uniform and nonuniform v-shaped planar arrays for 2-d direction...

Uniform and nonuniform V-shaped planar

arrays for 2-D direction-of-arrival estimation

T. Filik1 and T. E. Tuncer1

Received 26 June 2008; revised 28 January 2009; accepted 25 June 2009; published 22 September 2009.

[1] In this paper, isotropic and directional uniform and nonuniform V-shaped arrays areconsidered for azimuth and elevation direction-of-arrival (DOA) angle estimationsimultaneously. It is shown that the uniform isotropic V-shaped arrays (UI V arrays) haveno angle coupling between the azimuth and elevation DOA. The design of the UI V arraysis investigated, and closed form expressions are presented for the parameters of theUI V arrays and nonuniform V arrays. These expressions allow one to find the isotropicV angle for different array types. The DOA performance of the UI V array is comparedwith the uniform circular array (UCA) for correlated signals and in case of mutualcoupling between array elements. The modeling error for the sensor positions is alsoinvestigated. It is shown that V array and circular array have similar robustness for theposition errors while the performance of UI V array is better than the UCA for correlatedsource signals and when there is mutual coupling. Nonuniform V-shaped isotropic arraysare investigated which allow good DOA performance with limited number of sensors.Furthermore, a new design method for the directional V-shaped arrays is proposed. Thismethod is based on the Cramer-Rao Bound for joint estimation where the anglecoupling effect between the azimuth and elevation DOA angles is taken into account. Thedesign method finds an optimum angle between the linear subarrays of the V array.The proposed method can be used to obtain directional arrays with significantly betterDOA performance.

Citation: Filik, T., and T. E. Tuncer (2009), Uniform and nonuniform V-shaped planar arrays for 2-D direction-of-arrival

estimation, Radio Sci., 44, RS5006, doi:10.1029/2008RS003949.

1. Introduction

[2] DOA estimation has many applications in radar[Haykin, 1985], sonar, seismology [Bohme, 1995], andionospheric research [Black et al., 1993]. The perfor-mance of the direction finding (DF) system is signifi-cantly dependent to sensor array geometry. Therefore,the design of optimum array geometry for the best two-dimensional (2-D) DOA estimation performance is animportant problem. This problem is investigated inprevious works for the most general parameter settings.In these works, Cramer-Rao Bound (CRB) on errorvariance is used as the performance measure and objec-tive functions for the desired performance are minimized.The desired performance can change according to theapplication. In the work of Baysal and Moses [2003], the

goal is to find planar and volumetric arrays for uniformDOA performance in all directions. In the work of Okteland Randolph [2005], DOA of interest is an angularsector and the goal is to find optimum array geometry forthis scenario. It is seen that optimum array geometry forDOA estimation depends on many parameters includingthe number of sensors, number of sources and their DOAangles [Gershman and Bohme, 1997]. Furthermore it isnot easy to find a single optimum geometry since the costfunction changes depending on the number of sourcesand DOAs.[3] In this paper, the array geometry is fixed as V-shaped

in order to simplify the design and use certain advantages ofV-shaped arrays which are not well known in the literature.V-shaped planar arrays can be designed for good direc-tional DOA performance. It can also be designed forisotropic response such that the DOA performance isuniform for all directions. When the array intersensordistance is fixed to half of the wavelength, V-shaped arrayhas a larger aperture than circular array. The number ofsensors in V-shaped arrays can be decreased when the

RADIO SCIENCE, VOL. 44, RS5006, doi:10.1029/2008RS003949, 2009

1Department of Electrical and Electronics Engineering, Middle EastTechnical University, Ankara, Turkey.

Copyright 2009 by the American Geophysical Union.

0048-6604/09/2008RS003949

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sensors are placed nonuniformly for each subarray. Fur-thermore, it is possible to apply forward-backward spatialsmoothing [Pillai and Kwon, 1989] for each subarray inorder to deal with multipath signals. Fast algorithms canbe applied for these subarrays and the results can becombined as in the work of Hua et al. [1991]. It is alsoshown that joint accuracy of two subarrays is better thaneach subarray’s accuracy [Hua and Sarkar, 1991].[4] V-shaped arrays are not fully investigated in the

literature. In the work of Gazzah and Marcos [2006], V-shaped arrays are considered with limited scope. Statis-tical angle coupling between the azimuth and elevationangle estimation is ignored and V angle for uniformDOA performance is determined only for infinite numberof sensors. Until now there is no known method andexpression for finding the isotropic V angle. Furthermorethe directional characteristic of the V-shaped arrays is notfully exploited.[5] In this paper, closed form expressions are presented

for the V angle in order to obtain isotropic DOAresponse. The UI V-shaped array and UCA are comparedfor the same number of sensors and intersensor distances.The comparison is done in terms of sensor positionerrors, source signal correlation, and mutual couplingbetween antennas. It is shown that the DOA accuracy ofthe UI V-shaped array is better than UCA. V-shapedarrays and UCA have similar robustness for the sensorposition errors. The effect of source signal correlation issimilar for both arrays while the performance of UI Varray gets better as the correlation increases. A similarobservation is done for the mutual coupling. The perfor-mance of UI V array is better than UCA in case ofmultiple sources. Different nonuniform V-shaped isotro-pic arrays are considered where the numbers of sensors ateach subarray can be different. It is shown that the DOAperformance can be improved significantly when theisotropic nonuniform V-shaped arrays are used. A designprocedure for the directional uniform V-shaped arrays ispresented. The directional V-shaped arrays are alsocompared with UCA for different DOA scenarios.[6] The contribution of this paper for 2-D DOA

estimation with V-shaped planar array geometry can besummarized as follows. Closed form expressions for theisotropic V angle are presented. The expressions aregiven for both uniform and nonuniform V-shaped planararrays. The performances of V-shaped arrays, includinguniform isotropic and nonuniform arrays, are analyzedfor different cases. These involve correlated signals,mutual coupling, and sensor position errors. It is shownthat V-shaped arrays perform better than the UCA fordifferent types of error sources which is not well knownin the literature. A design procedure is presented foruniform directional V arrays which allow one to trade offthe isotropic characteristics for the better DOA perfor-mance for a given angular sector. The optimization of the

Vangle is done by defining a cost function over the CRBon DOA error variance which takes into account thecoupling effect of azimuth and elevation angles. Theproposed design considers two regions, namely, focusedand unfocused region. The limits of the regions deter-mine the angular accuracy and the performance of the Varray. It is shown that optimum V angle can be foundeasily with only a limited search due to the monotoniccharacteristics of the cost function for the worst and bestlevels specified in the design parameters of the regions.This design procedure can also find the V angle forisotropic DOA performance numerically [Filik andTuncer, 2008a, 2008b].[7] The paper is organized as follows. In section 2, we

describe the model of the array signals and CRB expres-sions are presented for 2-D DOA estimation. In section 3,closed form expressions for uniform and nonuniformV-shaped arrays for isotropic azimuth response are given.We present the directional V-shaped planar array designprocedure in section 4. In section 5, the effect of mutualcoupling between array elements is considered. Theperformances of the designed V-shaped arrays andUCA are presented in section 6.

2. Problem Formulation

2.1. Data Model

[8] We consider an array of M sensors located at thepositions [xl, yl], l = 1, . . ., M. We assume that there areL (L < M) narrowband signals impinging on the arrayfrom the directions Qi = [fi, qi] i = 1, . . ., L, where f andq are the azimuth and elevation angles, respectively, asshown in Figure 1. If the sensors are identical omnidi-rectional and far-field assumption is made, the sensoroutput, y(t), can be written as,

y tð Þ ¼ A Qð Þs tð Þ þ n tð Þ; t ¼ 1; . . . ;N ð1Þ

where N is the number of snapshots. It is assumed thatthe noise, n(t), is both spatially and temporally whitewith variance s2. It is also uncorrelated with the source

Figure 1. Coordinate system for 2-D angle estimationand V-shaped array.

RS5006 FILIK AND TUNCER: UNIFORM AND NONUNIFORM V-SHAPED ARRAYS

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signals. A(Q) = [a(f1, q1). . .a(fL, qL)] is the M � Lsteering matrix for the planar array and the vectors a(f, q)are given as,

a f; qð Þ ¼ e j2pl x1 cosf sin qþy1 sinf cos qð Þf g . . .he j

2pl xM cosf sin qþyM sinf cos qð Þf g

iT: ð2Þ

The output covariance matrix, R, is

E y tð Þy tð ÞHn o

¼ R ¼ ARsAH þ s2I; ð3Þ

where (.)H denotes the conjugate transpose of amatrix,Rs isthe source correlation matrix and I is the identity matrix.

2.2. CRB for 2-D DOA Estimation

[9] CRB shows the ultimate performance of an unbi-ased estimate for a given array geometry. When 2-DDOA estimation is considered, there is statistical cou-pling between the azimuth and elevation DOA perform-ances in general. The existence of coupling depends onarray geometry. Some of the array geometries likecircular arrays are uncoupled. V-shaped arrays showcoupling effects and therefore coupling should be takeninto account for the CRB. The proposed V-shaped arraydesign method uses the CRB in the cost function.Therefore, a review of the angle coupling effect for theCRB is considered in this section. The inequality for thevariance of the parameters is given as,

var q̂m� �

� F1� �

mm; ð4Þ

where the mnth element of the Fisher information matrix,F, is given by Weiss and Friedlander [1993] as

Fmn ¼ N � tr R1@R

@pmR1

@R

@pn

; ð5Þ

For 2-D angle estimation, the unknown parameter vectoris defined by p = [f, q]. Fisher information matrix (FIM)is given by

F ¼ Fff FfqFqf Fqq

� �ð6Þ

where

Fff ¼ 2NRe RsAHR1ARs

�

� _AHf P?AR

1 _Af

� �T

ð7Þ

Ffq ¼ 2NRe RsAHR1ARs

�

� _AHf P?AR1 _Aq� �T

ð8Þ

Fqq, can be written similar to (7). Ffq = Fqf and

P?A ¼ I A AHA

�1

AH ; _Af ¼XLn¼1

@A

@fn: ð9Þ

If the off-diagonal term, Ffq is zero, the estimates of theazimuth and elevation angles are uncoupled. But forarbitrary array geometries, this off-diagonal term, Ffq, isnonzero. For 2-D angle estimation, the CRB defined byMirkin and Sibul [1991] and Nielsen [1994] takes thecoupling effect into account and the CRB for the azimuthand elevation angles are given as,

CRBf ¼1

Fff

1

1 r2

� �ð10Þ

CRBq ¼1

Fqq

1

1 r2

� �ð11Þ

where

0 � r2 ¼F2fq

FffFqq� 1: ð12Þ

If r2 = 1, the estimates are said to be perfectly coupledand the unknown parameters (f, q) cannot be estimatedsimultaneously. If r2 6¼ 0, uncertainty in one parameterdegrades the other parameter’s accuracy. r2 = 0 isrequired for uncoupled 2-D DOA estimation. Thereforeit is important to consider the coupling effect when 2-DDOA estimation is done. The constraints on array sensorlocations for uncoupled DOA angle estimation arereviewed in the following part.[10] In order to have 2-D uncoupled DOA angle

estimation, the off-diagonal terms of FIM must be zero,i.e., Ffq = 0. The required conditions for uncoupled DOAestimation according to the array sensor locations arederived by Nielsen [1994] as,

Pxx ¼ Pyy and Pxy ¼ 0 ð13Þ

where Pxx, Pyy and Pxy depend on the sensor coordinates,

Pxx ¼XMl¼1

xl xcð Þ2; Pyy ¼XMl¼1

yl ycð Þ2 ð14Þ

Pxy ¼XMl¼1

xl xcð Þ yl ycð Þ ð15Þ

and xc, yc given as,

xc ¼1

M

XMl¼1

xl; yc ¼1

M

XMl¼1

yl ð16Þ


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(xl, yl) is the lth sensor position and (xc, yc) are arraycenter of gravity. Therefore (13) should be satisfied inorder to have uncoupled DOA angle estimation for aplanar array.

3. Isotropic Planar Array

[11] CRB for an isotropic planar array in case of asingle source is uniform for all azimuth DOA angles. Inthe works of Baysal and Moses [2003] and Gazzah andMarcos [2006], it is shown that the conditions for anisotropic array are the same as the conditions for an arrayto have uncoupled azimuth and elevation estimationwhich is given in the previous part.[12] In the following part, we present the closed form

expressions which return the V angle for uniform andnonuniform V-shaped isotropic planar arrays.

3.1. Isotropic Uniform V-Shaped Array

[13] Let M be an odd number for simplicity and k =Mþ12

is the index of the reference sensor at the origin. Thesensor positions for uniform V-shaped array in Figure 2can be expressed as,

xl ¼ l kð Þd sing2

� �; l ¼ 1; . . . ;M

yl ¼ jl kjd cosg2

� �; l ¼ 1; . . . ;M : ð17Þ

It is assumed that the sensor positions are symmetricaccording to the y axis and sensors are separated with adistance which is an integer multiple of a distance d. Inorder to design isotropic uniform V-shaped array,equation (13) should be satisfied. Since the sensorpositions are uniform and symmetric according to they axis, xc = 0 and Pxy = 0 for all g angles. The conditionPxx = Pyy should be satisfied for the isotropic angle giso.The derivation of giso formulation is presented in

Appendix A. The closed form expression for giso for auniform V-shaped array is given as,

giso ¼ 2� arctanffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM 2 þ 34M 2

r !: ð18Þ

3.2. Isotropic Nonuniform V-Shaped Array

[14] In this case, the distance from the reference sensoris nonuniform for the sensors in the array as shown inFigure 3. It is known that nonuniform arrays can performbetter than the same number of element uniform lineararray (ULA), in a variety of cases [Tuncer et al., 2007].There are M1 sensors at the left nonuniform linear subarray and M2 sensors at the right nonuniform linear subarray and a reference sensor at the origin. We can expressthe sensor positions for the nonuniform V-shaped arrayas,

xl ¼ dl sing2

� �;

yl ¼ dl cosg2

� �for l ¼ 1; . . . ;M1

xl ¼ dl sing2

� �;

yl ¼ dl cosg2

� �for l ¼ M1 þ 2; . . . ;M

xM1þ1 ¼ 0; yM1þ1 ¼ 0 ð19Þ

where dl is a real positive number. We have to place thesensors to satisfy (13) for isotropic performance. In orderto have Pxy = 0, we need to satisfy the followingequations.

d1 þ � � � þ dM1ð Þ ¼ dM1þ2 þ � � � þ dMð Þ

d21 þ � � � þ d2M1� �

¼ d2M1þ2 þ � � � þ d2M

� �ð20Þ

The details of the derivation of the isotropic angle arepresented in Appendix B. The closed form expression

Figure 2. Uniform V-shaped array geometry.

Figure 3. Nonuniform V-shaped array geometry.


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which returns the V angle for isotropic performance, giso,for nonuniform V-shaped arrays is given as,

giso ¼ 2 arctan

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 2

M

PM1i¼1

di

� �2PM1i¼1

d2i

vuuuuuut

!ð21Þ

Both (20) and (21) should be satisfied in order to obtainisotropic performance for a nonuniform V-shaped array.[15] We can design nonuniform V-shaped arrays for

isotropic DOA performance in two steps. In the first step,the sensor locations are selected to satisfy (20). Then theisotropic angle for the nonuniform V-shaped array isobtained as in (21).

4. Directional V-Shaped Planar Array

Design

[16] In the previous part, we derived analytic expres-sions for the design of isotropic V-shaped arrays. Whileit is useful to have isotropic response in many cases,directional arrays perform better when the array is con-strained to look more sensitively to a certain angularsector. In this part we present a design procedure for thedirectional V-shaped arrays.[17] The design procedure finds the optimum g� angle

to obtain the best DOA performance. Before goingthrough the design steps we need to understand thecharacteristics of the V-shaped arrays. CRB for differentDOA angles is shown in Figure 4 for different V angles.The characteristic is periodic by 180 degrees. The bestperformance is seen at 90 and 270 degrees and the worst

performance is seen at 0 and 180 degrees. This charac-teristic is observed when the V-shaped array is config-ured as shown in Figure 1, where the subarrays areplaced symmetrically with respect to y axis. Note thatsuch kind of configuration can always be realized bydefining x and y axis appropriately. When we change theV angle, g, the best and worst performance levels andthe width of these regions are changing. We need to findthe best V angle for the desired directional response.[18] In the design procedure, two regions are specified

as shown in Figure 5. Focused region is the angularsector where the best possible DOA accuracy is desired.Unfocused region is an angular sector where a DOAaccuracy below a certain level, H1, is acceptable. If afocused region different than the one shown in Figure 5,is desired, array and coordinate axis can be rotatedappropriately. Note that focused region is centered at90 degrees where the array shows the best performance.

Figure 4. DOA performance of nine-element V-shapedarrays with different g angles for a single source which isswept between all azimuth angles with 256 snapshotsand 20 dB SNR.

Figure 5. The design regions and parameters forV-shaped planar array geometry.

Figure 6. The best and worst performance levels of theazimuth CRB versus V angle, g, when a1 and a2 are90 degrees.


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The azimuth angles a1 and a2 determine the focusedregion limits. While a1 and a2 can be arbitrary ingeneral, the best performance is obtained if a1 + a2 =180�. Note that in this case a1 and a2 are symmetricallyplaced with respect to the 90 degrees. The target is tofind g� given the parameters a1, a2 and H1.[19] In Figure 6 the best and the worst performance

levels are plotted with respect to g angle when thecoupling effect of the azimuth and elevation angleestimation is taken into account. Note that if the couplingeffect is not taken into account, design of V array canconverge to a degenerate case, such as, a linear array (g =180 degrees). In this case H1 level goes to infinity andtherefore both azimuth and elevation angles cannot beresolved simultaneously. As it is seen, H1-g and H2-gcurves are monotonic. Once H1 is specified, thecorresponding angle in Figure 6 is an upper bound forthe best performance. Therefore optimum V angle, g�,should be less than this angle. The proposed designmethod has the following steps:[20] Step 1:H1,a1, anda2 values are specified (Figure 5).

We assume that a2 = 180 a1 for simplicity.[21] Step 2: From Figure 6, g angle (g1) corresponding

to H1 is found.[22] Step 3: CRB expression in (10) is evaluated for

the a1 azimuth angle corresponding to the V angle, gk,namely CRB(a1, gk). The cost for gk is e(k) = CRB(a1, gk).

[23] Step 4: Decrease gk angle by D, gk+1 = gk D,and repeat step 3 for k = 2, . . ., K. D is the step size andK = (g1 giso)/D[24] Step 5: Find the minimum e(k) and the corre-

sponding gk angle as the optimum V angle, g�

g� ¼ argmingk

e kð Þf g: ð22Þ

5. Analysis of Mutual Coupling Effects

[25] Mutual coupling between antenna elements is animportant factor which degrades the DOA estimationperformance. CRB for unknown mutual coupling matrix(MCM), C, is the fundamental tool in order to quantifythe DOA performance [Friedlander and Weiss, 1991; Yeand Liu, 2008]. In this paper, the CRB formulation of Yeand Liu [2008] is implemented in order to compare theUCA and isotropic V-shaped arrays. The array output incase of mutual coupling can be expressed as,

y tð Þ ¼ CA Qð Þs tð Þ þ n tð Þ: ð23Þ

The target in this part is to compare the DOAperformances for UCA and V-shaped arrays in a fairmanner. In order to achieve this target, both arrays areconstructed by employing dipole antennas with l/2 sizeand 50 ohm load in FEKO [EM Software and Systems

Table 1. Distance Between Sensors for Nine-Element UCA in Terms of l

UCA 1 2 3 4 5 6 7 8 9

1 0 0.5 0.939 1.266 1.439 1.439 1.266 0.939 0.52 0.5 0 0.5 0.939 1.266 1.439 1.439 1.266 0.9393 0.939 0.5 0 0.5 0.939 1.266 1.439 1.439 1.2664 1.266 0.939 0.5 0 0.5 0.939 1.266 1.439 1.4395 1.439 1.266 0.939 0.5 0 0.5 0.939 1.266 1.4396 1.439 1.439 1.266 0.939 0.5 0 0.5 0.939 1.2667 1.266 1.439 1.439 1.266 0.939 0.5 0 0.5 0.9398 0.939 1.266 1.439 1.439 1.266 0.939 0.5 0 0.59 0.5 0.939 1.266 1.439 1.439 1.266 0.939 0.5 0

Table 2. Distance Between Sensors for Nine-Element UI V-Shaped Array in Terms of l

V 1 2 3 4 5 6 7 8 9

1 0 0.5 1 1.5 2 1.753 1.627 1.649 1.8152 0.5 0 0.5 1 1.5 1.272 1.219 1.361 1.6493 1 0.5 0 0.5 1 0.814 0.908 1.219 1.6274 1.5 1 0.5 0 0.5 0.454 0.814 1.272 1.7535 2 1.5 1 0.5 0 0.5 1 1.5 26 1.753 1.272 0.814 0.454 0.5 0 0.5 1 1.57 1.627 1.219 0.908 0.814 1 0.5 0 0.5 18 1.649 1.361 1.219 1.272 1.5 1 0.5 0 0.59 1.815 1.649 1.627 1.753 2 1.5 1 0.5 0


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S.A. (Pty) Ltd., 2008]. The radius of the dipole is selectedas 1.5 � 103 l and the operating frequency is 30 MHz.There are 9 antennas and the intersensor distance is set tol/2 for both arrays. FEKO is an electromagneticsimulation tool which can model the antenna elementswith sufficient accuracy and close to the practicalsituation. Tables 1 and 2 present the distance betweenarray elements for UCA and UI V-shaped arrays,respectively. The mutual coupling between two antennasdepends on the distance between antennas. As thedistance increases, the magnitude of the couplingcoefficient decreases. In the literature, MCM for UCAis usually represented with only one coefficient[Friedlander and Weiss, 1991]. In addition, the coeffi-cients for the antennas with a distance greater than0.707l are ignored [Ye and Liu, 2008]. In this paper, weignored the coefficients when the distance betweenantennas is greater than l in order to have a moreaccurate evaluation. Tables 3 and 4 show the MCMmatrices and the mutual coupling coefficients for the twoarrays. It can be seen that UI V-shaped array uses sevencoefficients whereas the UCA array uses only twocoefficients. In addition, coupling coefficients for thesame distance may be different for the V-shaped arraydue to the different interaction between antennas. Thecoupling coefficients for two arrays are given in Table 5.[26] The real and imaginary parts of the elements of

the MCM contribute to the Fisher Information Matrix

(FIM). Therefore as the number of coefficients increases,the size of the FIM and its condition number increases. Itmay be no longer well conditioned [Svantesson, 1999].This also disturbs the smoothness of the CRB character-istics. As a result, the increase in the number of couplingcoefficients decreases the accuracy of DOA performance.[27] For a single source, the number of unknowns

is large compared to the number of equations for UIV-shaped array in (23). When some of the unknowns areignored and MCM is estimated, the DOA accuracydecreases. As a result, the DOA performance of UIV-shaped array is worse than the UCA for a singlesource. It is also observed that its performance gets betterthan the UCA when the number of coupling coefficientsis decreased. As the number of sources increases, thenumber of equations increases and MCM can be esti-mated accurately. In our simulations, we have found thatUI V-shaped arrays perform better than UCAwhen thereis more than one source. The comparisons of the per-formances of the two arrays are presented in the follow-ing section.

6. Simulation Results

[28] In this section, we consider the isotropic anddirectional V-shaped arrays in order to show the charac-teristics of the V array for different cases. Examples ofthe isotropic uniform and nonuniform V arrays areconsidered and compared with UCA. Furthermore theeffect of sensor position error is investigated for bothV-shaped and circular arrays by using the MUSICalgorithm.[29] In simulations, source angles are considered in

degrees where azimuth angles are between 0 and 360degrees and elevation angles are between 0 and 90 degrees(Figure 1). There are 1000 trials for each experiment andthe number of snapshots is 256.

6.1. Simulations for Uniform Isotropic V-ShapedArrays

[30] UI V-shaped planar arrays can be easily designedfrom equation (18) for a specified number of sensors, M.For example if M = 9, giso is 53.9681�. The performance

Table 3. Mutual Coupling Matrix for Nine-Element UCA

UCA 1 2 3 4 5 6 7 8 9

1 1 c1 c2 c2 c12 c1 1 c1 c2 c23 c2 c1 1 c1 c24 c2 c1 1 c1 c25 c2 c1 1 c1 c26 c2 c1 1 c1 c27 c2 c1 1 c1 c28 c2 c2 c1 1 c19 c1 c2 c2 c1 1

Table 4. Mutual Coupling Matrix for Nine-Element UI V-Shaped

Array

V 1 2 3 4 5 6 7 8 9

1 1 v42 v4 1 v23 v2 1 v3 v7 v64 v3 1 v5 v1 v75 v5 1 v56 v7 v1 v5 1 v37 v6 v7 v3 1 v28 v2 1 v49 v4 1

Table 5. Mutual Coupling Coefficients of UCA and UI V-

Shaped Array

UCA UI V Array

c1 = 0.1534 + 0.1019i v1 = 0.1334 + 0.2059ic2 = 0.0347 0.0960i v2 = 0.1386 + 0.1198i

v3 = 0.1549 + 0.0924iv4 = 0.1268 + 0.1210iv5 = 0.0876 + 0.1482iv6 = 0.0124 0.1490iv7 = 0.0722 0.0915i


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of this V-shaped array is compared with the UCA inFigure 7. There are three sources at the azimuth anglesf1 = 60, f2 = 100 and f3 = 120 degrees and elevationangles are fixed at q = 90 degrees for all sources. Sourcesignals are uncorrelated. As it is seen from Figure 7, UIV-shaped array shows better performance than the cir-cular array when both arrays have the same number ofsensors and intersensor distances. In Figure 7, the per-formances of V-shaped array and UCA are outlined whenthere is an error in sensor positions denoted by pe. pe isan error with respect to the intersensor distance, d = l/2where l is the wavelength. Therefore % 2 position errorcorresponds to

jpejd

= 0.02. Error displacement is on acircle with radius jpej and the circle center is at the true

sensor position. Figure 7 shows that both the UI V arrayand UCA have similar robustness for the various positionerrors (% 2, % 1, and % 0.2). Also it is evident that theUI V array has better performance for each of theposition errors.[31] Figure 8 shows the DOA performance of UI V

array and UCA for correlated source signals. There aretwo sources at the azimuth angles f1 = 80 and f2 =85 degrees, respectively, and the elevation angle is fixedat q = 90 degrees for each source. SNR is set to 15 dB forthe equi-power sources. The source covariance matrix,Rs is taken as,

Rs ¼1 rr 1

� �ð24Þ

Figure 7. Azimuth DOA performance for three sourcesat 60, 100, and 120 degrees, respectively, when UIV-shaped array and UCA are used without and withsensor position errors (% 2, % 1, and % 0.2).

Figure 8. Azimuth CRB DOA performance of nine-element UI V array and UCA for two sources when thesources are correlated with the correlation coefficient r.Sources are at 80 and 85 degrees, and elevations arefixed at 90 degrees. SNR is equal to 15 dB.

Figure 9. CRB DOA performance with and withoutunknown mutual coupling of UI V-shaped array andUCA for two sources when one source is swept between 0and 360 degrees while the other source is at 161 degrees.Elevation angles are fixed to 90 degrees.

Figure 10. CRB DOA performance with and withoutunknown mutual coupling for three sources at 60, 100,and 120 degrees, respectively, when UI Varray and UCAare used. Elevation angles are fixed to 90 degrees.


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where r is selected as a positive real value in [0,1] forsimplicity. It turns out that the UI V array has betterperformance for the correlated sources signals. Thedifference between V array and the UCA increases asthe value of r increases especially for the values close tor = 1. Note that r = 1 corresponds to the coherent sourcecase.[32] Figure 9 shows the DOA performance when there

are two sources fixed at 161 and 180 degrees and thethird source is swept between 0 and 360 degrees. Figure 9shows the CRB characteristics with and withoutunknown mutual coupling. The SNR is fixed at 20 dB.

It can be easily seen that the coupling decreases theDOA performance. However the DOA performance forUI V-shaped array is better than the UCA for all of theDOA angles.[33] Figure 10 shows the SNR performance of the

UCA and UI V-shaped array for three sources at 60,100 and 120 degrees, respectively, with and withoutunknown mutual coupling. It can be seen that theDOA performance degrades due to mutual couplingbut the performance of UI V-shaped array is better thanthe UCA.

6.2. Simulations for Nonuniform Isotropic V-ShapedArrays

[34] In case of nonuniform V-shaped array, we selectthe left arm as a nonredundant nonuniform linear array(NLA) for simplicity. The sensor locations for the NLA

Figure 11. CRB DOA performance of nonuniformisotropic (NUI) V array and UCA for a single source isswept between 0 and 360 degrees when M = 7, M = 10and elevation angles are fixed to � = 90� and SNR =20 dB.

Table 6. Isotropic Nonuniform V-Shaped Design Examples for

M1 = M2 and M1 6¼ M2

Nonuniform Sensor Positions gisoo (deg)

½6; 4; 1|fflffl{zfflffl}M1¼3

; 0; 1; 4; 6|fflffl{zfflffl}M2¼3

� 61.0530o

½6; 4; 1|fflffl{zfflffl}M1¼3

; 0; 1:691; 2:809; 6:5|fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}M2¼3

� 61.0530o

½17; 12; 10; 4; 1|fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}M1¼5

; 0; 5; 10; 13; 16|fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}M2¼4

� 57.0976o Figure 12. CRB DOA performance for two sources at�1 = 81�, �2 = 98�, respectively, when DU V array andUCA are used (elevation angles are fixed to � = 90�).

Figure 13. The elevation CRB for DU V array andUCA with different azimuth angles (for �1).


9 of 12

RS5006

with respect to d = l2are dNLA = [0, 1, 4, 6]. The right arm

can be adjusted to have M1 = M2 or M1 6¼ M2. Sensorpositions of the right subarray are selected in order tosatisfy (20). Then g�iso is determined from (21). Some ofthe examples for isotropic nonuniform arrays are pre-sented in Table 6. CRB levels of the designed isotropicnonuniform arrays are given in Figure 11. Figure 11shows that DOA accuracy can be significantly improvedwith nonuniform V-shaped arrays for the same number ofsensors. Note that this result is obvious due to the factthat array aperture is increased. However, NLA stillreturns unambiguous solutions since there is at leasttwo sensors with the intersensor distance less than l

2.

6.3. Simulations for Directional Uniform V-ShapedArrays

[35] In the directional case, sources are assumed to belocalized in an angular sector. We choose design param-eters as a1 = 80

o, a2 = 100o and H1 = 0.5�. Angular step

size is D = 1� for M = 9 sensors and the number ofsnapshots N = 256. If the design procedure is applied forthese parameters, the best DOA performance is obtainedfor g� = 119�. In Figure 12, there are two sources at f1 =81� and f2 = 98� degrees. Figure 12 shows that designeddirectional uniform (DU) V-shaped array has better DOAperformance than UCA and L-shaped array (g = 90�).The DOA performance for the elevation angle is shownin Figure 13 for f1. As it is seen from Figure 13,elevation performance of the directional V-shaped arraychanges depending on the azimuth angle. Circular arrayhas uncoupled azimuth and elevation angle response.Figure 14 shows the DOA performance when there aretwo sources fixed at 83 and 99 degrees and third sourceis swept between 0 and 360 degrees in one degree

resolution. SNR is fixed at 20 dB. Figure 14 shows thatDU V-shaped array has significantly better resolutionand DOA performance than UCA.

7. Conclusion

[36] We have investigated the uniform and nonuniformisotropic and directional V-shaped planar arrays. Closedform expressions for the isotropic performance arepresented for both uniform and nonuniform V arrays.V-shaped isotropic arrays are compared with UCA. Thecomparison is done for a variety of cases which includecorrelated sources, sensor position errors and mutualcoupling. It turns out that the isotropic V-shaped arrayhas better performance than UCA for the same numberof sensors and intersensor distance. The source signalcorrelation and sensor position error do not change thesuperiority of the UI V array. In case of mutual coupling,UI V-shaped array has better performance for multiplesources. It is shown that DOA performance can be im-proved significantly when isotropic nonuniform V-shapedarrays are used.[37] A design method for directional uniform V-shaped

array is proposed. The proposed method finds the opti-mum V angle, g�, for the specified design parameters.When the sources are in an angular sector, DU V-shapedarray performs significantly better compared to UCA. Itturns out that V-shaped arrays have the better perfor-mance for the same number of sensors and interelementdistance due to its effective aperture.

Appendix A: Isotropic VAngle for Uniform

V-Shaped Arrays

[38] In this Appendix A, we derive the closed formequation (18) which returns isotropic Vangle for uniformV-shaped arrays. The array center of gravity (xc,yc) foruniform and symmetric V-shaped arrays, is given as

xc ¼1

Md sin g=2ð Þ

XMl¼1

l kð Þ ¼ 0 ðA1Þ

yc ¼1

Md cos

g2

� �XMl¼1

jl kj

¼ 1M

d cosg2

� � M 2 1ð Þ4

: ðA2Þ

For isotropic V-shaped arrays Pxy, must be zero. Sincexc = 0,

Pxy ¼XMl¼1

xlyl XMl¼1

xlyc: ðA3Þ

Figure 14. CRB DOA performance of DU V array andUCA for three sources when one source is sweptbetween 0 and 360 degrees while the other sourcesare at 83 and 99 degrees. Elevation angles are fixed to90 degrees.


10 of 12

RS5006

If we open this equation,

Pxy ¼ d2 sing2

� �cos

g2

� � XMl¼1

l kð Þjl kj

M2 1ð Þ4M

XMl¼1

l kð Þ!

ðA4Þ

wherePM

l¼1(l k)jl kj = 0 andPM

l¼1(l k) = 0,so Pxy = 0. Pxx must be equal to Pyy for isotropicresponse. We can find Pxx as,

Pxx ¼XMl¼1

xlð Þ2¼ d2 sin2g2

� �XMl¼1

l kð Þ2

which gives

Pxx ¼ 2d2 sin2g2

� � M 2 1ð ÞM24

: ðA5Þ

Then we need to find Pyy

Pyy ¼XMl¼1

yl ycð Þ2¼XMl¼1

y2l þ y2c 2ylyc

�

¼XMl¼1

y2l þMy2c 2ycXMl¼1

yl ðA6Þ

where

XMl¼1

y2l ¼ 2d2 cos2g2

� �XM12l¼1

l kð Þ2

¼ 2d2 cos2 g2

� � M 2 1ð ÞM24

ðA7Þ

and

My2c ¼ d2 cos2g2

� � M 1ð Þ2 M þ 1ð Þ216M

ðA8Þ

2ycXMl¼1

yl ¼ d2 cos2g2

� � M2 1ð Þ2M

XMl¼1

jl kj

¼ d2 cos2 g2

� � M 1ð Þ2 M þ 1ð Þ28M

: ðA9Þ

Therefore if we combine the expressions in (A7), (A8)and (A9), with (A6), we get Pyy as,

Pyy ¼ d2 cos2g2

� � M 2 1ð Þ8

M 2 þ 36M

: ðA10Þ

Using the equations (A5) and (A10) in order to satisfy(13), V angle is found as,

gisoM ¼ 2� arctanffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM 2 þ 34M 2

r !: ðA11Þ

Appendix B: Isotropic V Angle for

Nonuniform V-Shaped Arrays

[39] In this part, the derivation of (20) and (21) fornonuniform V-shaped isotropic planar array is presented.

xc ¼1

Msin g=2ð Þ

XMl¼M1þ2

dl XM1l¼1

dl

!ðB1Þ

yc ¼1

Mcos g=2ð Þ

XM1l¼1

dl þXM

l¼M1þ2dl

!ðB2Þ

Pxy must be zero for isotropic response.

Pxy ¼XMl¼1

xl xcð Þ yl ycð Þ ¼ 0

¼XMl¼1

xlyl Mxcyc ¼ 0

XMl¼1

xlyl ¼ Mxcyc ðB3Þ

The above equation is satisfied only if

XM1l¼1

dl ¼XM

l¼M1þ2dl and

XM1l¼1

d2l ¼XM

l¼M1þ2d2l : ðB4Þ

So xc becomes zero and Pxy = 0. We need to equate Pxx toPyy in order to get isotropic response.

Pxx ¼XMl¼1

xl xcð Þ2¼XMl¼1

x2l

¼ sin2 g2

� � XM1l¼1

d2l þXM

l¼M1þ2d2l

!ðB5Þ

If (B4) is satisfied, Pxx and Pyy can be written as,

Pxx ¼ 2 sin2g2

� �XM1l¼1

d2l ðB6Þ


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RS5006

Pyy ¼XMl¼1

yl ycð Þ2¼XMl¼1

y2l My2c

¼ 2 cos2 g2

� �XM1l¼1

d2l My2c

where

My2c ¼4

Mcos2

g2

� � XM1l¼1

dl

!2: ðB8Þ

Therefore if we substitute (B8), into (B7), we get Pyy as,

Pyy ¼ 2 cos2g2

� � XM1l¼1

d2l 2

M

XM1l¼1

dl

!20@1A: ðB9Þ

If we equate (B6) and (B9) in order to satisfy isotropycondition (Pxx = Pyy), we get giso as in (21).

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ðB7Þ


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