On the accuracy limits of misspecified delay-Doppler estimation

Abstract

This work derives compact closed-form expressions of the misspecified Cramér–Rao bound and pseudo-true parameters of time-delay and Doppler for a high dynamics signal model. Those expressions are validated by analyzing the mean square error (MSE) of the misspecified maximum likelihood estimator. A noteworthy outcome of these MSE results is that, for some magnitudes of acceleration and signal-to-noise ratios, neglecting the acceleration is beneficial in the MSE sense. The variance performance improvement is obtained at the cost of a systematic error in the true parameter estimation. This can be seen as a specific case of the trade-off between bias and variance. Neglecting the acceleration can improve the Doppler estimation when the error induced on the misspecified model is less than the variance increase due to including an extra parameter to estimate. Then, for some non-zero acceleration magnitudes and short integration times, the Doppler estimation using a misspecified model outperforms a correctly specified model in the MSE sense.