Approximations to directivity for linear, planar, and volumetric apertures and arrays

Even after decades of sonar design, approximations to the directivity factor (DF) 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 array´s 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 (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 θ and a polar angle φ. Hence, calculation of the directivity requires a two-fold integration over the angular space defined by the azimuthal and polar angles. For complex, large-size 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 shading coefficients, steering angles, and directional array element responses