  Location awareness is now becoming a vital requirement for many practical
applications. In this paper, we consider passive localization of multiple
targets with one transmitter and several receivers based on time of arrival
(TOA) measurements. Existing studies assume that positions of receivers are
perfectly known. However, in practice, receivers' positions might be
inaccurate, which leads to localization error of targets. We propose factor
graph (FG)-based belief propagation (BP) algorithms to locate the passive
targets and improve the position accuracy of receivers simultaneously. Due to
the nonlinearity of the likelihood function, messages on the FG cannot be
derived in closed form. We propose both sample-based and parametric methods to
solve this problem. In the sample-based BP algorithm, particle swarm
optimization is employed to reduce the number of particles required to
represent messages. In parametric BP algorithm, the nonlinear terms in messages
are linearized, which results in closed-form Gaussian message passing on FG.
The Bayesian Cramer-Rao bound (BCRB) for passive targets localization with
uncertain receivers is derived to evaluate the performance of the proposed
algorithms. Simulation results show that both the sample-based and parametric
BP algorithms outperform the conventional method and attain the proposed BCRB.
Receivers' positions can also be improved via the proposed BP algorithms.
Although the parametric BP algorithm performs slightly worse than the
sample-based BP method, it could be more attractive in practical applications
due to the significantly lower computational complexity.
