Cramer-Rao Lower Bound for Joint Time Delay Estimation in aRandomly Distributed Antenna Array

In this paper, the Cramer-Rao lower bound (CRLB) for time delay estimation between signals received at multiple randomly distributed antennas is considered. Fourier Transform of the received signals is used to derive the probability density function (PDF) in the frequency domain. The Fisher information matrix (FIM) is then deduced and the CRLB is obtained in the end. Furthermore, the result is discussed in two special cases and is compared with the CRLB in two-antenna case. It is remarkable that the estimation accuracy increases as the number of antenna elements increases. Moreover, it is shown that that increasing the number of results in a small advantage at high SNR values. However, the advantage becomes more significant for low SNR conditions. Besides, the CRLB between two signals cannot decrease infinitely, which is limited by the SNRs of the antenna pair, even by employing more antennas.