Estimation of Consistent Time Delays in Subsample via Auxiliary-Function-Based Iterative Updates

Abstract

In this paper, we propose a new algorithm for the estimation of multiple time delays (TDs). Since a TD is a fundamental spatial cue for sensor array signal processing techniques, many methods for estimating it have been studied. Most of them, including generalized cross correlation (CC)based methods, focus on how to estimate a TD between two sensors. These methods can then be easily adapted for multiple TDs by applying them to every pair of a reference sensor and another one. However, these pairwise methods can use only the partial information obtained by the selected sensors, resulting in inconsistent TD estimates and limited estimation accuracy. In contrast, we propose joint optimization of entire TD parameters, where spatial information obtained from all sensors is taken into account. We also introduce a consistent constraint regarding TD parameters to the observation model. We then consider a multidimensional CC (MCC) as the objective function, which is derived on the basis of maximum likelihood estimation. To maximize the MCC, which is a nonconvex function, we derive the auxiliary function for the MCC and design efficient update rules. We additionally estimate the amplitudes of the transfer functions for supporting the TD estimation, where we maximize the Rayleigh quotient under the non-negative constraint. We experimentally analyze essential features of the proposed method and evaluate its effectiveness in TD estimation. Code will be available at https://github.com/onolab-tmu/AuxTDE.

Minimum-Spanning-Tree-Based Time Delay Estimation Robust to Outliers

“…The RMSE (W/O OLs) is thus limited by the accuracy of the given TD measurements. In practice, it is desirable to apply some accurate TDE algorithms, such as those in [40], as the postprocessing of MST.…”

“…The peak of the main lobe is basically expected to be close to the true TD, whereas the peak and the true TD do not match because they are affected by noise and other factors. Some methods have utilized the redundancy of multichannel observation to further improve the accuracy of GCC-based TDE [24], [39], [40].…”

Efficient Joint Optimization of Sampling Rate Offsets Using Entire Multichannel Signal

“…These algorithms can handle non-convex and non-linear optimization problems and guarantee the monotonic nonincrease in the cost function. Owing to these properties, the auxiliary function techniques have been successfully applied to various audio applications [28], [31], [39]- [43].…”

“…On the basis of the conjugate symmetry of Υ[t, f ] and Euler's formula [43], we can reformulate the cost function in (23) by the following sum of cosine functions:…”