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Sequential Bayesian estimation for dynamic state space models involves recursive estimation of hidden states based on noisy observations. The update of filtering and predictive densities for nonlinear models with non-Gaussian noise using Monte Carlo particle filtering methods is considered. The Gaussian particle filter (GPF) is introduced, where densities are approximated as a single Gaussian, an assumption which is also made in the extended Kalman filter (EKF). It is analytically shown that, if the Gaussian approximations hold true, the GPF minimizes the mean square error of the estimates asymptotically. The simulations results indicate that the filter has improved performance compared to the EKF, especially for highly nonlinear models where the EKF can diverge.

Abstract-We address optimal estimation of correlated multiple-input multiple-output (MIMO) channels using pilot signals, assuming knowledge of the second-order channel statistics at the transmitter. Assuming a block fading channel model and minimum mean square error (MMSE) estimation at the receiver, we design the transmitted signal to optimize two criteria: MMSE and the conditional mutual information between the MIMO channel and the received signal. Our analysis is based on the recently proposed virtual channel representation, which corresponds to beamforming in fixed virtual directions and exposes the structure and the true degrees of freedom in the correlated channel. However, our design framework is applicable to more general channel models, which include known channel models, such as the transmit and receive correlated model, as special cases. We show that optimal signaling is in a block form, where the block length depends on the signal-to-noise ratio (SNR) as well as the channel correlation matrix. The block signal corresponds to transmitting beams in successive symbol intervals along fixed virtual transmit angles, whose powers are determined by (nonidentical) water filling solutions based on the optimization criteria. Our analysis shows that these water filling solutions identify exactly which virtual transmit angles are important for channel estimation. In particular, at low SNR, the block length reduces to one, and all the power is transmitted on the beam corresponding to the strongest transmit angle, whereas at high SNR, the block length has a maximum length equal to the number of active virtual transmit angles, and the power is assigned equally to all active transmit angles. Consequently, from a channel estimation viewpoint, a faster fading rate can be tolerated at low SNRs relative to higher SNRs.

Abstract-In this paper, we use the Gaussian particle filter introduced in a companion paper to build several types of Gaussian sum particle filters. These filters approximate the filtering and predictive distributions by weighted Gaussian mixtures and are basically banks of Gaussian particle filters. Then, we extend the use of Gaussian particle filters and Gaussian sum particle filters to dynamic state space (DSS) models with non-Gaussian noise. With non-Gaussian noise approximated by Gaussian mixtures, the nonGaussian noise models are approximated by banks of Gaussian noise models, and Gaussian mixture filters are developed using algorithms developed for Gaussian noise DSS models. 1 As a result, problems involving heavy-tailed densities can be conveniently addressed. Simulations are presented to exhibit the application of the framework developed herein, and the performance of the algorithms is examined.Index Terms-Dynamic state-space models, extended Kalman filter, Gaussian mixture, Gaussian particle filter, Gaussian sum filter, Gaussian sum particle filter, Monte Carlo filters, nonlinear non-Gaussian stochastic systems, particle filters, sequential Bayesian estimation, sequential sampling methods.

Sequential Bayesian estimation for nonlinear dynamic state-space models involves recursive estimation of filtering and predictive distributions of unobserved time varying signals based on noisy observations. This paper introduces a new filter called the Gaussian particle filter 1 . It is based on the particle filtering concept, and it approximates the posterior distributions by single Gaussians, similar to Gaussian filters like the extended Kalman filter and its variants. It is shown that under the Gaussianity assumption, the Gaussian particle filter is asymptotically optimal in the number of particles and, hence, has much-improved performance and versatility over other Gaussian filters, especially when nontrivial nonlinearities are present. Simulation results are presented to demonstrate the versatility and improved performance of the Gaussian particle filter over conventional Gaussian filters and the lower complexity than known particle filters. The use of the Gaussian particle filter as a building block of more complex filters is addressed in a companion paper.Index Terms-Dynamic state space models, extended Kalman filter, Gaussian mixture, Gaussian mixture filter, Gaussian particle filter, Gaussian sum filter, Gaussian sum particle filter, Monte Carlo filters, nonlinear non-Gaussian stochastic systems, particle filters, sequential Bayesian estimation, sequential sampling methods, unscented Kalman filter. Jayesh H. Kotecha received the B.E. degree in electronics and telecommunications from the College of Engineering, Pune, India, in 1995 and the M.S. and Ph.D. degrees from the State University of New York at Stony Brook in 1996 and 2001, respectively.Since January 2002, he has been with the University of Wisconsin, Madison, as a post-doctoral researcher. His research interests are primarily in the fields of communications, signal processing, and information theory. Presently, he is working in statistical signal processing, adaptive signal processing, and particle filters and physical layer aspects in multi-input multi-ouput communication systems, including channel modeling, tranceiver design, space-time coding, and capacity analysis.Peter M. Djurić (SM'99) received the B.S. and M.S. degrees in electrical engineering from the