Fingerprint-based indoor localization holds great potential for the Internet of Things. Despite numerous studies focusing on its algorithmic and practical aspects, a notable gap exists in theoretical performance analysis in this domain. This paper aims to bridge this gap by deriving several lower bounds and approximations of mean square error (MSE) for fingerprint-based localization. These analyses offer different complexity and accuracy trade-offs. We derive the equivalent Fisher information matrix and its decomposed form based on a wireless propagation model, thus obtaining the Cramér-Rao bound (CRB). By approximating the Fisher information provided by constraint knowledge, we develop a constraint-aware CRB. To more accurately characterize nonlinear transformation and constraint information, we introduce the Ziv-Zakai bound (ZZB) and modify it for adapt deterministic parameters. The Gauss–Legendre quadrature method and the trust-region reflective algorithm are employed to make the calculation of ZZB tractable. We introduce a tighter extrapolated ZZB by fitting the quadrature function outside the well-defined domain based on the Q-function. For the constrained maximum likelihood estimator, an approximate MSE expression, which can characterize map constraints, is also developed. The simulation and experimental results validate the effectiveness of the proposed bounds and approximate MSE.

Indoor localization is of pivotal significance for a wide variety of services in the context of the Internet of Things (IoT). Both ranging-based and fingerprint-based localization techniques are promising for employment in harsh indoor environments. Hence, we propose a unified framework based on factor graphs for ubiquitous high-accuracy indoor localization. Our unified framework efficiently integrates ranging and fingerprinting for striking an appealing accuracy versus deployment cost tradeoff, where the crowdsourcing required for the construction of fingerprinting databases can also be addressed with little human intervention. By intrinsically amalgamating the global grid sampling and the regularized importance-resampling techniques, a nonparametric belief propagation algorithm is proposed for achieving the accurate position estimation at the cost of a moderate computational complexity. For improving the robustness to environmental variations, a likelihood-ratio-based approach is employed to detect ranging outliers. Moreover, a low-complexity serial scheduling scheme defined over factor graphs is designed for real-time localization. We design a hybrid ultrawide bandwidth and Wi-Fi localization system relying on off-the-shelf commercial devices and evaluate the proposed unified framework in a typical office building. Our experimental results show that the proposed algorithm outperforms the existing state-of-the-art methods and it is capable of achieving submeter localization accuracy.

Belief propagation (BP) is a popular message passing algorithm for distributed cooperative localization. However, due to the nonlinearity of measurement functions, BP implementation has no closed-form expression and requires message approximations. While nonparametric BP can be used, it suffers from a high computational complexity, thus being impractical in energy-constrained networks. In this paper, a parametric Bayesian method with Gaussian BP implementation is proposed for distributed cooperative localization. With linearization of the Euclidean norm in ranging measurements, the joint posterior distribution of agents’ locations is successively approximated with a sequence of high-dimensional Gaussian distributions. At each iteration of the successive Gaussian approximation, vector-valued Gaussian BP is further adopted to compute the marginal distributions of agents’ locations in a distributed way. It is proved by the principle of majorization-minimization that the proposed successive Gaussian approximation is guaranteed to converge, and the sequence of the estimated agents’ locations converges to a stationary point of the objective function of the maximum a posteriori estimation. Furthermore, although cooperative localization involves loopy network topologies, in which convergence property of Gaussian BP is generally unknown, it is proved in this paper that vector-valued Gaussian BP converges, making the proposed parametric BP-based method being the first one achieving convergence guarantee. Compared to the nonparametric BP counterpart, the proposed method has a much lower computational complexity and communication overhead. Simulation results demonstrate that the proposed method achieves a superior performance in localization accuracy compared to existing cooperative localization methods.

A low-complexity parametric bilinear generalized approximate message passing (PBiGAMP)-based receiver is conceived for multi-carrier faster-than-Nyquist (MFTN) signaling over frequency-selective fading channels. To mitigate the inherent ill-conditioning problem of MFTN signaling, we construct a segment-based frequency-domain received signal model in the form of a block circulant linear transition matrix, which can be efficiently calculated by applying a two dimensional fast Fourier transform. Based on the eigenvalue decomposition of the block circulant matrices, we can diagonalize the covariance matrix of the complex-valued colored noise process imposed by the associated two dimensional non-orthogonal matched filtering. Building on this model, a PBiGAMP-based parametric joint channel estimation and equalization (JCEE) algorithm is proposed for MFTN systems. In this algorithm, we introduce a pair of additive terms for characterizing the interferences arising from adjacent segments and employ the exact discretea prioriprobabilities of the transmitted symbols for improving the bit error rate (BER) performance. To further enhance the system’s robustness in the presence of ill-conditioned matrices, we develop a refined PBiGAMP-based JCEE algorithm by introducing a series of scaled identity matrices. Moreover, the proposed PBiGAMP-based JCEE algorithms may be readily decomposed into GAMP-based equalization algorithms, when the channel state information is perfectly known. The overall complexity of the proposed algorithms only increases logarithmically with the total number of transmitted symbols. Our simulation results demonstrate the benefits of the proposed PBiGAMP-based iterative message passing receiver conceived for MFTN signaling.

Gaussian belief propagation (BP) finds extensive applications in signal processing but it is not guaranteed to converge in loopy graphs. In order to determine whether Gaussian BP would converge, one could directly use the classical convergence conditions of Gaussian BP, such as diagonal dominance, walk-summability, and convex decomposition. These classical conditions assume that the convergence conditions for Gaussian BP precisions and means are the same, which has been proved to be unnecessary. Generally, the condition for guaranteeing the convergence of Gaussian BP precisions is looser than that of Gaussian BP means. Moreover, the convergence of Gaussian BP means could be improved by damping when Gaussian BP precisions converge. Therefore, the convergence of Gaussian BP precisions is a prerequisite for guaranteeing the convergence of Gaussian BP means. This paper derives a simple convergence condition for Gaussian BP precisions, which can be verified in a distributed way. Through numerical examples, it is found that there exists scenarios where the new condition is satisfied but the classical conditions are not.

Gaussian belief propagation (BP) is widely used for distributed inference. Its computational complexity, and communication overhead depend on the total number of messages updated, and transmitted among variable nodes, respectively. For large, and dense networks, both computational complexity, and communication overhead could be extremely high. To this end, a variant of Gaussian BP called Gaussian SPAWN (sum-product algorithm over a wireless network) could be applied, where outgoing messages are approximated by beliefs. Similar to Gaussian BP, the convergence of Gaussian SPAWN is not guaranteed for loopy graphs. Therefore, we analyze the convergence of belief means, and variances in Gaussian SPAWN. Numerical results are presented to corroborate the newly established theories and a comparison of Gaussian BP, and Gaussian SPAWN is illustrated.

Gaussian belief propagation (BP) is a low-complexity and distributed method for computing the marginal distributions of a high-dimensional joint Gaussian distribution. However, Gaussian BP is only guaranteed to converge in singly connected graphs and may fail to converge in loopy graphs. Therefore, convergence analysis is a core topic in Gaussian BP. Existing conditions for verifying the convergence of Gaussian BP are all tailored for one particular pairwise factorization of the distribution in Gaussian Markov random field (MRF) and may not be valid for another pairwise factorization. On the other hand, convergence conditions of Gaussian BP in pairwise linear Gaussian model are developed independently from those in Gaussian MRF, making the convergence results highly scattered with diverse settings. In this paper, the convergence condition of Gaussian BP is investigated under a general pairwise factorization, which includes Gaussian MRF and pairwise linear Gaussian model as special cases. Upon this, existing convergence conditions in Gaussian MRF are extended to any pairwise factorization. Moreover, the newly established link between Gaussian MRF and pairwise linear Gaussian model reveals an easily verifiable sufficient convergence condition in pairwise linear Gaussian model, which provides a unified criterion for assessing the convergence of Gaussian BP in multiple applications. Numerical examples are presented to corroborate the theoretical results of this paper.

In wireless sensor networks, acquiring accurate timing information is a crucial requirement for time-based sensor localization. Utilizing a joint localization and synchronization method in sensor networks can improve positioning speed and accuracy. In this paper, we present a unified factor graph framework based on time of arrival (TOA) measurements to solve the problem of joint localization and time synchronization. A novel distributed cooperative joint estimation method based on belief propagation (BP) is proposed. We linearize the nonlinear terms in messages on factor graph in order to obtain a closed Gaussian form solution of message update. Accordingly, only the means and variances have to be updated and transmitted, which significantly reduce the communication overhead and computational complexity. To further reduce the communication overhead, we propose a message passing schedule. Simulation results show that the proposed BP method reach close performance to particle-based approaches with lower complexity.

Localization in wireless sensor networks has become an attractive research field in recent years. Most studies focus on the mitigation of measurement noise by assuming the positions of anchors are perfectly known, which may become impractical due to some inevitable errors in the observations of anchors' positions. This paper addresses the problem by taking into account the a priori position information of inaccurate anchors. Considering that the maximum likelihood (ML) algorithm suffers from the intractable integrals involved, we resort to expectation maximization (EM) algorithm to solve this problem iteratively. The a posteriori probability of the anchor position is approximated by circularly symmetric Gaussian distribution, with parameters optimized by minimizing Kullback-Leibler divergence of the two distributions. Building on this approximation, we are able to derive the expectation step in closed form. Particle swarm optimization is then followed to perform the maximization step. Numerical results demonstrate that the proposed EM estimator is less sensitive to the anchors' uncertainties and it significantly outperforms the traditional ML estimator which ignores the prior information of anchors.

Localisation in wireless sensor networks (WSNs) has received much attention, where most studies focus on mitigating the effects of measurement noise under the assumption of accurate anchors’ positions. However, anchors’ positions could be inaccurate for the inevitable errors in practical observations. This paper studies the sensor localisation with both inaccurate anchors’ positions and noisy range measurements in WSNs. To solve the intractable integrals in likelihood function, the authors propose to use expectation‐maximisation (EM) algorithm to obtain the maximum likelihood (ML) estimation iteratively. The ‘a posteriori’ distribution of the anchor's position uncertainty is approximated to a circularly symmetric Gaussian distribution by minimising the Kullback‐Leibler divergence between them. Building on this, the authors derive the expectation step in a closed‐form expression. In the maximisation step, based on the Taylor expansion of the confluent hypergeometric function of the first kind presented in the expectation step, analytical solutions are obtained. Simulation results show that the proposed EM estimator significantly outperforms the approximated ML estimator. The performance gain by using the EM estimator becomes larger as the increase of anchors’ position uncertainties. Moreover, the performance of the EM estimator is close to that of the Monte Carlo‐based estimator with much less computational complexities.