Joint time-scale and TDOA estimation: analysis and fast approximation

Relative motion (rm) between a signal source and a receiver causes a time scaling of the signal arriving at the receiver. When estimating the time-difference-of-arrival (TDOA) of a signal at two receivers, time scaling, when not properly accounted for, can introduce a bias that could dominate the estimation errors. Segmentization processing cannot reduce this bias. Following the derivation of the formulae for bias and mean-square errors of TDOA estimation under rm, this paper moves on to the joint estimation of TDOA and time scale. It proposes an iterative search for the maximization of the cross-ambiguity function (CAF), which is also the maximum likelihood function for additive Gaussian bandlimited white noise disturbance. In addition, a quadratic Lagrange interpolator is also proposed to obtain the initial parameter values for the iterative search, which can increase the chance of converging to the global minimum solution. It is necessary to time scale a digital sequence by a noninteger in the maximization process. For an N-point sequence, this operation, which first interpolates the samples by sinc functions and then resamples, is in the order O(N²). Noting that the magnitude of the sinc function decreases rapidly from its peak, this paper uses a fast approximation (FA) method that applies only five sinc coefficients for the interpolation, reducing the computation to O(N). Simulation results have corroborated that the maximization of the CAF does provide estimates that reach the Cramer-Rao lower bound and that the degradation in accuracy is negligible when FA is applied.