Research Article An Effective Technique for Enhancing Direction Finding Performance...

Research ArticleAn Effective Technique for Enhancing Direction FindingPerformance of Virtual Arrays

Wenxing Li Xiaojun Mao Wenhua Yu and Chongyi Yue

College of Information and Communication Engineering Harbin Engineering University Harbin 150001 China

Correspondence should be addressed to Xiaojun Mao wwwmaoxiaojun126com

Received 8 July 2014 Revised 22 August 2014 Accepted 22 August 2014 Published 30 December 2014

Academic Editor Dau-Chyrh Chang

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

The array interpolation technology that is used to establish a virtual array from a real antenna array is widely used in directionfinding The traditional interpolation transformation technology causes significant bias in the directional-of-arrival (DOA)estimation due to its transform errors In this paper we proposed a modified interpolation method that significantly reduces biasin the DOA estimation of a virtual antenna array and improves the resolution capability Using the projection concept this paperprojects the transformation matrix into the real array data covariance matrix the operation not only enhances the signal subspacebut also improves the orthogonality between the signal and noise subspace Numerical results demonstrate the effectiveness of theproposed method The proposed method can achieve better DOA estimation accuracy of virtual arrays and has a high resolutionperformance compared to the traditional interpolation method

1 Introduction

Array signal processing has a wide range of applications inradar communications sonar and acoustics Interpolationor mapping technique from a real antenna array to a virtualantenna array is a popular topic in array signal processing [1]Virtual array interpolation (mapping) technique was intro-duced in 1980s [2ndash5] Virtual array transformation can mapan arbitrary planar structure antenna array to be a uniformlinear array- (ULA-) type array and can increase the degreesof freedom (DOF) of an antenna array which is widely usedin the DOA estimation and adaptive beamforming [6 7]

There exist some efficient DOA estimation algorithmssuch as ldquoroot-MUSICrdquo and ldquoroot-WSFrdquo which are normallyrestricted to the ULA geometry These algorithms cannotbe used for the conformal arrays or uniform circular array(UCA) directly however the interpolation transform tech-nique can solve this problem efficiently [8] In Friedlanderrsquosvirtual array transformation (VAT) method [9] an arbitraryshaped antenna array can be transformed into a desiredvirtual antenna array by interpolating the interested scanningsector The basic idea of Friedlanderrsquos VAT is to dividethe spatial region into several subregions and then bytaking interpolation in the array scanning sector of interest

the manifold of the virtual array can be obtained by the linearinterpolation of the manifold of the real array The transformmatrix is computed as the least squares solution

The performance of array interpolation in the DOA esti-mation has been reported in the literature [10ndash12] Althoughthe array interpolation approach has some attractive proper-ties [13 14] an essential shortcoming of the method is that itoften introduces mapping errors that cause bias in the DOAestimation Hence the DOA estimations are not statisticallyoptimal

Pesavento et al developed a robust interpolation methodin [15] The method ensures that the interpolation transformerror of the selected sectors is minimal while the conversionoutside region sets up multiple ldquostop bandsrdquo to make theconversion error less than a criterion

Hyberg et al proposed a geometrical interpretationmethod of a Taylor series expansion of the DOA estimatorcriterion function to derive an alternative design of themapping matrix [16 17] The proposed design considers theorthogonality between the manifold mapping errors andcertain gradients of the estimator criterion function Thisbias-minimizing theory was extended to not only minimizebias but also consider finite sample effects due to noisereducing the DOA mean-square error (MSE) DOA MSE is

Hindawi Publishing CorporationInternational Journal of Antennas and PropagationVolume 2014 Article ID 728463 7 pageshttpdxdoiorg1011552014728463

2 International Journal of Antennas and Propagation

2d0 d (N minus 1)dx

y

middot middot middot

Figure 1 The structure of ULA

not reduced by minimizing the size of the mapping errorsbut instead by rotating the errors and the associated noisesubspace into the optimal directions related to a certaingradient of the DOA estimator criterion function in [17]The two methods show the superior performance on theDOAMSE reduction of array interpolation but its derivationcomputed procedure is complex

This paper proposed a novel method to reduce the DOAbias of virtual interpolation Using the projection conceptproject the transformation matrix onto the real array covari-ance matrix to enhance the signal subspace which improvesthe orthogonality between the signal and noise subspaceTheproposed method is efficient and easy to realize Numericalsimulations verify that this method can get better estimationaccuracy and has a high resolution performance compared tothe traditional interpolation method

2 Signal Model

Considering an omnidirectional array with 119873 elements illu-minated by 119872 narrow band signals the distance of the arrayelements is 119889 as shown in Figure 1

The signal 119904119896(119905) is incident in the direction 120579119896 the receivedsignal X can be expressed as follows

X (119905) = AS (119905) + N (119905) (1)

where X(119905) = [1199091(119905) 1199092(119905) 119909119873(119905)]119879 is a 119873 times 1 snap

data vector S(119905) = [1199041(119905) 1199042(119905) 119904119872(119905)]119879 is a vector

containing the complex signal envelops of 119872 narrow-bandsignal sources N(119905) = [1198991(119905) 1198992(119905) 119899119873(119905)]

119879 is a vector ofzero-mean spatial white sensor noise of variance 120590

2

119899 A is an

array manifold matrix namely A = [a(1205791) a(1205792) a(120579119872)]where a(1205791) = [1 119890

119895120573119896 119890119895(119873minus1)120573119896]

119879 119896 = 1 2 119872

represents a steering vector in the 120579119896 direction and 120573119896 is thephase difference that can be represented as

120573119896 =2120587

120582119889 sin (120579119896) (2)

Assume that the signal and noise are linearly indepen-dent and then the data covariance is written in the formatbelow

R = 119864 X (119905)X119867 (119905) = AR119904A119867

+ 1205902

119899I (3)

where 119864 denotes the expectation operator R119904 =

119864119878(119905)119878119867(119905) represents the autocorrelation matrix of

signal complex envelops 1205902119899is the noise power I is the unit

matrix and (sdot)119867 denotes the matrix conjugate transposition

Real antenna unit

Virtual antenna unit

Figure 2 Real antenna and interpolated arrays

In practice the desired signal is often present in the snap-shots The sample array covariance matrix can be expressedas follows

R =1

119870

119870

sum

119894=1

X (119894)X119867 (119894) (4)

where 119870 is number of snapshots VAT is based on inter-polation technique [8] in which the entire antenna arrayscanning vector is divided into several subregions and thesubregion of interest will be segmented through a certaintransformation to realize themapping from the original arrayto the corresponding virtual array

3 Conventional Interpolated Array

Consider a real ULA transformed into a virtual ULA via arrayinterpolation as illustrated in Figure 2

Assume that there is a signal located in the region Θ weequally divide Θ into

Θ = [120579119897 120579119897 + Δ120579 120579119897 + 2Δ120579 120579119903 minus Δ120579 120579119903] (5)

where 120579119897 and 120579119903 are the left and right boundary of regionΘ respectively Δ120579 is size of the interpolation step which isdetermined by the specified accuracy

The real array manifold matrix in the chosen area can beexpressed as follows

A = [a (120579119897) a (120579119897 + Δ120579) a (120579119897 + 2Δ120579)

a (120579119903 minus Δ120579) a (120579119903)] (6)

where a(120579119897) represents the steering vector of a real array in the120579119897 directionThe array manifold matrix of virtual array in thesame area Θ is expressed as follows

A = [a (120579119897) a (120579119897 + Δ120579) a (120579119897 + 2Δ120579)

a (120579119903 minus Δ120579) a (120579119903)] (7)

where a(120579119897) represents the steering vector of a virtual arrayin the 120579119897 direction There must exist a mapping relationshipbetween the real and the virtual array vectors Then aninterpolation matrix B is designed to satisfy the least squarethat is

minB

10038171003817100381710038171003817B119867A minus A10038171003817100381710038171003817

2

119865 (8)

International Journal of Antennas and Propagation 3

where sdot 119865 denotes the Frobenius norm for amatrixThe realand the virtual array manifold vectors satisfy the followingrelationship

B119867A (120579) = A (120579) 120579 isin Θ (9)

And their steering vectors satisfy the following equation

B119867a (120579) = a (120579) 120579 isin Θ (10)

When the number of a transformed array is greater thanthe actual number of antenna elements and the matrix Ahas a nonzero condition value by solving (8) the virtualtransformation matrix B is

B = (AA119867)minus1AA119867 (11)

Define the transformation error

119864 (B) =

minB10038171003817100381710038171003817B119867A minus A1003817100381710038171003817100381711986510038171003817100381710038171003817A10038171003817100381710038171003817119865

(12)

In an ideal case there is no error in the virtual trans-formation 119864(B) should be zero However in practice sinceinterpolation points in the transformation area infinities arelimited the interpolation operation often introduces map-ping errors These preprocessing techniques often introducemapping bias and excess variance in the DOA estimationsHence the estimations are not statistically optimal [6]

4 Modified Interpolated Method

In this section we describe a modified interpolation algo-rithmWe set the data covariancematrix of the real array R asa projection matrix After obtaining the transform matrix Baccording to (11) we reconstruct the transformationmatrixBby projecting it to the sample array covariance matrix

B = RB (13)

For a given transformationmatrix B we can compute thecovariance matrix of a virtual antenna array

R = BAR119904A119867B119867 + B (120590

2

119899I)B119867

= RBAR119904A119867B119867R119867 + 120590

2

119899RBB119867R119867

(14)

The above procedure can enhance the signal componentsin the virtual covariance matrix R and improve the orthogo-nality between the signal and the noise subspace

We can clearly see that BB119867 = I which implies thatthe original white noise turns into the colored noise after thevirtual transformation For most DOA estimation algorithmscan only work when the background noise is Gaussian whitenoise and the colored noise must be prewhitened Define thetransformation matrix as

T = (B119867B)minus12

B119867 (15)

Virtual subarray interpolator

Projection process

Prewhiten noise

Virtual array output

Real antenna array data X

X

BX

B = RB

Figure 3 Construction of modified interpolated approach

The real antenna array steering vector a(120579) and the virtualarray steering vector a(120579) have the following relationship

Ta (120579) = (B119867B)minus12

a (120579) =

_a (120579) 120579 isin Θ (16)

After the noise prewhitening above the covariancematrixof the virtual antenna can be computed by using the transfor-mation matrix T

Consider_R = TRT119867 =

_A R119904

_A119867

+ 1205902

119899I (17)

Therefore the covariance matrix of the virtual antenna isobtained and the application of a direction finding estimatorto (17) is straightforward [6]

To summarize the modified interpolation transforma-tion technique the transformation procedure is shown inFigure 3

In this paper the multiple signal classification (MUSIC)algorithm is used to estimate the DOA MUSIC algorithmis a high resolution technique based on exploiting theeigenstructure of an input covariance matrix We decomposethe autocorrelation matrix into signal and noise subspaces

The covariance matrix_R can be written as

_R = U119878Σ119878U

119867

119878+ U119873Σ119873U

119867

119873 (18)

where U119878 represents the signal subspace U119873 represents thenoise subspace Σ119878 = diag1205821 1205822 120582119872 is the signaleigenvalue Σ119873 = diag120582119872+1 120582119872+2 120582119873 is the noiseeigenvalue The noise subspace U119873 is orthogonal to all119872 signal steering vectors The spectrum of the MUSICalgorithm is given by

119875MUSIC =1

a119867 (120579)U119873U119867119873a119867 (120579)=

1

1003817100381710038171003817U119867119873a (120579)1003817100381710038171003817

(19)


Table 1 Comparison of the two methods

Notion of DOA RMSE reduction Computational complexityof mapping matrix T

Additional priorinformation compared to

[8]

Array interpolation of [17]Rotate the mapping errors and noise

subspace into optimal directions relativeto a certain gradient of the DOAestimator criterion function

119874(2(2119873 + 1)2

1198732

cal119873119873)

Complex gradient ofcriterion of the used

estimator

Proposed methodProject the transformation matrix on thereal array covariance matrix to strengthen

thesignal subspace119874(119873119873

3) None

If 120579 is equal to DOA the noise subspaceU119873 is orthogonalto the signal steering vectors and U119867

119873a(120579) becomes zero

when 120579 is a signal direction and the denominator is identicalto zero It is obvious that in practice U119867

119873a(120579) = 0 due to

finite samples If this happens the performance of MUSICalgorithm will not be optimal

Now we can summarize the modified VAT procedure asfollows

Step 1 Compute the real array covariance matrix R

Step 2 Compute the real array manifold A(120579) and virtualarray manifold A(120579) and then compute the transformationmatrix B using (11)

Step 3 Take the projection operation to get the new transfor-mation matrix B using B = RB

Step 4 Compute the covariance matrix_R of the virtual array

from the covariance matrix R of the real array

Step 5 Apply the MUSIC algorithm to the covariance_R in

(19)

We compare the proposedmethod with Hybergrsquos methodin [17] which is selected for comparison because of itssuperior performance on DOA mean-square error (MSE)reduction of array interpolation In [17] the authors proposeda design algorithm for the mapping matrix that minimizedthe DOA estimate bias The MSE-minimizing mappingmatrix 119879 is designed as

Topt = arg minT

119873cal

sum

119894=1

(1 minus 120583)10038171003817100381710038171003817Δ119890(119894)

V10038171003817100381710038171003817

2

+120583

2 (120579(119894) 119890(119894)V )

times[4 (Re 119892(119894)V Δ119890(119894)

V )2

+

119898

sum

119896=2

2120572(119894) 10038161003816100381610038161003816

119892(119894)

V T119867119890(119894)119896

10038161003816100381610038161003816

2

]

(20)

where 120583 is a weighting factor (0 le 120583 lt 1) and 119892(119894)

V ≜ 119892V(120579(119894))

are the gradient vectors 119873cal is the number of calibrationdirections In general the superscript (119894) means that thecorresponding quantity should be computed as if there werea single source in the 119894th calibration direction (see [17] fordetails) DOA MSE is not reduced by minimizing the size ofthe mapping errors but instead by rotating these errors andthe associated noise subspace into optimal directions relativeto a certain gradient of the DOA estimator criterion functionWe can clearly see that criterion in (20) is a quadratic functionof the elements of T The characteristics of the method of [17]and the proposed method are listed in Table 1

The comparisons of the two methods are given in Table 1where we can see that method in [17] is much more complexthan our method which need to calculate the complexgradient of criterion of the used estimator at first compared to[8]The calculation of references [17] is more than eight timeshigher than the proposedmethod Simulation is conducted toevaluate the performance of the different methods

5 Numerical Examples

In this section the estimation accuracy of the proposedinterpolationmethod and the conventional approach [3 8 9]is evaluated through numerical simulations

Numerical Experiment 1 The real array is uniform andlinear with 4 elements and the element space is 120582 Thenondirectional noise is spatial white Gaussian with a unitvariance The virtual antenna array is uniform and lineararray with 8 elements and element space 1205822 There arefour independent signals arriving from the directions minus550∘minus500∘ 100∘ and 150∘ The signal-to-noise ratio (SNR) ofthe six signals is 10 dB and the virtual transform sector is[minus60∘ minus40∘] cup [0

∘ 20∘] The step size is 01

∘ The number ofsnapshots is 128 200 Monte Carlo runs are used to obtaineach point

The simulation results of conventional method and theproposed method are shown in Figure 4 It can be seenfrom the Figure 4 that the real array has 3 DOFs whichcan process 3 signals at most while the virtual antennaarray has 8 elements with 7 DOFs and can process morethan 3 signals But the conventional interpolation methodfails to distinguish the two close signals (minus550∘ minus500∘) and(100∘ 150∘) for its large transform errors The modified


0 20 40 60 80

0

20

40

60

80

100

Angle (deg)

Pmus

ic (d

B)

Conventional interpolation approachModified interpolation approach

minus20

minus20minus80 minus60 minus40

Figure 4 Spatial spectrum of MUSIC algorithm for two methodscomparison

interpolation method can distinguish the two very closesignalsTheDOA finding results is (minus554∘ minus492∘ 101∘ and144∘) and the result is accurate We also can see that the

modified interpolation method can still work in the case ofthe number of signals exceeding the DOFs of actual arrayand the resolution is improved compared to the conventionalinterpolation method

Numerical Experiment 2 We consider a uniform and lineararray with 4 elements and the element space 120582 The nondi-rectional noise is spatial white Gaussian with a unit varianceThe virtual antenna array is also uniform and linear arraywith 8 elements and element space 1205822 Two independentsignals arrive from the directions 00∘ and 50

∘ and the virtualtransformation area is [minus5∘ 10∘] and the step size is 01∘ TheSNR of the two signals is 10 dB All SNR values are referredas per antenna element and the number of snapshots is 128and once again 200MonteCarlo runs are used to obtain eachpoint

Figure 5 shows the performance of conventional methodand the proposed method we can clearly see that theproposed method can distinguish the two signals while theconventional VAT fails for the SNR = 10 dBTheDOAfindingresults of proposed method is (01∘ 52∘) The resolution andaccuracy has been greatly improved compared conventionalinterpolated method This is because the proposed methodenhanced the signal subspace and improved the orthogo-nality between the signal and noise subspace by projectionprocessThe proposed method is considerably more accuratethan the conventional methods (also see Figure 4)

Figure 6 shows the root-mean-square errors (RMSEs) forthe MUSIC-based DOA estimators versus SNR by using theconventional interpolation approach themodified interpola-tion approach and the real eight-antenna arrayThe Cramer-Rao bounds (CRB) [18] of a real four-element array (CRB1)and eight-element array (CRB2) are plotted as a benchmark

0 50 100

0

20

Angle (deg)

Pmus

ic (d

B)


minus20

minus80

minus100 minus50

minus60

minus40


0 5 10 15 20 25 30 35 40SNR (dB)

RMSE

(deg

)

Conventional interpolation (virtual eight elements)Modified interpolation (virtual eight elements)Real eight elements arrayCRB1 (four elements)CRB2 (eight elements)

10minus2

10minus1

100

101

Figure 6 RMSE versus SNR for MUSIC-based DOA

It can be observed from this figure that the proposed methodhas better RMSE performance than the conventional methodfor the entire range of SNR values Since the interpolationtransformation can increase the DOF of an antenna arraythe DOF of the four-antenna array increases to seven inthis example When SNR gt 15 dB the RMSE of modifiedinterpolation is lower than CRB1 but is still larger than CRB2The RMSE of the conventional interpolation approach islarger than CRB1

The probabilities of source resolution versus SNR areshown in Figure 7 for different methods It can be observed



0 5 10 15 20 25 300

02

04

06

08

1

12

SNR (dB)

Prob

abili

ty o

f tar

get r

esol

utio

n

minus5

Figure 7 Probability of target resolution versus SNR

from Figure 7 that with the SNR increase the probabilities ofsource resolution of the methods increase but the proposedmethod has better source resolution capabilities than thetraditional method

Numerical Experiment 3 We consider that an eight-elementUCA with element space is one wave length The mapping isfrom theUCAonto an eight-element half-wavelength-spacedULA Virtual transformation area is [0

∘ 60∘] One signal

arrives from the direction 300∘ The number of snapshots is100 and 200Monte Carlo runs are used to obtain each pointAll SNR values are referenced per antenna element which aremodeled as isotropic unity gain and nonpolarized

The RMSE values as well as the CRB for the unmappedUCA data were plotted versus the SNR in Figure 8 It canbe seen that the method in [17] has the highest RMSEperformance which is slightly larger than the CRB It alsocan be seen that the proposed technique outperforms thetraditional interpolated method though it is still worse thanthe method in [17] but it has low complexity and is easy torealize The lower complexity makes our proposed algorithmsuitable for real-time implementation

6 Conclusion

The array interpolation technique is widely used in array sig-nal processing The interpolation preprocessing techniquesintroduce mapping errors that cause large bias in DOAestimation This paper proposed a modified interpolationmethod based on the covariance matrix projection thetransformation matrix is reconstructed by using the covari-ance matrix projection The modified interpolation methodenhances the signal subspace and improves the orthogonalitybetween the signal and the noise subspaces Simulation


0 5 10 15 20 25 30 35 40SNR (dB)

RMSE

(deg

)

Array Interpolation of [17]CRB

minus5minus10

10minus2

10minus3

10minus1

100

101

102

Figure 8 RMSE versus SNR

results show that the proposed approach offers better esti-mation accuracy and has a high resolution performancecompared to the tradition interpolation method

Conflict of Interests

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

Acknowledgments

This work was partially supported by ldquo973rdquo Basic ResearchDevelopment Program of China (no 6131380101) This paperis also supported by Pre-Research Fund of the 12th Five-YearPlan (no 4010403020102) and Fundamental Research Fundsfor the Central Universities (HEUCFT1304)

References

[1] K M Reddy and V Reddy ldquoAnalysis of interpolated arrays withspatial smoothingrdquo Signal Processing vol 54 no 3 pp 261ndash2721996

[2] T P Bronez ldquoSector interpolation of nonuniform arrays forefficient high resolution bearing estimationrdquo in Proceedings ofthe IEEE International Conference on Acoustics Speech andSignal Processing vol 5 pp 2885ndash2888 April 1988

[3] A J Weiss and M Gavish ldquoDirection finding using ESPRITwith interpolated arraysrdquo IEEE Transactions on Signal Process-ing vol 39 no 6 pp 1473ndash1478 1991

[4] A J Weiss ldquoPerformance analysis of spatial smoothing withinterpolated arraysrdquo EEE Transactions on Signal Processing vol41 no 5 pp 1881ndash1892 1993


[5] D N Swingler and R S Walker ldquoLine-array beamformingusing linear prediction for aperture interpolation and extrap-olationrdquo IEEE Transactions on Acoustics Speech and SignalProcessing vol 37 no 1 pp 16ndash30 1989

[6] F Belloni A Richter and V Koivunen ldquoDoA estimationvia manifold separation for arbitrary array structuresrdquo IEEETransactions on Signal Processing vol 55 no 10 pp 4800ndash48102007

[7] W Li Y Li andW Yu ldquoOn adaptive beamforming for coherentinterference suppression via virtual antenna arrayrdquo Progress inElectromagnetics Research vol 125 pp 165ndash184 2012

[8] B Friedlander ldquoDirection finding using an interpolated arrayrdquoin Proceedings of the International Conference on AcousticsSpeech and Signal Processing (ICASSP rsquo90) vol 5 pp 2951ndash2954Albuquerque NM USA April 1990

[9] B Friedlander ldquoThe root-MUSIC algorithm for direction find-ing with interpolated arraysrdquo Signal Processing vol 30 no 1 pp15ndash29 1993

[10] Y S Kim and Y S Kim ldquoModified resolution capability viavirtual expansion of arrayrdquo Electronics Letters vol 35 no 19 pp1596ndash1597 1999

[11] Y Wang H Chen and S Wan ldquoAn effective DOA methodvia virtual array transformationrdquo Science in China Series ETechnological Sciences vol 44 no 1 pp 75ndash82 2001

[12] B Friedlander and A J Weiss ldquoDirection finding using spatialsmoothing with interpolated arraysrdquo IEEE Transactions onAerospace and Electronic Systems vol 28 no 2 pp 574ndash5871992

[13] B K Lau G Cook and Y H Leung ldquoA modified arrayinterpolation approach to DOA estimation in correlated signalenvironmentsrdquo in Proceedings of the IEEE International Confer-ence on Acoustics Speech and Signal Processing vol 2 no 2 pp237ndash240 2004

[14] P Yang F Yang and Z P Nie ldquoDOA estimation with sub-arraydivided technique and interporlated esprit algorithmon a cylin-drical conformal array antennardquo Progress in ElectromagneticsResearch vol 103 pp 201ndash216 2010

[15] M Pesavento A B Gershman and Z Q Luo ldquoRobust arrayinterpolation using second-order cone programmingrdquo SignalProcessing Letters vol 9 no 1 pp 8ndash11 2002

[16] P Hyberg M Jansson and B Ottersten ldquoArray interpolationand bias reductionrdquo IEEE Transactions on Signal Processing vol52 no 10 pp 2711ndash2720 2004

[17] P Hyberg M Jansson and B Ottersten ldquoArray interpolationand DOAMSE reductionrdquo IEEE Transactions on Signal Process-ing vol 53 no 12 pp 4464ndash4471 2005

[18] P Stoica and A Nehorai ldquoMUSIC maximum likelihood andCramer-Rao bound further results and comparisonsrdquo IEEETransactions on Acoustics Speech and Signal Processing vol 38no 12 pp 2140ndash2150 1990

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Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



2d0 d (N minus 1)dx

y

middot middot middot

Figure 1 The structure of ULA

not reduced by minimizing the size of the mapping errorsbut instead by rotating the errors and the associated noisesubspace into the optimal directions related to a certaingradient of the DOA estimator criterion function in [17]The two methods show the superior performance on theDOAMSE reduction of array interpolation but its derivationcomputed procedure is complex

This paper proposed a novel method to reduce the DOAbias of virtual interpolation Using the projection conceptproject the transformation matrix onto the real array covari-ance matrix to enhance the signal subspace which improvesthe orthogonality between the signal and noise subspaceTheproposed method is efficient and easy to realize Numericalsimulations verify that this method can get better estimationaccuracy and has a high resolution performance compared tothe traditional interpolation method

2 Signal Model

Considering an omnidirectional array with 119873 elements illu-minated by 119872 narrow band signals the distance of the arrayelements is 119889 as shown in Figure 1

The signal 119904119896(119905) is incident in the direction 120579119896 the receivedsignal X can be expressed as follows

X (119905) = AS (119905) + N (119905) (1)

where X(119905) = [1199091(119905) 1199092(119905) 119909119873(119905)]119879 is a 119873 times 1 snap

data vector S(119905) = [1199041(119905) 1199042(119905) 119904119872(119905)]119879 is a vector

containing the complex signal envelops of 119872 narrow-bandsignal sources N(119905) = [1198991(119905) 1198992(119905) 119899119873(119905)]

119879 is a vector ofzero-mean spatial white sensor noise of variance 120590

2

119899 A is an

array manifold matrix namely A = [a(1205791) a(1205792) a(120579119872)]where a(1205791) = [1 119890

119895120573119896 119890119895(119873minus1)120573119896]

119879 119896 = 1 2 119872

represents a steering vector in the 120579119896 direction and 120573119896 is thephase difference that can be represented as

120573119896 =2120587

120582119889 sin (120579119896) (2)

Assume that the signal and noise are linearly indepen-dent and then the data covariance is written in the formatbelow

R = 119864 X (119905)X119867 (119905) = AR119904A119867

+ 1205902

119899I (3)

where 119864 denotes the expectation operator R119904 =

119864119878(119905)119878119867(119905) represents the autocorrelation matrix of

signal complex envelops 1205902119899is the noise power I is the unit

matrix and (sdot)119867 denotes the matrix conjugate transposition

Real antenna unit

Virtual antenna unit

Figure 2 Real antenna and interpolated arrays

In practice the desired signal is often present in the snap-shots The sample array covariance matrix can be expressedas follows

R =1

119870

119870

sum

119894=1

X (119894)X119867 (119894) (4)

where 119870 is number of snapshots VAT is based on inter-polation technique [8] in which the entire antenna arrayscanning vector is divided into several subregions and thesubregion of interest will be segmented through a certaintransformation to realize themapping from the original arrayto the corresponding virtual array

3 Conventional Interpolated Array

Consider a real ULA transformed into a virtual ULA via arrayinterpolation as illustrated in Figure 2

Assume that there is a signal located in the region Θ weequally divide Θ into

Θ = [120579119897 120579119897 + Δ120579 120579119897 + 2Δ120579 120579119903 minus Δ120579 120579119903] (5)

where 120579119897 and 120579119903 are the left and right boundary of regionΘ respectively Δ120579 is size of the interpolation step which isdetermined by the specified accuracy

The real array manifold matrix in the chosen area can beexpressed as follows

A = [a (120579119897) a (120579119897 + Δ120579) a (120579119897 + 2Δ120579)

a (120579119903 minus Δ120579) a (120579119903)] (6)

where a(120579119897) represents the steering vector of a real array in the120579119897 directionThe array manifold matrix of virtual array in thesame area Θ is expressed as follows

A = [a (120579119897) a (120579119897 + Δ120579) a (120579119897 + 2Δ120579)

a (120579119903 minus Δ120579) a (120579119903)] (7)

where a(120579119897) represents the steering vector of a virtual arrayin the 120579119897 direction There must exist a mapping relationshipbetween the real and the virtual array vectors Then aninterpolation matrix B is designed to satisfy the least squarethat is

minB

10038171003817100381710038171003817B119867A minus A10038171003817100381710038171003817

2

119865 (8)



B119867A (120579) = A (120579) 120579 isin Θ (9)


B119867a (120579) = a (120579) 120579 isin Θ (10)


B = (AA119867)minus1AA119867 (11)


119864 (B) =

minB10038171003817100381710038171003817B119867A minus A1003817100381710038171003817100381711986510038171003817100381710038171003817A10038171003817100381710038171003817119865

(12)




B = RB (13)


R = BAR119904A119867B119867 + B (120590

2

119899I)B119867

= RBAR119904A119867B119867R119867 + 120590

2

119899RBB119867R119867

(14)



T = (B119867B)minus12

B119867 (15)


Projection process

Prewhiten noise



X

BX

B = RB



Ta (120579) = (B119867B)minus12

a (120579) =

_a (120579) 120579 isin Θ (16)



_A R119904

_A119867

+ 1205902

119899I (17)





_R = U119878Σ119878U

119867

119878+ U119873Σ119873U

119867

119873 (18)


119875MUSIC =1

a119867 (120579)U119873U119867119873a119867 (120579)=

1

1003817100381710038171003817U119867119873a (120579)1003817100381710038171003817

(19)





[8]



119874(2(2119873 + 1)2

1198732

cal119873119873)


estimator



3) None




119873a(120579) = 0 due to









(19)


Topt = arg minT

119873cal

sum

119894=1

(1 minus 120583)10038171003817100381710038171003817Δ119890(119894)

V10038171003817100381710038171003817

2

+120583

2 (120579(119894) 119890(119894)V )

times[4 (Re 119892(119894)V Δ119890(119894)

V )2

+

119898

sum

119896=2

2120572(119894) 10038161003816100381610038161003816

119892(119894)

V T119867119890(119894)119896

10038161003816100381610038161003816

2

]

(20)


V ≜ 119892V(120579(119894))










0 20 40 60 80

0

20

40

60

80

100

Angle (deg)

Pmus

ic (d

B)


minus20









0 50 100

0

20

Angle (deg)

Pmus

ic (d

B)


minus20

minus80

minus100 minus50

minus60

minus40


0 5 10 15 20 25 30 35 40SNR (dB)

RMSE

(deg

)


10minus2

10minus1

100

101






0 5 10 15 20 25 300

02

04

06

08

1

12

SNR (dB)

Prob

abili

ty o

f tar

get r

esol

utio

n

minus5







6 Conclusion



0 5 10 15 20 25 30 35 40SNR (dB)

RMSE

(deg

)


minus5minus10

10minus2

10minus3

10minus1

100

101

102





Acknowledgments


References






















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











B119867A (120579) = A (120579) 120579 isin Θ (9)


B119867a (120579) = a (120579) 120579 isin Θ (10)


B = (AA119867)minus1AA119867 (11)


119864 (B) =

minB10038171003817100381710038171003817B119867A minus A1003817100381710038171003817100381711986510038171003817100381710038171003817A10038171003817100381710038171003817119865

(12)




B = RB (13)


R = BAR119904A119867B119867 + B (120590

2

119899I)B119867

= RBAR119904A119867B119867R119867 + 120590

2

119899RBB119867R119867

(14)



T = (B119867B)minus12

B119867 (15)


Projection process

Prewhiten noise



X

BX

B = RB



Ta (120579) = (B119867B)minus12

a (120579) =

_a (120579) 120579 isin Θ (16)



_A R119904

_A119867

+ 1205902

119899I (17)





_R = U119878Σ119878U

119867

119878+ U119873Σ119873U

119867

119873 (18)


119875MUSIC =1

a119867 (120579)U119873U119867119873a119867 (120579)=

1

1003817100381710038171003817U119867119873a (120579)1003817100381710038171003817

(19)





[8]



119874(2(2119873 + 1)2

1198732

cal119873119873)


estimator



3) None




119873a(120579) = 0 due to









(19)


Topt = arg minT

119873cal

sum

119894=1

(1 minus 120583)10038171003817100381710038171003817Δ119890(119894)

V10038171003817100381710038171003817

2

+120583

2 (120579(119894) 119890(119894)V )

times[4 (Re 119892(119894)V Δ119890(119894)

V )2

+

119898

sum

119896=2

2120572(119894) 10038161003816100381610038161003816

119892(119894)

V T119867119890(119894)119896

10038161003816100381610038161003816

2

]

(20)


V ≜ 119892V(120579(119894))










0 20 40 60 80

0

20

40

60

80

100

Angle (deg)

Pmus

ic (d

B)


minus20









0 50 100

0

20

Angle (deg)

Pmus

ic (d

B)


minus20

minus80

minus100 minus50

minus60

minus40


0 5 10 15 20 25 30 35 40SNR (dB)

RMSE

(deg

)


10minus2

10minus1

100

101






0 5 10 15 20 25 300

02

04

06

08

1

12

SNR (dB)

Prob

abili

ty o

f tar

get r

esol

utio

n

minus5







6 Conclusion



0 5 10 15 20 25 30 35 40SNR (dB)

RMSE

(deg

)


minus5minus10

10minus2

10minus3

10minus1

100

101

102





Acknowledgments


References






















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













[8]



119874(2(2119873 + 1)2

1198732

cal119873119873)


estimator



3) None




119873a(120579) = 0 due to









(19)


Topt = arg minT

119873cal

sum

119894=1

(1 minus 120583)10038171003817100381710038171003817Δ119890(119894)

V10038171003817100381710038171003817

2

+120583

2 (120579(119894) 119890(119894)V )

times[4 (Re 119892(119894)V Δ119890(119894)

V )2

+

119898

sum

119896=2

2120572(119894) 10038161003816100381610038161003816

119892(119894)

V T119867119890(119894)119896

10038161003816100381610038161003816

2

]

(20)


V ≜ 119892V(120579(119894))










0 20 40 60 80

0

20

40

60

80

100

Angle (deg)

Pmus

ic (d

B)


minus20









0 50 100

0

20

Angle (deg)

Pmus

ic (d

B)


minus20

minus80

minus100 minus50

minus60

minus40


0 5 10 15 20 25 30 35 40SNR (dB)

RMSE

(deg

)


10minus2

10minus1

100

101






0 5 10 15 20 25 300

02

04

06

08

1

12

SNR (dB)

Prob

abili

ty o

f tar

get r

esol

utio

n

minus5







6 Conclusion



0 5 10 15 20 25 30 35 40SNR (dB)

RMSE

(deg

)


minus5minus10

10minus2

10minus3

10minus1

100

101

102





Acknowledgments


References






















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 20 40 60 80

0

20

40

60

80

100

Angle (deg)

Pmus

ic (d

B)


minus20









0 50 100

0

20

Angle (deg)

Pmus

ic (d

B)


minus20

minus80

minus100 minus50

minus60

minus40


0 5 10 15 20 25 30 35 40SNR (dB)

RMSE

(deg

)


10minus2

10minus1

100

101






0 5 10 15 20 25 300

02

04

06

08

1

12

SNR (dB)

Prob

abili

ty o

f tar

get r

esol

utio

n

minus5







6 Conclusion



0 5 10 15 20 25 30 35 40SNR (dB)

RMSE

(deg

)


minus5minus10

10minus2

10minus3

10minus1

100

101

102





Acknowledgments


References






















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











0 5 10 15 20 25 300

02

04

06

08

1

12

SNR (dB)

Prob

abili

ty o

f tar

get r

esol

utio

n

minus5







6 Conclusion



0 5 10 15 20 25 30 35 40SNR (dB)

RMSE

(deg

)


minus5minus10

10minus2

10minus3

10minus1

100

101

102





Acknowledgments


References






















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation


























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Research Article An Effective Technique for Enhancing Direction Finding Performance...

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