Localizing uniformly moving single-frequency sources using an inverse 2.5D approach

Abstract

The localization of 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 approach is used and most methods operate at least partially in the time domain. In contrast, this manuscript presents an inverse source localization algorithm for uniformly moving single-frequency sources that acts entirely in the frequency domain. 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 measured data at the microphone grid, the unknown source distribution in the moving frame can be determined. First, the measured time signals need to be transformed from the time into the frequency domain using a windowed discrete Fourier transform (DFT), which leads to an effect called spectral leakage that depends on the length of the time interval and the analysis window used.

To include the spectral leakage effect in the numerical model, the calculation of the transfer matrix is modified using the Fourier transform of the analysis window used in the DFT applied to the measurements. Currently, this approach is limited to single-frequency sources as this restriction allows for 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 moving sources are considered, utilizing the Doppler effect enhances 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, the distance between source and receiver, and the robustness with respect to noise.