Introduction: The use of left ventricular assist device (LVAD) has grown rapidly. Adverse events do continue to occur. In recent years, analysis of LVAD sound recordings emerged as a means to monitor pump function and detect pump thrombosis. The aim of this study was to characterize the sounds from HeartMate II and to evaluate the use of handheld iOS devices for sound recordings. Method: Signal analysis of LVAD sound recordings, with dedicated recording equipment and iOS devices, was performed. Two LVADs running in mock loop circuits were compared to an implanted LVAD. Spectral analysis and parametric signal models were explored to quantify the sound and potentially detect changes in it. Results: The sound recordings of two LVADs in individual mock loop circuits and a third one implanted in a patient appeared to be similar. Qualitatively, sound characteristics were preserved following changes in pump speed. Recordings using dedicated equipment showed that HeartMate II sound comprises low-frequency components corresponding to pump impeller rotation, as well as high-frequency components due to a pulse width modulation of the electric power to the pump. These different signal components interact and result in a complicated frequency spectrum. The iPhone and iPod recordings could not reproduce the sounds as well as the dedicated equipment. In particular, lower frequencies were affected by outside disturbances. Discussion: This article outlines a systematic approach to LVAD sound analysis using signal processing methods to quantify and potentially detect changes, and describes some of the challenges, for example, with the use of inexpensive recording devices.

Nonlinear Kalman filters are algorithms that approximately solve the Bayesian filtering problem by employing the measurement update of the linear Kalman filter (KF). Numerous variants have been developed over the past decades, perhaps most importantly the popular sampling based sigma point Kalman filters.In order to make the vast literature accessible, we present nonlinear KF variants in a common framework that highlights the computation of mean values and covariance matrices as the main challenge. The way in which these moment integrals are approximated distinguishes, for example, the unscented KF from the divided difference KF.With the KF framework in mind, a moment computation problem is defined and analyzed. It is shown how structural properties can be exploited to simplify its solution. Established moment computation methods, and their basics and extensions, are discussed in an extensive survey. The focus is on the sampling based rules that are used in sigma point KF. More specifically, we present three categories of methods that use sigma-points 1) to represent a distribution (as in the UKF); 2) for numerical integration (as in Gauss-Hermite quadrature); 3) to approximate nonlinear functions (as in interpolation). Prospective benefits and downsides are listed for each of the categories and methods, including accuracy statements. Furthermore, the related KF publications are listed.The theoretical discussion is complemented with a comparative simulation study on instructive examples.

The Ensemble Kalman filter (EnKF) is a standard algorithm in oceanography and meteorology, where it has got thousands of citations. It is in these communities appreciated since it scales much better with state dimension n than the standard Kalman filter (KF). In short, the EnKF propagates ensembles with N state realizations instead of mean values and covariance matrices and thereby avoids the computational and storage burden of working on n×n matrices. Perhaps surprising, very little attention has been devoted to the EnKF in the signal processing community. In an attempt to change this, we present the EnKF in a Kalman filtering context. Furthermore, its application to nonlinear problems is compared to sigma point Kalman filters and the particle filter, so as to reveal new insights and improvements for high-dimensional filtering algorithms in general. A simulation example shows the EnKF performance in a space debris tracking application.

We consider the filtering problem in linear state space models with heavy tailed process and measurement noise. Our work is based on Student's t distribution, for which we give a number of useful results. The derived filtering algorithm is a generalization of the ubiquitous Kalman filter, and reduces to it as special case. Both Kalman filter and the new algorithm are compared on a challenging tracking example where a maneuvering target is observed in clutter.

The second order extended Kalman filter (EKF2) is based on a second order Taylor expansion of a nonlinear system, in contrast to the more common (first order) extended Kalman filter (EKF1). Despite a solid theoretical ground for its approximation, it is seldom used in applications, where the EKF and the unscented Kalman filter (UKF) are the standard algorithms. One reason for this might be the requirement for analytical Jacobian and Hessian of the system equations, and the high complexity that scales with the state order $n_x$ as $n_x^5$. We propose a numerical algorithm which is based on an extended set of sigma points (compared to the UKF) that needs neither Jacobian nor Hessian (or numerical approximations of these). Further, it scales as $n_x^4$, which is an order of magnitude better than the EKF2 algorithm presented in literature.

This thesis is on filtering in state space models. First, we examine approximate Kalman filters for nonlinear systems, where the optimal Bayesian filtering recursions cannot be solved exactly. These algorithms rely on the computation of certain expected values. Second, the problem of filtering in linear systems that are subject to heavy-tailed process and measurement noise is addressed.

Expected values of nonlinearly transformed random vectors are an essential ingredient in any Kalman filter for nonlinear systems, because of the required joint mean vector and joint covariance of the predicted state and measurement. The problem of computing expected values, however, goes beyond the filtering context. Insights into the underlying integrals and useful simplification schemes are given for elliptically contoured distributions, which include the Gaussian and Student’s t distribution. Furthermore, a number of computation schemes are discussed. The focus is on methods that allow for simple implementation and that have an assessable computational cost. Covered are basic Monte Carlo integration, deterministic integration rules and the unscented transformation, and schemes that rely on approximation of involved nonlinearities via Taylor polynomials or interpolation. All methods come with realistic accuracy statements, and are compared on two instructive examples.

Heavy-tailed process and measurement noise in state space models can be accounted for by utilizing Student’s t distribution. Based on the expressions forconditioning and marginalization of t random variables, a compact filtering  algorithm for linear systems is derived. The algorithm exhibits some similarities with the Kalman filter, but involves nonlinear processing of the measurements in form of a squared residual in one update equation. The derived filter is compared to state-of-the-art filtering algorithms on a challenging target tracking example, and outperforms all but one optimal filter that knows the exact instances at which outliers occur.

The presented material is embedded into a coherent thesis, with a concise introduction to the Bayesian filtering and state estimation problems; an extensive survey of available filtering algorithms that includes the Kalman filter, Kalman filters for nonlinear systems, and the particle filter; and an appendix that provides the required probability theory basis.