The combined effect of signal decorrelation and random noise on the variance of time delay estimation

Tissue motion and elasticity imaging techniques commonly use time delay estimation (TDE) for the assessment of tissue displacement. The performance of these techniques is limited because the signals are corrupted by various factors including electronic noise, quantization, and speckle decorrelation. Speckle decorrelation is caused by changes in the coherent interference among scatterers when the tissue moves relative to the ultrasound beam. In time delay estimation, the effect of noise is usually addressed through the signal-to-noise ratio (SNR) term. Decorrelation, often a significant source of error in medical ultrasound, is commonly described in terms of the correlation coefficient. A relationship between the correlation coefficient and the SNR was previously derived in the literature, for identical signals corrupted by uncorrelated random noise. In this paper, we derive the relationship between the peak of the correlation coefficient function and the SNR for two jointly stationary signals when a delay is present between the signals. Recently, an expression for the Cramer-Rao lower bound (CRLB) has been derived in the literature for partially decorrelated signals in terms of the SNR and the correlation coefficient. Since the applicability of the CRLB is determined not only by the SNR, but also by the correlation coefficient, it is important to unify the expression for the CRLB for partially correlated signals. In this paper, we derive an expression for the CRLB in term of an equivalent SNR converted from the correlation coefficient using an SNR-p relationship, and show this expression to be equivalent to the expression for CRLB. We also corroborate the validity of the SNR-p expression with a simulation. Using this formulation, correlation measurements can be converted to SNR to obtain a composite SNR. The use of this composite SNR in lieu of those in the CRLB expression in the literature allows the extension of the literature results to the solution of the common TDE problems that involve signal decorrelation.