A Covariance Calculation Method Based on the Joint Estimation of the Time Delay and the Doppler Factor

The measurement accuracy of multidimensional information is a crucial factor of localization accuracy, and it is typically characterized by the covariance matrix. Therefore, obtaining an accurate covariance matrix is of great significance to analyze the performance of bistatic localization. To address the difficulty in calculating the precise covariance matrix caused by the strong cross-correlation between the time delay and the Doppler factor, a covariance matrix calculation method based on the joint estimation of time delay and Doppler factor is proposed. Specifically, a received signal model under a wideband underwater acoustic channel is constructed, and the Cramer-Rao lower bound (CRLB) for channel parameter estimation under this model is derived. Then, by utilizing the mathematical relationships between the channel parameters and the time delay as well as the Doppler factor, the CRLB for their estimations are deduced. Finally, based on the property that the estimation error of an efficient estimator approaches the CRLB, the covariance matrix for the joint estimation of the time delay and the Doppler factor is obtained. Simulation results demonstrate that, compared with the short-time Fourier transform, Wigner distribution, fractional Fourier transform (FrFT), and the traditional experience methods, the proposed method improves the calculation accuracy of the covariance matrix by explicitly accounting for the cross-correlation between the time delay and the Doppler factor.