Abstract
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.
Highlights
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
We have developed a new sequential Monte Carlo (SMC) sampling method—the multilevel mixture Kalman filter (MKF)—under the framework of MKF for conditional dynamic linear systems
This new scheme generates random streams using sequential importance sampling (SIS), based on the multilevel or hierarchical structure of the indicator random variables. This technique can be used in conjunction with the delayed estimation methods, resulting in a delayed multilevel MKF
Summary
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. By using the marginalization technique for Monte Carlo computation [17], the MKF focuses on the sampling of the indicator variable λt other than the whole state variable {xt, λt} This method can drastically reduce Monte Carlo variances associated with a standard sequential importance sampler applied directly to the space of the state variable. We need not sample from the entire original sampling space, and many Kalman filter update steps associated with the standard MKF can be saved This kind of hierarchical structure imposed on the indicator space is employed in the partitioned sampling strategy [18], which greatly improved the efficiency and the accuracy for multiple target tracking over the original SMC methods.
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