Fast iteration algorithms for implementing the acoustic beamforming of non-synchronous measurements

Acoustic beamforming is a technique to measure the sound field and then achieve the localization of the sound sources. It usually works at a high frequency because the size and density of the microphone array are limited. Differing from the traditional beamforming, non-synchronous measurements beamforming can widen the working frequency bandwidth with a movable microphone array. It generates a cross-spectral matrix of all the microphones in the sequential positions of measurements, and this matrix is incomplete due to the missing phase relationships between non-synchronous measurements. Thus the non-synchronous measurements beamforming can be regarded as a cross-spectral matrix completion problem. One application of the non-synchronous measurements beamforming is that it can immediately obtain a completed noise image by scanning a large sound source; then the primary sound sources are expected to be identified by the global image. However, it is still a challenge for the current non-synchronous measurements beamforming algorithm since the computation time is long when more numbers of measurements are involved. In this paper, two algorithms based on the augmented Lagrange multiplier method and alternating direction method of multipliers are proposed for achieving fast implementation. The proposed algorithms can improve the iteration speed of the non-synchronous measurements beamforming which is proved first in the simulation, and they are further applied in the on-site measurement of the vehicle engine compartment (the data missing cross-spectral matrix of 15 non-synchronous measurements for a given frequency can be completed only in a few seconds). This industrial application has shown the potential of the proposed methods to be further applied for the sound source imaging of large mechanical systems.