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Directivity is a parameter of sound source that is key to obtain accurate results from numerical simulations of room acoustics. Unfortunately, it is very time-consuming to obtain, as it requires measurements of the sound pressure level on multiple points on a sphere around sound source located in an anechoic chamber. Current standard for loudspeakers’ formats used in the simulations is a 5 degree resolution equiangular grid, which consists of over 2500 values and some software producers opt for even higher resolution, since the finer the mesh, the more accurate the results. Spherical harmonics are an orthogonal set of special functions defined on the surface of a sphere. As they can represent any spherical function, one can use them to describe the sound source directivity. Being continuous functions, their values can be determined for any point on the sphere, thus there is no problem with data resolution. The paper aims to investigate the accuracy of the source directivity approximation by means of Discrete Spherical Harmonic Transform (DSHT) in the least-squares sense for sparse measurement data. Two types of grids are discussed - equiangular and equiareal. Measurements using equiangular grid are executed at points located at intersections of meridians and parallels spaced equally angularly on the surface of a sphere. Disadvantage of this grid is the unequal points distribution. The closer to the poles of a sphere, the bigger point density along the parallel, which causes having more measurement data around the poles than at the equator. Such approach is not efficient and has to be compensated by use of weights for determining least-squares error fitting. Equiareal grid is based off of a partition of a unit sphere’s surface into a finite number of regions of equal area. The partition of the surface of a sphere is based on Zhou construction for S2 and the set of points used in the measurement consists of the center points of obtained regions. As a result, points are evenly distributed on the sphere, thus there is no need to use weights for the least-squares error calculation using this type of grid. For both types of grid, the directivity data was approximated as continuous spherical function using DSHT. Values calculated at set of angles corresponding to 5 degree resolution equiangular grid were compared with data obtained directly from measurement to estimate the approximation error. Accuracy of the method, in the form of area-weighted root-mean-square error (AWRMSE), was calculated for all considered one-third octave bands and for different maximum orders of spherical harmonics used for approximation. Comparison between approximation by means of DSHT and alternative forms of interpolation such as nearest neighbour, bilinear, spline or bicubic was carried out and discussed.
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