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

Localizing linearly moving sound sources using microphone arrays is 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, 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 data at the microphone grid, the unknown source distribution in the moving frame can be determined. First, the time signals need to be transformed from time into frequency domain using a windowed discrete Fourier transform (DFT), which leads to spectral leakage that depends on the length of the time interval and the analysis window used. To include spectral leakage in the numerical model, the calculation of the transfer matrix is modified using the Fourier transform of the analysis window in the DFT applied to the measurements. Currently, this approach is limited to single-frequency sources as this restriction allows for simplified calculations and reduces the computational effort. The least squares problem is solved using a Tikhonov regularization and an L-curve approach. 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 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 source frequency, the distance between source and receiver, and the robustness with respect to noise.