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Effect of Geometry of Planar Antenna Arrays on

Cramer-Rao Bounds for DOA Estimation
Xiaopeng Yang*,1, Teng Long1 and Tapan K. Sarkar2
1
School of Information and Electronics, Beijing Institute of Technology, Beijing 100081, China
2
Department of Electrical Engineering and Computer Science, Syracuse University, Syracuse, NY 13244, USA
* E-mail: xiaopengyang@bit.edu.cn

Abstract—In this paper, the effect of the geometry of planar The rest of this paper is organized as follows. The basic
antenna arrays on the accuracy of Direction of Arrival (DOA) derivations of CRBs for 2-D DOA estimation are introduced in
estimation of impinging signals is studied. The expressions of Section 2. The geometry and the received voltage expressions
Cramer-Rao Bounds (CRB) for the two-dimensional (2-D) DOA of the hexagonal, square, L, cross and Y shaped antenna arrays
estimation are derived. Based on the hexagonal, square, L, cross are given in Section 3. The simulation results are presented in
and Y shaped antenna arrays, the CRBs of the elevation and Section 4. Finally, this paper is summarized in Section 5.
azimuth angles of estimated DOA are investigated according to
the received power ratio of signal to noise (SNR) and the fixed
and the changing direction of impinging signal. II. CRAMER-RAO BOUNDS FOR 2-D DOA ESTIMATION
The received signals of the planar antenna array (2-D) can
Keywords-Cramer-Rao Bounds (CRB); Direction of Arrival be formulated as
(DOA) Estimation; Antenna Arrray
x ' n, m x n, m  w n , m (1)
I. INTRODUCTION
where x(n, m) denotes the signals received by each antenna
Direction of Arrival (DOA) estimation of signal source has
elements; w(n, m) denotes the additive noise on each receiving
been an extensively investigated topic in the signal processing
branch and x'(n, m) denotes the received signal contaminated
and received considerable attentions in many fields such as
with noise. n and m are the index of antenna elements in the 2-
radar, sonar, radio astronomy and mobile communications [1].
D antenna array. One snapshot of the received signals is
The goal of DOA estimation is to use the received data to
usually expressed in one column vector as
determine the angles of arrival of a number of signals
impinging on the antenna array. The techniques to estimate the x vec ^ x(n, m); n, m` (2)
DOA can be categorized as the conventional techniques (such
as delay-and-sum method and Capon’s Minimum Variance w vec ^w(n, m); n, m` (3)
method), the high resolution subspace based techniques (such
as MUSIC and ESPRIT), the maximum likelihood techniques x' vec ^ x '(n, m); n, m` (4)
and the Matrix Pencil method [2]. It has been well known that
the quality of DOA estimation not only depends on the selected where vec{·} denotes a vector filled with the corresponding
estimation algorithm but also depends on the geometry of elements. The general expression of the probability density
antenna array. In contrast to the abundant development of DOA function (pdf) of received signals is
estimation algorithm, the effect of the geometry of antenna 1
array has not yet been sufficiently investigated. The uniformly p x'|M exp  x 'H  E x '
H
k xˆ 1 M x ' E x ' (5)
det S k x' M
spaced linear antenna array (ULA) has been applied in some
works to estimate the 1-D (azimuth only) DOA [3]. But, for the where H, det(·) and E(·) denote the transpose conjugate, the
2-D DOA estimation problem, a planar antenna array is at least
determinant and the expectation, respectively. kx’(M) is the
required. In this paper, the hexagonal, square, L, cross and Y
shaped planar antenna arrays are used for the DOA estimation. covariance matrix of received signals. M is the 4Ph1column
vector of unknown parameters, which is defined by
The effect of the geometry of planar antenna arrays on the
>M1 M2 " M P @
T
accuracy of DOA estimation is investigated by using the M (6)
Cramer-Rao Bounds (CRB). CRB is widely used to evaluate
> Ai J i Ti Ii @
T
the ultimate attainable performance in an estimation problem, Mi (7)
which is a lower bound on the variance of any unbiased
estimation of a deterministic parameter [4]. In this paper, the where T denotes the transpose operation. P is the number of
expressions of CRB for 2-D DOA estimation are firstly derived. impinging signals. Ai, Ji, Ti, and Ii denote the amplitude, phase,
Based on the planar antenna array, the CRBs of the azimuth elevation angle and azimuth angle of the ith impinging signal
and elevation angles of estimated DOA are investigated on the 2-D antenna array.
according to the received power ratio of signal to noise (SNR)
and the fixed and the changing direction of impinging signal.
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978-1-4244-5900-1/10/$26.00 ©2010 IEEE

389
 In most cases, the noise on each receiving branch is varCRB Aˆi ª ª F 1 M º ii º 11
assumed to be the zero mean additive Gaussian white noise ¬¬ ¼ ¼
with variance V2, i.e. E(w)=0, then E(x')=x. Therefore, the varCRB Jˆi ª ª F 1 M º ii º 22
covariance matrix of received signals can be simplified as ¬¬ ¼ ¼
(13)
varCRB Tˆi ª ª F 1 M º ii º 33
kx ' M E x ' E x ' x ' E x '
H ¬¬ ¼ ¼
varCRB Mˆi ª ª F 1 M º ii º 44
H ¬¬ ¼ ¼
E x ' x x ' x (8)
E ww H III. GEOMETRY AND RECEIVED VOLTAGE EXPRESSIONS OF
PLANAR ANTENNA ARRAYS
V 2 I NuN In this paper, the hexagonal, square, L, cross and Y shaped
where N is the total number of antenna elements of the 2-D antenna arrays, are applied for the 2-D DOA estimation. The
array. Consequently, the pdf of received signals (Eq. 5) can be antenna element in all arrays is an identical omni-directional
expressed as sensor. The geometry and the received voltage expressions of
each antenna array are introduced in this Section, which are
1 § 1 2 · prepared for the CRB calculation. Firstly, the standard
p x'|M exp ¨  2 x ' x ¸ (9) hexagonal array [4] is introduced. The 3-D view of hexagonal
SV 2 N © V ¹ array including the definition of the elevation and azimuth
angles is shown in Fig.1 (a) and the geometry of standard
The corresponding 4Ph4P Fisher Information Matrix F is hexagonal array (total number of antenna element NH is 19) is
defined as illustrated in Fig.1 (b).
§ w 2 log p x ' | M ·
ª¬ F M º¼ ij  E ¨ ¸ (10)
¨ wMi wM j ¸
© ¹
where Fij is the (i, j)th block element of F, the dimension of
which is 4h4 in the 2-D DOA estimation. w wMi is the partial
derivative with respect to the ith element Miof M, and log()
denotes the natural logarithm. By applying Eq. (9) into Eq. (10),
Fij can be expressed as
(a) 3-D view (Definition (b) Standard hexagonal
§ wx H wx · of direction) array (NH =19)
1
¨   
¨ wM wM ¸¸
Fij 2 Re
V2 © i j ¹
Fig.1: Geometry of standard hexagonal array

where Re() denotes the real part. By applying Eq. (7) into Eq. The array manifold vector of hexagonal array is denoted by
(11), Fij can be expressed in detail as ­° ª § N  m 1 · dx º ½°
3 dx
A n, m exp ® jS « m sin T sin M  ¨ n  x ¸ sin T cos M » ¾
§ ª wx « O O »¼ ¿°
wx wx º · ¯° ¬ 2 © 2 ¹
H
wx wx wx H
wx wxH H
(14)
¨« »¸ Nx 1 Nx 1 Nx 1
¨ « wAi wAj wAi wJ j wAi wT j wAi wM j » ¸ m  ,  1," , , n 0,1," , N x  m  1
2 2 2
¨« H ¸
¨ « wx wx wx H wx wx H wx wx H wx » ¸
» where Nx is the number of antenna elements in the horizontal
1 ¨ « wJ i wAj wJ i wJ j wJ i wT j wJ i wM j » ¸   
Fij 2 Re ¨ « H »¸
row through the origin, which is odd (3, 5, 7…). The received
V2 ¨ « wx wx wx H wx wx H wx wx H wx » ¸ voltage without noise on each antenna element can be obtained
¨ « wT wAj wTi wJ j wTi wT j wTi wM j » ¸ as
¨« i »¸
¨ « wx H wx wx H wx wx H wx wx H wx » ¸ P
°­ ª 3 dx § N  m  1 · dx º °½
¨« »¸ V n, m ¦ A exp jJ i exp ® jS «m
«¬ 2 O
sin T i sin Ii  ¨ n  x ¸ sin T i cos Ii » ¾
¹O
(15)
¨ « wMi wMi wM j »¼ ¸¹
i

wMi wJ j wMi wT j »¼ °¿
2
©¬ wAj i 1 °¯ ©

Based on the received signals of the planar antenna array where the Gaussian white noise is not included in the received
and calculating the corresponding partial derivatives in Eq. (12), voltages expression, because the noise is independent of
unknown parameters, when the partial derivative is applied, the
then the 4Ph4P Fisher Information Matrix F can be formed
noise part will become zero. Using Eq. (2) to stack each entry
straightforwardly. The CRB on the variance of the unbiased
of V(n, m) into one column, the NH h1 column vector x is
estimator of the ith unknown parameters Mi is the ith diagonal
formed. The partial derivative of x with respect to the
block element of the inverse matrix of F, namely [F -1(M)]ii. amplitude, phase, elevation angle and azimuth angle of the ith
Therefore, the CRBs of the amplitude, phase and elevation impinging signal can be calculated by
angle and azimuth angle of the estimated impinging signals are
the corresponding diagonal elements of [F -1(M)]ii, respectively, ª wx º
« »
­° ­° ª
vec ®exp jJ i exp ® j 2S « m
3 dx § N  m 1 · dx
sin Ti sin Ii  ¨ n  x
º ½° ½°
¸ sin Ti cos Ii » ¾ ; n, m ¾ (16)
¬ wAi ¼ NH u1 °¯ °¯ «¬ 2 O © 2 ¹O »¼ °¿ °¿
which are shown by

390
ª wx º °­ °­ ª 3 dx § N  m 1 · dx º °½ °½ ­P ª dy º
« »
¬ wJ i ¼ NH u1
vec ® jAi exp jJ i exp ® j 2S « m
«¬ 2 O
sin Ti sin Ii  ¨ n  x ¸ sin Ti cos Ii » ¾ ; n, m ¾
¹O
(17) °¦ Ai exp jJ i exp « j 2S m sin T i sin Ii »  N y d m d N y , n 0
¯° ¯° © 2 »¼ °¿ °¿ °i 1 ¬ O ¼
V m, n ® P (22)
° A exp jJ exp ª j 2S n d x sin T cos I º  N d n d N , m 0
°¦
­ ª § N  m 1 · dx º ½
° j 2S « m
3 dx
cos Ti sin Ii  ¨ n  x ¸ cos Ti cos Ii » ˜ ° i i « O i i » x x
«¬ 2 O ¯i 1 ¬ ¼
ª wx º °° © 2 ¹O »¼ °°
« »
¬ wTi ¼ N u1
vec ®
° ­° ª 3 dx § Nx  m 1 · dx º ½°
¾
°
(18)
H
° Ai exp jJ i exp ® j 2S « m 2 O sin Ti sin Ii  ¨ n 
°¯ ° «
¬ © 2
¸ sin Ti cos Ii » ¾ ; n, m °
¹ O »
¼¿ ° °¿
The received voltage of Y shaped array is
¯
­ ª 3 dx § N  m 1 · dx º ½
° j 2S « m sin T i cos Ii  ¨ n  x ¸ sin T i sin Ii » ˜ ° ­ P ª dx º 0 d m d N1
ª wx º °° «¬ 2 O © 2 ¹O »¼ °° ° ¦ Ai exp jJ i exp « j 2S m sin Ti cos Ii »
¬ O ¼ n 0, k 0
« »
¬ wIi ¼ N H u1
vec ®
° °­ ª 3 dx § Nx  m 1 · dx º °½ °
¾ (19) ° i1
°P
° Ai exp jJ i exp ® j 2S « m 2 O sin T i sin Ii  ¨ n  ¸ sin T i cos Ii » ¾ ; n, m ° ° A exp jJ exp ­° j 2S n d x 2 ªsin T sin I  sin T cos I º ½° 0 d n d N2
¯° ¯° ¬« © 2 ¹O »¼ ¿° ¿° V m, n, k ®¦ i i ®
O 2 ¬ i i i i ¼¾ (23)
°i 1 ¯° ¿° m 0, k 0
°P ­ ½ 0 d k d N3
°¦ Ai exp jJ i exp °® j 2S k d x 2 ª  sin Ti sin Ii  sin Ti cos Ii º °¾
By applying Eqs. (16-19) into Eq. (12), the Fisher Information °̄ i 1 ¯° O 2 ¬ ¼
¿° m 0, n 0

Matrix F is formed straightforwardly. Then the CRBs of the

amplitude, phase and elevation angle and azimuth angle of the After stacking the received voltages to form one column vector,
estimated signals are obtained by using Eq. (13). the corresponding partial derivatives are calculated. Then the
Fisher Information Matrix can be formed by using Eq. (12).
Consequently, the corresponding CRBs of estimated
parameters can be obtained by analysis of the inverse of the
Fisher Information Matrix.

IV. SIMULATION RESULTS

By analyzing the CRB expressions in Section 2, it has been
found that the number of antenna elements has effect on the
accuracy of the CRB calculation. Therefore, the same number
(a) Square array (b) L array of antenna elements for the planar antenna arrays should be
adopted for the reasonable comparison. In the simulation, 169
antenna elements which are the suitable minimum number are
applied, which also make the hexagonal, square, L, cross and Y
shaped antenna arrays be the corresponding standard shaped
array. Moreover, the array spacing of half wavelength is used
for all the planar antenna arrays.
The CRBs of the elevation and azimuth angles of estimated
DOA versus the received SNR based on the hexagonal, square,
(c) Cross array (d) Y array L, cross and Y shaped antenna arrays are respectively shown in
Fig.2: Geometries of square, L, cross and Y shaped antenna Fig.3 (a) and (b), where the CRBs are expressed in 10log10
arrays format. In the simulation, the elevation and azimuth angles of
the impinging signal on the antenna array (DOA) are assumed
The geometry of square, L, cross and Y shaped planar to be 45°, respectively.
antenna arrays are shown in Fig.2. The same procedure as the
processing of hexagonal array can be applied to these planar
antenna arrays to calculate the corresponding CRBs. In the
analysis of the calculation of CRB based on the hexagonal
array, it has been found that the received voltage is the main
factor for the whole CRB calculation. Because of the length
limitation of this paper, only the equations of received voltages
of square, L, cross and Y shaped planar antenna arrays are
listed as follows. The received voltage of square array is
P ­° ª dy d º ½°
V m, n ¦ A exp jJ i exp ® j 2S « m  1 sin T i sin Ii  n  1 x sin T i cos Ii » ¾ (a) (b)
O O
i
i 1 °¯ ¬ ¼ °¿ (20) Fig.3: CRBs of estimated DOA versus the received SNR
1 d m d Ny , 1 d n d Nx
(Incident angle: T=45°,I= 45°)
The received voltage of L shaped array is
It is found that the CRBs of estimated DOA become smaller
­P ª dy º with the increase of the received SNR, which is obviously
°¦ Ai exp jJ i exp « j 2S m  1 sin Ti sin Ii » 1 d m d N y , n 1
°i 1 ¬ O ¼ correct based on the common sense knowledge of estimation
V m, n ® P (21)
° A exp jJ exp ª j 2S n  1 dx º theory. The similar accuracy of DOA estimation is obtained by
°¦ i i «
¬ O
sin T i cos Ii » 1 d n d N x , m 1
¼
¯i 1 the hexagonal and square arrays. The best estimation accuracy
of elevation and azimuth angles are respectively obtained by
The received voltage of cross shaped array is the Y and L shaped array. The same CRBs of elevation angles

391
are produced by the L and cross shaped array, because the It is found that the Y, cross and L shaped antenna array give
geometrical configurations of these two antenna arrays are very the better estimation accuracy than the hexagonal and square
similar in the observation quadrant and the elevation angles of antenna array. From Fig.5 (a), it is found that the cross and L
incident signal are 45°. shaped antenna array have almost the same estimation accuracy
for the elevation angle when the azimuth angles of the incident
The effect of the elevation angle of incident signal on the wave are between 30eand 60e. The estimation accuracy of L
CRBs has also been investigated and the corresponding results and Y shaped antenna array is affected by the azimuth angles
are shown in Fig.4, where the azimuth angel of incident signal of incident wave because of the unsymmetrical geometry.
is 45°, and the elevation angels of incident signal are changed
from 1° to 89°. From Fig.4 (a), it is found that the CRBs of the Except the analysis of the performances of planer antenna
estimated elevation angels become larger with the increase of arrays, the sizes of the planar antenna arrays have also to be
the elevation angel of the incident signal, namely the considered for the deployment of the practical DOA estimation
estimation accuracy is reduced. This is because the resolution system. The sizes of the corresponding planar antenna arrays
of planer antenna array becomes low when the direction of the used in the simulation are shown in Table 1.
incident signal is near the plane of antenna array. On the other
hand, it is found that the CRBs of the estimated azimuth Table 1: Sizes of planer antenna arrays
become smaller with the increase of the elevation angel of
Hexagonal Square L Cross Y
incident signal from Fig.4 (b), namely the estimation accuracy array array shaped shaped shaped
is improved. This is because the equivalent aperture of planer
antenna array is improved for the azimuth direction when the Size 127.31d2 144 d2 7056 d2 3528 d2 4073.8 d2
direction of the incident signal is near the plane of antenna
array. The hexagonal and square arrays also give the similar
where d denotes the array spacing of all planar antenna arrays.
estimation accuracy. The Y and L shaped array also produce
It is found that the size of L shaped array is almost 55 times
the best estimation accuracy for the elevation and azimuth
than that of hexagonal array which is the smallest.
angles, respectively.
V. CONCLUSIONS
The impact of the geometry of planar antenna arrays on the
accuracy of DOA estimation has been studied, and the basic
expressions of CRB for the 2-D DOA estimation have been
derived. Based on the hexagonal, square, L, cross and Y shaped
planar antenna arrays, CRBs of the estimated DOA have been
investigated according to the received SNR and the fixed and
the changing direction of impinging signal. It has been found
that the L, cross and Y shaped antenna arrays have the better
(a) (b) performance than the hexagonal and square antenna arrays.
Fig.4: CRBs of estimated DOA versus the incident elevation angle However, the estimation accuracy of L and Y shaped antenna
array is affected by the azimuth angles of incident wave
(Incident angle: T=45°,I= 1°, 2°, …, 89°)
because of the unsymmetrical geometries. Although the
performance of the hexagonal and square antenna arrays is not
as good as the other planner array, the sizes of which are much
smaller than the other planner array. These results obtained in
this paper can give some valuable references for the antenna
array design of the practical DOA estimation systems.

REFERENCES
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(a) (b) [2] N. Yilmazer, “ Efficient Direction of Arrival Estimation and the Three-
Dimensional Matrix Pencil Mehtod to Find DOA of the Signals Along
Fig.5: CRBs of estimated DOA versus the incident azimuth angle with Their Wavelengths,” Ph.D dissertation, Syracuse University, May
(Incident angle: T=1e, 2e, Ă, 89e,I= 45e) 2006.
[3] X. Xu, Z. Ye, Y. Zhang and C. Chang, “A Deflation Approach to
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investigated and the corresponding results are shown in Fig.5, [4] Harry L. Van Trees, “Optimum Array Processing: Part IV of Detection,
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azimuth angels of incident signal are changed from 1eto 89e.

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