5.4 Fundamental Limits for Time-Based Ranging

Fundamental lower bounds such as the Cramer Rao Lower Bound (CRLB) are commonly used for setting a lower bound on an estimator s MSE. Use of the CRLB is justified by the asymptotic optimality properties of the ML estimator. In other words, the ML estimator ^[11] asymptotically achieves the CRLB (and is asymptotically unbiased) under certain regularity conditions [140, 236]. For the problem of time-delay estimation, the CRLB yields a tight bound commonly at large SNRs and/or for sufficiently long observation intervals [235].

For moderate and low SNRs, or when the observation interval is not sufficiently long, CRLBs may not be very tight. This is because the cross-correlation outputs for a received signal are quasi-periodic with respect to f _c, where f _c is the center frequency of the received signal. In other words, while there is a unique correlation peak at the true TOA, there are also weaker correlation peaks (which may be comparable to the global peak for narrowband signals) that oscillate with a period of 1 /f _c. Therefore, the receiver may erroneously lock onto one of those weaker peaks at low/medium SNRs. In order to get close to the CRLB, the receiver must unambiguously distinguish between adjacent peaks of the correlation function. Otherwise, the achievable MSE may be drastically inferior to that predicted by the CRLB [236].