In this paper, we consider the tracking of a radar target with unknown range and range rate at low signal-to-noise ratio (SNR). For this nonlinear estimation problem, the Cramér-Rao lower bound (CRLB) provides a bound on an unbiased estimator's mean-squared error (MSE). However, there exists a threshold SNR at which the estimator variance deviates from the CRLB. We consider the Barankin bound (BB) on the range and range-rate variance in order to obtain a tighter lower bound at low SNR, and we use the BB to predict the SNR threshold for a transmitted signal. We demonstrate that the BB with the additional information provided by the threshold SNR has an advantage over the CRLB in selecting the optimal transmit waveform at low SNRs. We also develop a waveform parameter configuration method that uses the BB and the ambiguity function resolution cell measurement model to optimize the SNR threshold.
MOTIVATION AND RELATION TO PRIOR WORKIn narrowband radar and tracking problems, we are interested in simultaneously measuring the time delay and Doppler shift of a received target reflected signal for estimating the target's range and range rate [1]. The estimates of the target's location and velocity at the receiver can vary over time, partly due to the presence of clutter, environmental effects or general interference, and variance or resolution of these estimates can be affected by the choice of the transmitted waveform parameters [1-3]. Thus, techniques have been proposed to opportunistically transmit the best waveform [2, 3] in order to improve estimation error performance. Waveform configuration for low SNRs has also been considered with the track-before-detect algorithm and using the CRLB to predict estimates of the target's location and velocity [4,5].The CRLB provides a lower bound on an estimator's variance performance. However, in target tracking problems with the receiver operating under low SNR conditions, the CRLB may not provide a tight bound, leading to inaccurate predictions of the position and velocity estimates [1, 6-8]. For SNRs below a certain threshold SNR, the estimator variance deviates from the CRLB [6,9]. As the SNR approaches this threshold region, the estimator's performance approaches an ambiguity or threshold region, where the estimator performance deteriorates rapidly. The errors in the threshold region are attributed to the fact that the estimator has dominant sidelobes due to unusually high noise fluctations. These fluctuations drive outlier values that are not, in general, the true values located at the mainlobe [1,7,10] The Barankin bound (BB) is the greatest lower bound (infimum) on the variance of an unbiased parameter estimate [11]. The BB has This work was sponsored in part by DARPA under the SSPARC program. The views expressed are those of the authors and do not reflect the official policy or position of the Department of Defense or the U.S. Government. been applied to many statistical signal processing problems to provide a tighter lower bound on the variance of an unbiased estimator...
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