Methods for Estimation of Subsample Time Delays of Digitized Echo Signals

Time delay estimation (TDE) is commonly performed in practice by crosscorrelation of digitized echo signals. Since time delays are generally not integral multiples of the sampling period, the location of the largest sample of the crosscorrelation function (ccf) is an inexact estimator of the location of the peak. Therefore, one must interpolate between the samples of the ccf to improve the estimation precision. Using theory and simulations, we review and compare the performance of several methods for interpolation of the ccf. The maximum likelihood approach to interpolation is the application of a reconstruction filter to the discrete ccf. However, this method can only be approximated in practice and can be computationally intensive. For these reasons, a simple method is widely used that involves fitting a parabola (or other curve) to samples of the ccf in the neighborhood of its peak. We describe and compare two curve-fitting methods: parabolic and cosine interpolation. Curve-fitting interpolation can yield biased time-delay estimates, which may preclude the use of these methods in some applications. The artifactual effect of these bias errors on elasticity imaging by elastography is discussed. We demonstrate that reconstructive interpolation is unbiased. An iterative implementation of the reconstruction procedure is proposed that can reduce the computation time significantly.