. Measurements are taken with one source in theX band at each half degree between�20� and 20�. Foreach measurement, the principal eigenvector of thespatial covariance matrix is used as the experimentalsteering vector, while the assumed steering vector iscreated based on the measured source position. Thisoperation gives us the input/output training set.

Once the training set is obtained, the experimen-tal steering vectors obtained from the users’ signalscan be calibrated to obtain their accurate DOAs.

Fig. 13 presents the excellent calibration result forone source at array broadside ð0�Þ with a training withall sources. We can see that the best performance, thehighest peak which corresponds to the shortestdistance between the signal subspace and the calibra-tion steering vector, is obtained by RBF, followed byMLP, ADALINE, G matrix and finally the I/Qcalibration procedure. These results indicate that for aDOA in the training set, all trained algorithms give agood solution and we also conclude that the I/Qimbalances are less important for the performance ofDOA algorithms than the branch-to-branch ones.

To compare the performance of these techniqueswhen the DOA is not part of the training set, wecalculate the mean angle yaverage between thecalibrated acal and the theoretical a steering vectorsfor different sizes L of the training set

yaverage ¼1

L

XL

i¼1

arccoskaHðiÞacalðiÞk

kaHðiÞkkacalðiÞk

� �. (29)

ARTICLE IN PRESS

Fig. 8. The antenna array.

Fig. 9. Receiver chain block diagram in one antenna branch.



These results are presented in Fig. 14. We canconclude that the performance of all these trainedalgorithms is much poorer when the training set issmaller (as usual) and the best tradeoff between

performance and ease of implementation is pro-vided by ADALINE.

A neural networks comparison based on their sizefollows in Table 1. In summary, the size of the

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0 20 40 60 80 100 120 140 160 180 200

0

0.05

0.1

0.15

Snapshots

Am

plit

ud

e

Signal 1Signal 3Signal 5Signal 7Signal 9Signal 11Signal 13Signal 15

–0.2

–0.15

–0.1

–0.05

Fig. 10. The 8 uncalibrated in-line (I-branch) signals with a source at 0�.

0 20 40 60 80 100 120 140 160 180 200

0

1

2

3

4

5x 10–3

Snapshots

Am

plit

ud

e

Signal 1

Signal 3

Signal 5

Signal 7

Signal 9

Signal 11

Signal 13

Signal 15

–1

–2

–3

–4

–5

Fig. 11. The 8 calibrated in-line (I-branch) signals with a source at 0�.



ADALINE network depends only on the number ofantennas N; the RBF network depends on thenumber of antennas N and the size of the trainingset L; the size of the second layer of the MLPnetwork depends on the number of antennas N andthe first layer can be of any size, but empirically weconclude that to obtain good results, we have to fixthis first layer the same size as the second one.

To demonstrate the good generalization capacityof the neural networks, they were used to calibratenew experimental steering vectors from the spatialcovariance data matrix of two uncorrelated sources.

The results of two independent experiments ob-tained with the neural networks are compared inFigs. 15 and 16 with the same obtained with themethod of the calibration matrix G in (21).

The experiments are limited to two typical trialswith two sources and the results are sufficientlysatisfactory for our purposes: we can see in Figs. 15and 16 that the proposed technique based onADALINE generalizes well and outperforms theG matrix method while its computational complex-ity is of the same order. But the two other types ofneural networks, MLP and RBF, are less attractivedue to the presence of more peaks in the pseudos-pectrum than the real number of sources as can beseen in Figs. 15(a) and 16(a).

This can be explained given the fact that thetraining of a neural network can be seen as afunction approximation of the input vectors. Hence,neural networks with too many degrees of freedomare probably too non-linear for this application and

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0–0.02

–0.02

–0.04

–0.04

–0.06–0.06

0.02 0.04 0.06

0

0.02

0.04

0.06

I

Q

Fig. 12. Lissajou plot of antenna 1’s in-line (I) and quadrature

(Q) signals.

0 5 100

5

10

15

20

35

25

30

ADALINEMLPRBFGI/QNo-calib

Pm

usic(�

) (d

B)

� (degrees)

–5–10

Fig. 13. Pseudospectrum obtained by MUSIC algorithm with

calibrated and uncalibrated data of a source at 0�.

1 1/2 1/81/51/41/3 1/6 1/70

35

40

5

10

15

20

25

30

ADALINEMLPRBFG

Angle

Betw

een V

ecto

rsTraining Ratio

Fig. 14. Angle between calibrated and theoretical steering vectors

for training set sizes. A unit training ratio implies a half-degree

grid, while a training ratio of 1Nimplies an N

2degree grid, i.e. one

out of N training positions are retained from the original grid.

Table 1

Neural networks size comparison

W1 b1 W2 b2

ADALINE 2N 2N 2N 1 – –

RBF L 2N L 1 2N L 2N 1

MLP 2N 2N 2N 1 2N 2N 2N 1



will introduce unnecessary features, like curve fittingwith a high-degree polynomial.

6. Conclusion

An original calibration technique based on neuralnetworks was presented herein, trained by theore-tical and experimental steering vectors. We find thatthe ADALINE neural network provides better—or

very similar—results on average than all the othertypes of calibration techniques while being easy toimplement in real-time. After many experimentaltrials it seems to always converge experimentally tothe global minimum. The two other types of neuralnetworks seem to be too non-linear for thisproblem, while being 1—longer to train (MLP) or2—having many more neurons (RBF). Also, experi-ments have shown that branch-to-branch errors are

ARTICLE IN PRESS

0 50

35

5

10

15

20

25

30

ADALINEMLPRBFG

� (degrees)

Pm

usic(�

) (d

B)

� (degrees)

Pm

usic(�

) (d

B)

0 50

2

4

6

8

10

12ADALINEG

–10 –5–15–20

–10 –5–15–20

Fig. 15. Pseudospectrum obtained via the MUSIC algorithm

with calibrated data of uncorrelated sources at �11� and �6�

using various calibration techniques: (a) all techniques, (b) two

best techniques only (ADALINE and G matrix).

0 5 10

0

5

10

15

20

25

30ADALINEMLPRBFG

� (degrees)P

mu

sic(�

) (d

B)

Pm

usic(�

) (d

B)

0

2

4

6

8

10

12ADALINE

G

–10 –5–5

–15

0 5 10

� (degrees)

–10 –5–15

Fig. 16. Pseudospectrum obtained via the MUSIC algorithm

with calibrated data of uncorrelated sources at �5� and �0�

using various calibration techniques: (a) all techniques, (b) two

best techniques only (ADALINE and G matrix).



more critical for the performance of DOA algo-rithms than the I/Q one in an experimental context.

Moreover, the real-time capacity of the betteralgorithm, based on ADALINE neural networks,will be tested with a VLSI implementation with anew setup currently being developed comprising 3antenna arrays of 8 elements equipped with FPGAsand DSPs for real-time signal processing. This newsetup will allow us to obtain more results indifferent cases (e.g. various antenna array config-urations, training grids with different angularspacings, etc.) to generalize the behavior of thealgorithm facing the selection of different para-meters and derive practical insights with respect tothe desired resolution and the variability of thearray response in certain specific directions. Also, amore exhaustive study implying Monte Carlosimulations will permit us to obtain the perfor-mance statistics and compare them with othercalibration methods.

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