The effects of contaminating noise on the calculation of active acoustic intensity for pressure gradient methods

Bias errors for two-dimensional active acoustic intensity using multi-microphone probes have been previously calculated for both the traditional cross-spectral and the Phase and Amplitude Gradient Estimator (PAGE) methods [Whiting, Lawrence, Gee, Neilsen, and Sommerfeldt, J. Acoust. Soc. Am. 142, 2208-2218 (2017)]. Here, these calculations are expanded to include errors due to contaminating noise, as well as probe orientation. The noise can either be uncorrelated at each microphone location or self-correlated; the self-correlated noise is modeled as a plane-wave with a varying angle of incidence. The intensity errors in both magnitude and direction are dependent on the signal-to-noise ratio (SNR), frequency, source properties, incidence angles, probe configuration, and processing method. The PAGE method is generally found to give more accurate results, especially in direction; however, uncorrelated noise with a low SNR (below 10-15 dB) and low frequency (wavelengths more than 1/4 the microphone spacing) can yield larger errors in magnitude than the traditional method-though a correction for this is possible. Additionally, contaminating noise does not necessarily impact the possibility of using the PAGE method for broadband signals beyond a probe's spatial Nyquist frequency.