The Statistics of Superdirective Beam Patterns

Abstract

Superdirective arrays have been extensively studied because of their considerable potential accompanied, unfortunately, by a high sensitivity to random errors that affect the responses and positions of array elements. However, the statistics of their actual beam pattern (BP) has never been systematically investigated. This paper shows that the Rician probability density function (PDF), sometimes adopted to study the impact of errors in conventional arrays, is a valid approximation for superdirective BP statistics only where some mathematical terms are negligible. The paper also shows that this is the case for all linear end-fire arrays considered. A similar study is proposed concerning the correlation between BP lobes, showing that for the superdirective arrays considered the lobes, especially non-adjacent ones, are almost independent. Furthermore, knowledge of the PDF of the actual BP allows one to define quantile BP functions, whose probability of being exceeded, at any point, is fixed. Combining the lobes' independence with quantile BP functions, an empirical equation for the probability that the entire actual BP will not exceed a quantile function over an interval larger than a given size is obtained. This new knowledge and these tools make it possible to devise new methods to design robust superdirective arrays via optimization goals with clearer and more relevant statistical meaning.

Design of Planar Differential Microphone Array Beampatterns with Controllable Mainlobe Beamwidth and Sidelobe Level

“…So, the point (x 0 , R) on the T N (x) curve is the position of the mainlobe maximum. According to (22), we have…”

“…To overcome these drawbacks, many broadband beamforming methods have been developed, including the narrowband decomposition methods [ 12 , 13 ], the nested-array framework [ 14 , 15 , 16 ], the modal beamforming techniques [ 17 , 18 , 19 , 20 ], the super directive algorithms [ 21 , 22 , 23 ] and the differential beamformers (or differential microphone arrays, DMAs) [ 24 , 25 , 26 , 27 , 28 , 29 ]. Among those, the DMAs have been widely studied as they exhibit frequency-invariant beampatterns and high DF.…”

Efficient Design of Super-Directive Antenna Array using Schelkunoff Method and Genetic Algorithm

“…However, by reducing the separation distance and using appropriate excitation functions, the mutual coupling effect between array elements can be compensated through the optimized excitations of the super-directive feed network. Composite polynomials have been widely utilized to generate efficient array excitation functions [1] - [5] that lead to a 20-30% reduction in the half-power beam width. However, for an N-element antenna array with an overall length of several wavelengths, the computational time and design challenges in realizing the array excitation functions have hindered the application of this class of super-directive arrays [6] - [14].…”