Tight-Frame Reconstruction for Acoustic Intensity Estimation Using Cardioid Microphone Pairs

This paper investigates acoustic intensity estimation using pairs of cardioid microphones based on the cardioid-cardioid (C-C) method. Unlike conventional pressure-difference techniques, the C-C method is intrinsically less sensitive to the relationship between microphone spacing and acoustic wavelength. However, practical microphones inevitably deviate from ideal cardioid directivity, producing direction-dependent estimation errors. To improve robustness against such errors, a measurement framework based on spherical tight-frame microphone configurations is proposed. Directional intensity components measured along multiple axes are combined to reconstruct the three-dimensional acoustic intensity vector. Furthermore, directivity errors are represented using Legendre polynomial and spherical harmonic expansions, and a geometry-dependent leakage metric is introduced to quantify the error-suppression capability of different microphone arrangements. Theoretical analysis and numerical simulations demonstrate that tight-frame configurations effectively suppress direction-dependent errors through geometric averaging. The proposed leakage metric successfully predicts the influence of microphone directivity imperfections on the reconstructed intensity vector. The results further indicate that accurate wide-band acoustic-intensity estimation can be achieved even with relatively large microphone spacings, which are generally impractical in conventional pressure-difference approaches. % The proposed framework provides a physically interpretable and practically useful approach for acoustic intensity measurement using directional microphone arrays.