First-order bias and second-order variance of maximum likelihood time delay and Doppler shift estimates

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Exact expressions for the bias and variance of a maximum likelihood time delay and Doppler shift estimate are difficult to obtain analytically. It has become popular in such nonlinear problems to compute limiting bounds on mean-square estimation error, such as the Cramer–Rao bound (CRB), since these are easier to obtain than the true variance. Recently it has been shown [Makris and Naftali, J. Acoust. Soc. Am. 106, 2 (1999)] that the first-order bias and second-order variance of a general maximum-likelihood estimator (MLE) can be computed analytically, using higher-order asymptotics. This approach is applied to the classic radar/sonar problem of estimating the time delay and Doppler shift of a deterministic signal in additive white Gaussian noise. The MLE is shown to be unbiased to first order for this problem. By evaluating the asymptotic expansion for the variance for three specific signal waveforms, Gaussian, LFM, and HFM, it is found that (a) the CRB yields an unrealistically optimistic variance estimate for SNR’s lower than 20–30 dB, (b) both first- and second-order time-delay variance terms decrease with increasing signal bandwidth, and (c) both first and second Doppler-shift variance terms of LFM and HFM signals approach constant values as bandwidth increases.