Exact Fisher Information Matrix With State-Dependent Probability of Detection

In this paper, the exact Fisher information matrix (EFIM) is derived for target localization with range measurements under imperfect detection. We treat the probability of detection (PD) as a target state-dependent parameter, and take a partial derivative of PD with respect to the target state during the calculation of the EFIM. By introducing an additional information impact parameter (AIIP), we define a parameter to reveal the divergence between EFIM and the original Fisher information matrix. We analytically prove that the AIIP is bandwidth invariant, and subsequently we know that the divergence parameter is inversely linear with the square of the signal bandwidth. Moreover, we also analyze the connection of the divergence parameter with the signal-to-noise ratio (SNR) and the false alarm rate by simulation. The results suggest that approximating PD as a state-free parameter is feasible for most of the real radar applications with low false alarm rate, large bandwidth, and moderate SNR, except for some specific continuous wave radar systems, which operate with a small bandwidth and high false alarm rate.