Crame´R-Rao Bound Of Direction Finding Using Uniform Circular Array And 2-Circle Concentric Uniform Array

ABSTRACT

Source direction-of-arrival estimation problem has received much attention in recent years

following its significant role in array-signal processing and wide range of applications such

as radar, wireless communication, sonar, seismology among others. Direction finding has

been solved by several techniques such as Maximum likelihood estimator, MUltiple Signal

Classification, Estimation of Parameters via Rotational Invariance Technique and Cram´er-

Rao bound using array of sensors in both uniformly-spaced and non-uniformly-spaced.

The sensors have further been arranged in different geometric patterns ranging from onedimension

to three-dimensional. However, little effort has been made in direction finding

using concentric planar arrays with fixed centers at the Cartesian origin. In this study, a

new planar sensor-array geometry (the 2-circle concentric uniform array geometry) centered

at the Cartesian origin, that maximizes the array’s spatial aperture mainly for bivariate

azimuth-polar resolution of direction-of-arrival estimation problem was proposed. The

proposed geometry provides almost invariant azimuth angle coverage and offers the advantage

of full rotational symmetry (circular invariance) while maintaining an inter-sensor

spacing not exceeding half wavelength (for non-ambiguity with respect to the Cartesian

direction cosines) among other advantages. The study adopted Cram´er-Rao bound technique

of direction finding via a uniform circular array (single ring array) and the proposed

geometry to estimate the bivariate azimuth-polar angles-of-arrival. Both the array manifolds

and the Cram´er-Rao bounds for the uniform circular array and that of the proposed

array grid were derived. Further, a better-accurate performance in direction finding of the

proposed array grid over that of the single ring array grid was analytically verified under

different constraints of investigation. It was found that the proposed sensor-array geometry

has better estimation accuracy than a single ring array and the 2-circle concentric uniform

array geometry would have the best estimation accuracy for minimal number of sensors

hence reducing the hardware cost. The study therefore recommends that the 2-circle concentric

uniform array geometry should be used for direction finding with minimal number

of sensors and with an inter-sensor spacing not exceeding half a wavelength as opposed to

a uniform circular array geometry.