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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.
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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).
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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.
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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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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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06531 Ankara, Turkey. ([email protected])
ðB7Þ
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