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

TL;DR

This work develops an inverse source localization method for uniformly moving single-frequency sources that operates entirely in the frequency domain using a 2.5D Helmholtz framework. By deriving a transfer function that accounts for motion along the x-axis and incorporating the spectral leakage from windowed DFTs directly into the forward model, the authors solve a Tikhonov-regularized least-squares problem to recover the moving-frame source distribution. The approach leverages the Doppler effect to widen the observation frequency range and explores different strategies for selecting observation frequencies (single-bin, regular grids, random bins), finding that random binization mitigates periodic artifacts and improves horizontal localization, especially for longer analysis windows. Validation with simulated moving-point sources demonstrates localization within a single grid point, with the method showing stronger performance along the direction of motion and clear sensitivity to window length, source frequency, speed, and presence of ground reflections. The work lays groundwork for extensions to broadband and multi-frequency sources and potential integration with boundary element methods for more complex environments.

Abstract

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.

Figures (17)

• Figure 1: DFT of the signal of a moving point source at a stationary receiver. The panels show the spectral amplitudes for a single-frequency source moving at different speeds $v_s$ (a), for different window lengths $T_g$ for a Hanning-window (b), and for a rectangular window (c). The distance between source and receiver plane was set to $r_2=4$ m.
• Figure 2: Overview over the integrand. Shown is the magnitude of $\hat{q}_{}\! \left(\cdot \right)$ in Eq. (\ref{['Equ:Monoxom']}) for ${f_0}=1000$ Hz and source speeds of $v_s=25$ m s$^{-1}$ (dashed orange) and $v_s=50$ m s$^{-1}$ (solid blue). The thin gray lines show the Fourier transforms of a Hanning window: solid for $T_g=1000$ ms, dash-dotted for $T_g=50$ ms. The thin dotted gray line shows the Fourier transform of a rectangular window with $T_g=1000$ ms. The Fourier transforms of the windows are normalized to their peak amplitude for easier comparison.
• Figure 3: Area covered by the source grid (black rectangle) at time $t = 0$ s and the area covered by the stationary microphone array (orange circle) which was used for the test cases.
• Figure 4: Source maps based on a single DFT bin $\Omega_n=\left\{f_0\right\}$ for each microphone. Shown are the amplitudes of the regularized solutions for a source with $f_0=1000$ Hz moving with two velocities: $v_s=1$ m s$^{-1}$ in panels (a) and (b), $v_s=50$ m s$^{-1}$) in panels (c) and (d). A short window ($T_g=50$ ms) was used in panels (a) and (c), a long window with $T_g=1000$ ms was used in panels (b) and (d). The black x denotes the true source position, the red circle the position of the detected maximum. The dynamic range is set to 20 dB and the maps are normalized to the respective peak amplitude.
• Figure 5: Source maps based on the same set of regularly spaced frequencies for each microphone. Shown are the amplitudes of the regularized solutions for ${f_0}=1000$ Hz and $v_s=50$ m s$^{-1}$ using a short window ($T_g=50$ ms) in panels (a) and (b), and a long window ($T_g=1000$ ms) in panels (c) and (d). Panels (a) and (c) show the case of $M=5$; panels (b) and (d) show the case of $M=11$ regularly spaced frequency bins. The black x denotes the true source position, the red circle the position of the detected maximum. The dynamic range is set to 20 dB and the maps are normalized to the respective peak amplitude.