Laplace entropy and its application to time delay estimation for speech signals

Time delay estimation (TDE) is a basic technique for numerous applications where there is a need to localize and track a radiating source. It is particularly challenging in the presence of noise and reverberation, and when the, source signal is: speech which is inherently nonstationary and random. The most important TDE algorithms for two sensors are based on the generalized cross-correlation (GCC) method. These algorithms perform reasonably well when reverberation or noise is not too high. In an earlier study of the authors, a more sophisticated approach was proposed. It employs more sensors and takes advantage of their delay redundancy to improve the precision of the TDOA (time difference of arrival) estimate between the first two sensors. The approach is, based on the multichannel, cross-correlation coefficient (MCCC) and was found more robust to noise and reverberation. In this paper, we show that this approach can also be developed on a basis of joint entropy. For Gaussian signals, we show that, in the search of the TDOA estimate, maximizing MCCC is equivalent to minimizing joint entropy. But with the generalization of the idea to non-Gaussian speech signals, the joint entropy based new multichannel TDE algorithm manifests a potential to outperform the MCCC-based method. Since there is no rigorous mathematical, formula for speech entropy, we use the assumption that speech can be plausibly modeled by a Laplace distribution and develop a practical approximation of Laplace entropy for TDE of speech signals. The performance of the proposed new algorithm is investigated via simulations.