Approximations to Directivity for Linear, Planar, and Volumetric Apertures and Arrays
NAVAL UNDERSEA WARFARE CENTER NEWPORT DIV RI
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Even after decades of sonar design, approximations to the directivity factor or index of an array are often used inappropriately. Many of the approximations commonly used provide accurate directivity approximations for only the simplest of array geometries. As the arrays size, shape, weighting, and complexity increase, there is a renewed need for better directivity approximations. Directivity is defined as the ratio of the output signal-to-noise ratio SNR of an array to the input SNR at an omnidirectional element in a spherically isotropic noise field. Calculation of directivity is obtained by integrating the magnitude squared response of the array over all angles of incidence. In spherical coordinates, these arrival angles are denoted by an azimuthal angle theta and a polar angle phi. Hence, calculation of the directivity requires a two fold integration over the angular space defined by the azimuthal and polar angles. For complex, large aperture arrays consisting of thousands of array elements, directivity calculations using numerical integration procedures can be time consuming, even on state of the art computing systems. This report provides a number of accurate formulas for estimating the directivity of linear, planar, and volumetric apertures and arrays, which are allowed to have arbitrary amplitude shading coefficients, steering angles, and directional array element responses.
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