Cramer-Rao Lower Bounds for Estimation of Doppler Frequency in Emitter Location Systems

This paper derives Cramer-Rao bounds on estimates of the Doppler-shifted frequency of a coherent pulse-train intercepted at a single moving antenna. Such estimates are used to locate the emitter that transmitted the pulse train. Coherency from pulse to pulse allows much better frequency accuracy and is considered to be necessary to support accurate emitter location. Although algorithms for estimating the Doppler-shifted frequency of a coherent pulse train have been proposed, previously no results were available for the Cramer-Rao lower bound (CRLB) for frequency estimation from a coherent pulse train. This paper derives the CRLB for estimating the Doppler-shifted frequency of a coherent pulse train as well as for a non-coherent pulse train; a comparison of these two cases is made and the bound is compared to previously published accuracy results. It is shown that a general rule of thumb is that the frequency CRLB for coherent pulse trains depends inversely on pulse on-time, number of pulses, variance of pulse times, and the product of signal-to-noise-ratio and sampling frequency SNR×Fs; pulse shape and modulation have virtually no impact on the frequency accuracy. For the case that the K intercepted pulses are equally spaced by the pulse repetition interval (PRI), then the CRLB decreases as 1/PRI and as 1/K. It is also shown that roughly the gain in coherent accuracy vs. the non-coherent case is K times the ratio of pulse on-time to PRI; since PRI is typically much larger than pulse on-time, the coherent scenario allows much better frequency estimation accuracy.