Sub-Sample Time Delay Estimation via Auxiliary-Function-Based Iterative Updates

We propose an efficient iterative method to estimate a sub-sample time delay between two signals. We formulate it as the optimization problem of maximizing the generalized cross correlation (GCC) of the two signals in terms of a continuous time delay parameter. The maximization is carried out with an auxiliary function method. First, we prove that when written as a sum of cosines, the GCC can be lower bounded at any point by a quadratic function. By repeatedly maximizing this lower-bound, an alternative update algorithm for the estimation of the time delay is derived. We follow through with numerical experiments highlighting that given a reasonable initial estimate, the proposed method converges quickly to the maximum of the GCC. In addition, we show that the method is robust to noise and attains the Cramér-Rao lower bound (CRLB).