Multilevel Mixture Kalman Filter
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Multilevel Mixture Kalman Filter Dong Guo Department of Electrical Engineering, Columbia University, New York, NY 10027, USA Email: [email protected]
Xiaodong Wang Department of Electrical Engineering, Columbia University, New York, NY 10027, USA Email: [email protected]
Rong Chen Department of Information and Decision Sciences, University of Illinois at Chicago, Chicago, IL 60607-7124, USA Email: [email protected] Department of Business Statistics & Econometrics, Peking University, Beijing 100871, China Received 30 April 2003; Revised 18 December 2003 The mixture Kalman filter is a general sequential Monte Carlo technique for conditional linear dynamic systems. It generates samples of some indicator variables recursively based on sequential importance sampling (SIS) and integrates out the linear and Gaussian state variables conditioned on these indicators. Due to the marginalization process, the complexity of the mixture Kalman filter is quite high if the dimension of the indicator sampling space is high. In this paper, we address this difficulty by developing a new Monte Carlo sampling scheme, namely, the multilevel mixture Kalman filter. The basic idea is to make use of the multilevel or hierarchical structure of the space from which the indicator variables take values. That is, we draw samples in a multilevel fashion, beginning with sampling from the highest-level sampling space and then draw samples from the associate subspace of the newly drawn samples in a lower-level sampling space, until reaching the desired sampling space. Such a multilevel sampling scheme can be used in conjunction with the delayed estimation method, such as the delayed-sample method, resulting in delayed multilevel mixture Kalman filter. Examples in wireless communication, specifically the coherent and noncoherent 16-QAM over flat-fading channels, are provided to demonstrate the performance of the proposed multilevel mixture Kalman filter. Keywords and phrases: sequential Monte Carlo, mixture Kalman filter, multilevel mixture Kalman filter, delayed-sample method.
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INTRODUCTION
Recently there have been significant interests in the use of the sequential Monte Carlo (SMC) methods to solve online estimation and prediction problems in dynamic systems. Compared with the traditional filtering methods, the simple, flexible—yet powerful—SMC provides effective means to overcome the computational difficulties in dealing with nonlinear dynamic models. The basic idea of the SMC technique is the recursive use of the sequential importance sampling (SIS). There also have been many recent modifications and improvements on the SMC methodology [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]. Among these SMC methods, the mixture Kalman filter (MKF) [3] is a powerful tool to deal with conditional dynamic linear models (CDLMs) and finds important applications in digital wireless communications [3, 13, 14]. A similar method is also discussed in [15] for CDLM system. The CDLM is a direct generalization of the dynamic linear model
(DLM) [16] and it can be gen
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