On Analytic Methods for 2.5-D Local Sound Field Synthesis Using Circular Distributions of Secondary Sources

Local sound field synthesis allows for synthesizing a given desired sound field inside a limited target region such that the field is free of considerable spatial aliasing artifacts. Spatial aliasing artifacts are a consequence of overlaps due to unavoidable repetitions of the space-spectral coefficients of the secondary source driving function. We analyze various conceivable analytic ways of restricting the bandwidth of the spatial spectrum of the driving function such that considerable overlapping is prevented: local spatial bandlimitation (A), spectral windowing (B), and local spatial band-limitation plus spectral windowing (C). While solution B is computationally significantly more efficient than A and C, it provides only limited control over the spatial location around which the aliasing-free region evolves. Solutions A and C provide more flexibility and higher accuracy whereby both achieve largely identical results so that the spectral windowing after the local spatial bandlimitation may be skipped. We present a detailed analysis of the properties of the spatial aliasing artifacts arising in the synthesis of a virtual plane wave. We establish a procedure for predicting the maximum possible size of the aliasing-free target region depending on its location and on the propagation direction of the desired sound field. The results can help reducing regularization in numerical solutions as they represent physical limitations that can be considered in the choice of parameters.