Azimuth and Elevation Direction Finding with Planar Arrays Using
Thompson's Adaptive Method
Sourabh Biyani*, Randy Haupt, Tamal Bose
Utah State University
Electrical and Computer Engineering
4120 Old Main Hill
Logan, UT 84322-4120
Abstract. This paper presents a method for resolving the directions of arrival in
azimuthal and elevation plane simultaneously using a planar array. An adaptive algorithm
based on Pisarenko's harmonic retrieval method is presented and implemented.
1. Introduction
A crucial problem in the area of wireless and mobile communication is the
estimation of direction-of-arrival of user signals and planar waves impinging on
the array elements. In [I J, the azimuthal direction finding capabilities of different planar
array geometries are compared. Direction finding in the azimuthal and elevation plane
using the ESPRIT algorithm is shown in [2]. In this paper, we present an adaptive version
of Pisarenko's harmonic retrieval method to determine the elevation and azimuthal angle
simultaneously using a planar array. The method is based on the LMS algorithm and has
a low computational complexity.
11. Problem Formulation
Consider an array consisting of M uniformly spaced antenna elements and p @<M)
narrowband uncorrelated sources impinging on the array from directions
(O,,h),...,(6p,+p),where sand # are the elevation and azimuthal angles
respectively. The bf x 1 vector of a snapshot received by the array can be expressed as
X(
t ) = A (0,
a)). ( t )+ n ( t )
(1)
where A (a,@)
is the M x p matrix of the steering vectors,
U
(6,,
4, ) is the steering vector of the array in the direction of
(e,,4, ) . Our problem can
be stated as follows: Given the sampled data x , determine the directions of amval
(6,, h ) ,. ..,( 6p,+p)of the signals impinging on the array.
To solve this problem we assume that the steering vector { u ( Q , # ) }is known and the
signals and noise are stationary and ergodic complex valued random processes with zero
mean. In addition, the noise is assumed to be uncorrelated with the signals. It can be
shown that the problem of estimating the directions-of-arrival of multiple signals is
0-7803-8302-8/04/$20.00
02004 IEEE
399
similar to the harmonic retrieval problem [3]. One such solution to this problem known as
Pisarenko's Harmonic Retrieval Method is presented in [4]. It consists of finding the
andithe
nassociated eigenvector q,," of the correlation matrix
minimum eigenvalue ;Im
R,, and computing the zeros (roots)
of the polynomial whose coefficients are the
components of eigenvector, q,,, ,given by
It was shown in [5] that by constraining the impulse response (w ) of an adaptive filter to
have a unit norm, w converges to qm,.. Using an adaptive LMS algorithm with weight
normalization,we can steer the nulls of an antenna array in the direction of signals whose
direction-of-arrival is to be estimated. When this algorithm converges, the roots are on
the unit circle in the direction corresponding to the electrical angles. The algorithm is
given as follows:
For ~ = 0 . 1 .....
2
e ( . ) = -w(n)' x ( n >
(4)
w ( n + 1) = w (n)+ ye' ( n ) x( n )
w ( n + l ) =-
w(n+l)
llw(n+1)ll
The electrical angles are computed by finding the roots of the polynomial whose
coefficients are the updated weights of antenna elements as given by (4). The direction of
arrivals (DOAs) can then be obtained from the estimated electrical angles.
The array factor for a planar array having M elements in the x-direction separated by
distance d, and N elements in the y-direction separated by d, is given by
where wm, is the weight on the antenna element at position ( m n ) and k = 2n/ A .
If the planar array is separable, then the analysis of a planar array can be seen as a
analysis of two linear arrays in orthogonal directions [3]. The steering vectors,
U, and U, in x- and y-direction respectively are given by
400
The LMS algorithm is performed in two orthogonal directions with a unit norm
constraint. The roots of the polynomial formed by the updated element weights give an
estimate of electrical angles as
qX=M,sin(8)cos(&
G, =kdYsin(G))sin@)
(8)
'
We can extract the azimuthal angles from (8) as
and the elevation angle can then be obtained from (8) and (9) as
Ill. Results
An L-shaped planar may with 8 elements in the x and y directions has 2 sources
impinging at an elevation of (12",25") and an azimuth of (0",30"). The LMS
algorithm worked with 1500 snapshots of data. The estimated elevation and azimuthal
DOAs obtained were (11.85",25.19")and (0.31",29.82"). The nulls of the antenna
pattem on the unit circle are shown in Figures 1 and 2. The antenna pattem is plotted with
respect to {(ux,vy)
and shown in Figure 3.
0
1
J)
i
O
5[i
I
.I!
%--.--..
-1
_.-.
*
-0.5
0
_.---__..-..-_'
0.5
1
Real Pad
Figure 1. Root locations of polynomial Figure 2. Root locations of polynomial
Q ( z ) for the x-direction. The location Q ( z ) for the y-direction. The location
correspond to the electrical angle,
(ux.
correspond to the electrical angle,
401
(uy .
Figure 3. 3-D antenna pattem of a 8 x 8 L-shaped planar array, used to resolve the
directions of 2 source signals at different azimuth and elevation angles.
IV. Conclusion
It is shown that a separable L-shaped planar array can he considered as two linear arrays
in orthogonal directions. This allows us to estimate the DOAs in azimuthal and elevation
planes simultaneously. The LMS algorithm implemented has a computational complexity
of O(M) and can be performed in real time. Root finding involves many computations, so
it is not a real time process.
References
[ l ] A. Manikas, A. Alexiou, H. R. Karimi, “Comparison of the ultimate direction finding
capabilities of a number of planar array geometries,” IEE Proceedings - Radar ,
Sonar andNavigation, Vol. 144, Issue 6, pp. 321-329, Dec. 1997.
[2] A. Swindlehurst, T. Kailath, “Azimuth / Elevation direction finding using regular
array geometries,” IEEE Trans. on Aerospace and Electronic Systems, Vol. 29, Issue
1, pp. 145-156, Jan. 1993
[3] S. Biyani, “Direction Finding Arrays,” Masters Thesis, Dept. Of Electrical
and Computer Eng., Utah State University, Logan, Dec. 2003
[4] V.F. Pisarenko, “The retrieval of harmonics from a covariance function”,
Geophysical Journal of Royal Astronomical Society, Vol. 33, No. 3, 1973.
[5] P. A. Thompson, “An adaptive spectral analysis technique for unbiased frequency
estimation in the presence of white noise”, 13‘* Asilomar Conference on Circuits,
Systems and Computers, Vol. 13, Nov. 1979.
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