Local minima-free adaptive tdoa estimation for acoustic sources in presence of high spatial aliasing

Acoustic source time-difference of arrivals (TDOAs) estimation is a basic task required by source localization and separation techniques. Standard wide-band methods, e.g. the GCC-PHAT, estimate the TDOAs by evaluating the cross-correlation at multiple time lags and performing a direct-search of the global maximum. A more convenient formulation is to model the time-delay in the frequency domain and casting the estimation to an adaptive problem. By minimizing an integral cost function between the observed cross-power spectrums and frequency-dependent phasors, the TDOA can be adaptively estimated by a normalized gradient descent approach without requiring any explicit direct search and providing a continuous TDOA estimate in the space. However, in presence of spatial aliasing the cost function is non-convex and a gradient-based adaptation is not guaranteed to converge to the global optimum. In this work we propose a structure of interleaved gradients derived on cost functions with a progressive degree of non-convexity. The adaptations proceed in parallel propagating the belief from the most to the least convex approximations in order to induce the overall optimization to escape from local minima. Through extensive Monte Carlo simulations we show that the proposed approach is virtually insensitive to local minima and can converge to the global optimum independently on the initialization even with a microphone spacing of 1 meter.