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.

翻译：使用麦克风阵列定位运动声源通常基于对信号进行多普勒效应补偿。在时域中，这种补偿逐样本进行；在频域中，则需采用短时片段（假设该时间内多普勒效应近似恒定）并对每个片段进行离散傅里叶变换。与此不同，作者针对匀速运动的单频声源开发了一种在谱域中工作的逆2.5D定位方法，允许使用更长的时窗。该方法通过修改2.5D正演模型，直接计算运动在静态观测者位置产生的影响，既无需修正测量信号，也无需假设测量在所用时窗内满足准稳态条件。遗憾的是，该方法不直接适用于宽带随机声源。在本工作中，我们将研究静态观测者观测匀速运动随机源时其统计特性的变化规律。基于2.5D框架，推导了运动声源的功率谱密度与静态接收器处互谱密度广义形式——Loève谱之间的关系。通过最高速度达100 m·s⁻¹的仿真数据，本文验证了基于多窗估计的Loève谱方法定位运动宽带随机声源的概念可行性。目前该方法要求源信号平稳，且谱密度在关注频率附近一定范围内保持平坦，同时暂未考虑声源间的相关性。