An overview on the time delay estimate in active and passive systems for target localization

TL;DR: The analysis shows that in the case of low SNR and when signal and noise autospectra are constants over the band or signal and noises fall off at the same rate, the minimum standard deviation of the time delay estimate varies inversely to the SNR, to the square root of the product of observation time and bandwidth, and to the center frequency.

Abstract: Sonar and radar systems not only detect targets but also localize them. The process of localization involves bearing and range estimation. These objectives of bearing and range estimation can be accomplished actively or passively, depending on the situation. In active sonar or radar systems, a pulsed signal is transmitted to the target and the echo is received at the receiver. The range of the target is determined from the time delay obtained from the echo. In passive sonar systems, the target is detected from acoustic signals emitted by the target, and it is localized using time delays obtained from received signals at spacially separated points. Several authors have calculated the variance of the time delay estimate in the neighborhood of true time delays and have presented their results in terms of coherence function and signal and noise autospectra. Here we analyze these derivations and show that they are the same for the case of low signal-to-noise ratio (SNR). We also address a practical problem with a target-generated wide-band signal and present the Cramer-Rao lower bound on the variance of the time delay estimate as a function of commonly understood terms such as SNR, bandwidth, observation time, and center frequency of the band. The analysis shows that in the case of low SNR and when signal and noise autospectra are constants over the band or signal and noise autospectra fall off at the same rate, the minimum standard deviation of the time delay estimate varies inversely to the SNR, to the square root of the product of observation time and bandwidth, and to the center frequency (provided W^{2}/12 f\min{0}\max{2} \ll 1 , where W = bandwidth and f_{0} = center frequency of the band). The only difference in the case of a high SNR is that the standard deviation varies inversely to the square root of the SNR, and all other parameter relationships are the same. We also address the effects of different signal and noise autospectral slopes on the variance of the time delay estimate in passive localization.