Efficient Calculation of Directivity Indices for Certain Three-Dimensional Arrays

摘要

Array directivity is defined as the ratio of the output signal-to-noise ratio ofan array to the input signal-to-noise ratio at an omni-directional element in an 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, theat, 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. Except for a few simple array geometries and unique frequencies, analytical solutions for this two-fold integration can generally not be obtained. Numerical integration techniques must therefore be used to calculate directivity. For large-aperture arrays, consisting of thousands of array elements, conventional integration procedures, such as a double application of Simpson's rule over azimuth and elevation angles, is woefully time consuming and can be potentially inaccurate. This study presents a number of procedures which dramatically decrease the time required to numerically calculate directivity for certain array shapes and weights. Algorithms, written on a SUN SPARC station 10 in Fortran 77, are provided in the appendices. However, the factorization procedures described here can be implemented equally well in other programming languages, such as MATLAB. TM The enclosed programs, which incorporate all of the procedures described herein, can be used to calculate the directivity of certain planar arrays, as well as three-dimensional arrays which are conformal to the cylindrical portion of a submarine hull, for example.