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.
Highlights
IN data-independent beamforming [1], superdirectivity theory is often used [2-5] when the array aperture is comparable to or less than the wavelength
Constrained optimization techniques have been proposed [1, 3-5, 12], as well as techniques that leverage the probability density function (PDF) of random errors to optimize the expected beam power pattern (EBPP, i.e., the mean power of the ABP) or other statistical functions [18-22]
This paper introduces new knowledge regarding the statistics of beam pattern (BP) that characterize superdirective arrays
Summary
IN data-independent beamforming [1], superdirectivity theory ( called supergain) is often used [2-5] when the array aperture is comparable to or less than the wavelength. Mathematical tools are provided to test whether the Ricianity assumption for the PDF of the ABP modulus is acceptable, given the array geometry, the weight coefficients and the statistics of random errors. Because of the peculiar importance of linear end-fire arrays [4, 6] in the field of superdirectivity, this paper confirms the Ricianity of their ABPs and opens the possibility of evaluating the robustness of a specific design by the quantile BP functions and the probability of no overruns larger than a fixed size. The paper is organized into two parts: in the first (Sections II to V) Ricianity is studied and quantile functions are introduced; in the second (Sections VI and VII) the independence between side lobes is investigated and the probability of no overruns larger than a certain size is derived
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