Optimal three dimensional antennas for adaptive applications (such as the SAGE Algorithm)

This study investigates in a first part optimal three dimensional antennas for adaptive applications. Spherical arrays provide wide scan coverage with low grating lobe levels. The choice of the antenna element distribution on the spherical surface is governed by the desire to maintain a nearby constant pattern. We calculate the uniform distribution of a set of antennas on a sphere. Based on formulas of the spherical trigonometry, we show the only five possible exact uniform distributions. Because the number of possible uniform spherical distributions is limited, the number of elements that can be placed uniformly on a spherical surface is also limited to 20. In a second part, the behaviour of the Space Alternating Generalised Expectation maximisation (SAGE) algorithm with an uniform three dimensional antenna distribution is calculated in a multipath environment. This optimisation algorithm is used to replace the high dimensional optimisation procedure necessary to compute the joint maximum likelihood estimate of the parameters by several separate maximisation processes which can be performed sequentially. The calculated parameters in this paper are the relative delay, the azimuth angle, elevation angle and the complex amplitude of the waves. Convergence and performance analysis are based on numerical simulations. We calculate the resolution of this algorithm, the convergence speed, and the Bit Error Rate (BER) as function of Signal to Noise ratio (SNR). The investigations demonstrate that the SAGE algorithm is a powerful tool that can be succesfully applied for parameter extraction with high resolution