Localizing broadband noise sources using the Loève spectrum and a 2.5D approach

The localization of moving sound sources using a microphone array is typically based on modifying the signal to compensate for the Doppler effect. In the time domain this compensation is done on a sample-by-sample basis. In the frequency domain short time segments need to be used in which the Doppler effect is assumed to be approximately constant and a discrete Fourier transform is done on each segment. In contrast, the authors developed an inverse 2.5D localization method for uniformly moving single-frequency sources that works in the spectral domain and allows for the use of longer windows. This was achieved by modifying the 2.5D forward model to directly compute the effect of the motion in the static observer position. The method does neither require to modify the measured signal nor does it require quasi-stationary of the measurements within the window used. Unfortunately, this approach is not directly suitable for broad-band stochastic sources, and in the present work we will investigate how the statistical properties of a uniformly moving stochastic source change when observed at a static observer. Using a 2.5D setting, the relation between the power spectral density of the moving source and the Loève spectrum, which is a generalization of the cross-spectral density at the static receivers, was derived. Based on simulated data with speeds up to 100 m\,s$^{-1}$, the work presented here provides a proof of concept for a method based on multi-taper estimates for the Loève spectrum to localize moving broad-band stochastic sources . Currently, the method requires a stationary source signal and that the spectral density is flat within a certain range around the frequency of interest. Also, correlations between sources are currently not considered.

Localizing uniformly moving mono-frequent sources using an inverse 2.5D approach

Localizing linearly moving sound sources using microphone arrays is particularly challenging as the transient nature of the signal leads to relatively short observation periods. Commonly, a moving focus is used and most methods operate at least partially in the time domain. In contrast, here an inverse source localization algorithm for mono-frequent uniformly moving sources that acts entirely in the frequency domain is presented. For this, a 2.5D approach is utilized and a transfer function between sources and a microphone grid is derived. By solving a least squares problem using the data at the microphone grid, the unknown source distribution in the moving frame can be determined. For that the measured time signals need to be transformed into the frequency domain using a windowed discrete Fourier transform (DFT), which leads to effects such as spectral leakage that depends on the length of the time interval and the analysis window used. To include these effects in the numerical model, the calculation of the transfer matrix is modified using the Fourier transform of the analysis window. Currently, this approach is limited to mono-frequent sources as this allows a simplification of the calculation and reduces the computational effort. The least squares problem is solved using a Tikhonov regularization employing an L-curve approach to determine a suitable regularization parameter. As a moving source is considered, the Doppler effect allows to enhance the stability of the system by combining the transfer functions for multiple frequencies in the measured signals. The performance of the approach is validated using simulated data of a moving point source with or without a reflecting ground. Numerical experiments are performed to show the effect of the choice of frequencies in the receiver spectrum, the effect of the DFT, the frequency of the source, and the distance of source and receiver.