Calculation of the cross spectral matrix with Daniell's method and application to acoustical beamforming

In microphone array techniques the cross spectral matrix is used to localize acoustic sources. In this paper Daniell's method is revisited to average the cross spectral matrix of the sensor signals in the frequency domain. It is shown that this method leads to similar results as an averaging in the time domain with Welch's method concerning the reduction of variance, the correlations between different sources and the localization of multiple sources. Furthermore, the phase information of the sensor signals during the averaging process is analyzed. The correct estimation of the cross spectral matrix is necessary for calculating the true sound pressure level of multiple sources with standard and advanced beamforming methods. Lastly, the application to acoustical beamforming is shown by using simulated and experimental data of uncorrelated sources for different averaging window sizes. Simulated and measured, stationary and rotating sound sources are examined. Especially for rotating sound sources an averaging in the frequency domain is useful due to different source positions in different time segments as a result of the source motion.