Joint TDOA and FDOA Estimation: A Conditional Bound and Its Use for Optimally Weighted Localization

Abstract

Modern passive emitter-location systems are often based on joint estimation of the time-difference of arrival (TDOA) and frequency-difference of arrival (FDOA) of an unknown signal at two (or more) sensors. Classical derivation of the associated Cramér-Rao bound (CRB) relies on a stochastic, stationary Gaussian signal-model, leading to a diagonal Fisher information matrix with respect to the TDOA and FDOA. This diagonality implies that (under asymptotic conditions) the respective estimation errors are uncorrelated. However, for some specific (nonstationary, non-Gaussian) signals, especially chirp-like signals, these errors can be strongly correlated. In this work we derive a "conditional" (or a "signal-specific") CRB, modeling the signal as a deterministic unknown. Given any particular signal, our CRB reflects the possible signal-induced correlation between the TDOA and FDOA estimates. In addition to its theoretical value, we show that the resulting CRB can be used for optimal weighting of TDOA-FDOA pairs estimated over different signal-intervals, when combined for estimating the target location. Substantial improvement in the resulting localization accuracy is shown to be attainable by such weighting in a simulated operational scenario with some chirp-like target signals.Index Terms-Chirp, conditional bound, confidence ellipse, frequency-difference of arrival (FDOA), passive emitter location, time-difference of arrival (TDOA).

Wideband Signal De-Interleaving (WSD)

“…But in Appendix B.4 it is shown that the null space of I θ is not empty and so it is not invertible. This is because the log-likelihood function is not uniquely defined by the model in (23). To eliminate the over parameterization we use the following transformations.…”

“…Stein [9] on the other hand modeled the signal as deterministic but unknown and derived the MLE for the differential delay and Doppler for a two sensor case. Under similar assumptions for the signal, Yeredor and Angel [23] have derived the CRLB for the TDOAs. After the TDOAs are estimated they are used to estimate the location of the source [24,25,26,27].…”

“…Recently, Yeredor and Angel [23] have proposed a method where the FIM is used as the weighting matrix. In order to compute the FIM, the emitter location must be known.…”

TDOA, Frequency and Phase Offsets Estimation Taking Into Account Carrier Phase of Arrival

“…The problem of time, frequency and phase shifts estimation in dual-channel systems has been extensively studied in the literature, see [6][7][8][9][10][11][12]. An estimator for TDOA, phase and discrete Doppler shifts using Gaussian random signals was proposed in [7].…”

“…An estimator for TDOA, phase and discrete Doppler shifts using Gaussian random signals was proposed in [7]. A signal model with TDOA, arbitrary Doppler shifts, and random phase using unknown deterministic signals was analyzed in [8]. In these papers and in a number of TDOA/FDOA localization related papers such as [10], the carrier phase of arrival (CPOA) term was modeled as a part of an unknown or random phase term or complex attenuation.…”