Abstract—Many distributed inference problems in wireless sensor networks can be represented by probabilistic graphical models, where belief propagation, an iterative message passing algorithm provides a promising solution. In order to make the algorithm efficient and accurate, messages which carry the belief information from one node to the others should be formulated in an appropriate format. This paper presents two belief propagation algorithms where non-linear and non-Gaussian beliefs are approximated by Fourier density approximations, which significantly reduces power consumptions in the belief computation and transmission. We use self-localization in wireless sensor networks as an example to illustrate the performance of this method. I. INTRODUCTION DVANCES in sensor technology and telecommunications make wireless sensor network (WSN) an appropriate solution for a wide variety of applications[1][2]. In a WSN, sensor nodes are spatially distributed to monitor the physical or environmental states. Information can be exchanged through the wireless channel so that the whole network works in a cooperative fashion. Many estimation problems in WSNs can be represented by probabilistic graphical models and solved by belief propagation methods. Belief propagation (BP) is an iterative message passing algorithm in which each node calculates its belief about other nodes and communicates with them to exchange their beliefs about each other. Compact messages that are transmitted between nodes carry the necessary information of the beliefs, based on which the receiver can reconstruct the transmitter’s belief about it. For discrete beliefs, messages can be a short vector of probabilities. For continuous beliefs with Gaussian distribution, it is enough to ensemble the mean and variance in the message. However, in many applications, beliefs have non-linear and non-Gaussian distributions so that belief calculation and transmission consumes a lot of power. That limits its application in WSNs which have strong power constraints. Hence, an appropriate representation of beliefs which reduces the complexity while keeping the accuracy is necessary but non-trivial. Monte Carlo methods can be used where messages contain samples that are drawn from the distribution to represent the C. Na, H. Wang and D. Obradovic are with Siemens AG, Corporate Technology, Munich, Germany (phone: +49-89-636-49499; e-mail: {na.chongning.ext, dragan.obradovic, hui.wang.ext}@ siemens.com). U. D. Hanebeck is with Intelligent Sensor-Actuator-Systems Laboratory (ISAS), Institute of Computer Science and Engineering, Universität Karlsruhe (TH), Germany (e-mail: uwe.hanebeck@ieee.org). beliefs. Gibbs sampling is a popular method in this case. However, this is only possible for sufficiently small networks. Authors of [3] used non-parametric BP method where beliefs are represented by Gaussian mixtures. It generalizes particle filtering for inference in non-linear, non-Gaussian time series. In this paper, we introduce Fourier density approximation (FDA) method to represent the beliefs. Fourier series were first employed to estimate probability densities in [4]. Recently, [5] and [6] ensured the non-negativity of Fourier series by approximating the square root of the density instead of the density itself. The usage of Fourier series in nonlinear Bayesian filtering is also derived in [5] and [6]. Using Fourier density approximation, the belief can be represented sufficiently by only a small number of Fourier coefficients. Hence, the transmission power and time between sensor nodes are significantly saved. Compared to other density representations like Gaussian mixture or Monte Carlo methods, the optimal number of coefficients under a required approximation error with respect to a density distance metric is more efficiently obtained. Furthermore, the sum-product operations in BP algorithms can be more effectively calculated in Fourier domain since some convolution-like integral operations are more easily calculated than in space domain. Since the Fourier series are orthogonal expansions, the coefficients are derived independently and effectively [5]. In practice, this is done by efficient Fast Fourier Transform (FFT). In this paper, the self-localization in WSNs, a common practice of brief propagation, is used to evaluate the performance of Fourier density approximation. Two Fourier based algorithms are proposed, which are simplified transmission based on Fourier density approximation (ST-FDA) and simplified computation and transmission based on Fourier density approximation (SCT-FDA). ST-FDA reduces the size of the belief message to save radio transmission power, which is a critical factor for WSNs. SCT-FDA further simplifies the sum-product algorithm (SPA) to reduce computation power. The paper is organized as follows. Section II presents BP as a general approach to the inference problems in WSNs. Fourier density approximation method will be introduced in Section III. Section IV uses a sensor self-calibration example to illustrate the use of Fourier density approximation for BP. ST-FDA and SCT-FDA algorithms are proposed. Their performances will be evaluated through simulation and the results will be shown in Section V. Finally, Section VI concludes the paper. Fourier Density Approximation for Belief Propagation in Wireless Sensor Networks Chongning Na, Hui Wang, Dragan Obradovic, and Uwe D. Hanebeck A