Uncertainty analysis of the standard delay-and-sum beamformer and array calibration

Beamforming has become an ubiquitous task in aeroacoustic noise measurements for source localization and power estimation. The standard delay-and-sum (DAS) beamformer is the most commonly used beamforming algorithm due to its simplicity and robustness and also serves as the basis for more sophisticated algorithms, such as the deconvolution approach for the mapping of acoustic sources (DAMAS). The DAS data reduction equation is a function of many parameters including the microphone locations, microphone transfer functions, temperature and the cross-spectral matrix (CSM), where each one of these parameters has a unique uncertainty associated with it. This paper provides a systematic uncertainty analysis of the DAS beamformer and Dougherty's widely used calibration procedure under the assumption that the underlying mathematical model of incoherent, monopole sources is correct. An analytical multivariate method based on a first-order Taylor series expansion and a numerical Monte-Carlo method based on assumed uncertainty distributions for the input variables are considered. The uncertainty of calibration is analyzed using the Monte-Carlo method, whereas the uncertainty of the DAS beamformer is analyzed using both the complex multivariate and the Monte-Carlo methods. It is shown that the multivariate uncertainty analysis method fails when the perturbations are relatively large and/or the output distribution is non-Gaussian, and therefore the Monte-Carlo analysis should be used in the general case. The calibration procedure is shown to greatly reduce the uncertainties in the DAS power estimates. In particular, 95 percent confidence intervals for the DAS power estimates are presented with simulated data for various scenarios. Moreover, the 95 percent confidence intervals for the integrated DAS levels at different frequencies are computed using experimental data. It is shown that with experimental data, the 95 percent confidence intervals for the integrated power levels are within of the mean levels when the component uncertainties are set at low but achievable values.