Crame´R-Rao Bound of Direction Finding using Uniform Circular Array And 2-Circle Concentric Uniform Array

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. x