Fast Algorithms for Displacement and Low-Rank Structured Matrices

This tutorial provides an introduction to the development of fast matrix algorithms based on the notions of displacement and various low-rank structures

Active control of time-varying broadband noise and vibrations using a sliding-window Kalman filter

Recently, a multiple-input/multiple-output Kalman filter technique was presented to control time-varying broadband noise and vibrations. By describing the feed-forward broadband active noise control problem in terms of a state estimation problem it was possible to achieve a faster rate of convergence than instantaneousgradient least-mean-squares algorithms and possibly also a better tracking performance. A multiple input/ multiple output Kalman algorithm was derived to perform this state estimation. To make the algorithm more suitable for real-time applications, the Kalman filter was written in a fast array form and the secondary path state matrices were implemented in output normal form. The resulting filter implementation was verified in simulations and in real-time experiments. It was found that for a constant primary path the filter had a fast rate of convergence and was able to track time-varying spectra. For a forgetting factor equal to unity the system was robust but the filter was unable to track rapid changes in the primary path. A forgetting factor lower than unity gave a significantly improved tracking performance but led to a numerical instability for the fast array form of the algorithm. To improve the numerical behavior, while enabling fast tracking and convergence, several variants are described in this paper. Results will be shown for a sliding window Recursive Least Squares filter in fast array form, which will later be extended to a full Kalman filter implementation by taking into account the uncertainty of the secondary path between the control sources and the error sensors. Multiple variants will be discussed in this paper. The first variant is the standard sliding window technique, which applies both updates and downdates to the filter coefficients. The second variant is an algorithm which only applies an update step to the filter coefficients and interprets the downdate step as an addition of a covariance matrix to the Riccati equation. The third variant uses an implicit forgetting factor. These implementations use a factorized form of the hyperbolic orthogonal transformation matrix. The different techniques will be applied to measured data of noise in houses near the runway of an airport. Results are given of the performance regarding tracking, convergence and numerical stability of the algorithms

Efficient implementation of a structured total least squares based speech compression method

AbstractWe present a fast implementation of a recently proposed speech compression scheme, based on an all-pole model of the vocal tract. Each frame of the speech signal is analyzed by storing the parameters of the complex damped exponentials deduced from the all-pole model and its initial conditions. In mathematical terms, the analysis stage corresponds to solving a structured total least squares (STLS) problem. It is shown that by exploiting the displacement rank structure of the involved matrices the STLS problem can be solved in a very fast way. Synthesis is computationally very cheap since it consists of adding the complex damped exponentials based on the transmitted parameters.The compression scheme is applied on a speech signal. The speed improvement of the fast vocoder analysis scheme is demonstrated. Furthermore, the quality of the compression scheme is compared with that of a standard coding algorithm, by using the segmental signal-to-noise ratio

Array algorithms for H^2 and H^∞ estimation

Currently, the preferred method for implementing H^2 estimation algorithms is what is called the array form, and includes two main families: square-root array algorithms, that are typically more stable than conventional ones, and fast array algorithms, which, when the system is time-invariant, typically offer an order of magnitude reduction in the computational effort. Using our recent observation that H^∞ filtering coincides with Kalman filtering in Krein space, in this chapter we develop array algorithms for H^∞ filtering. These can be regarded as natural generalizations of their H^2 counterparts, and involve propagating the indefinite square roots of the quantities of interest. The H^∞ square-root and fast array algorithms both have the interesting feature that one does not need to explicitly check for the positivity conditions required for the existence of H^∞ filters. These conditions are built into the algorithms themselves so that an H^∞ estimator of the desired level exists if, and only if, the algorithms can be executed. However, since H^∞ square-root algorithms predominantly use J-unitary transformations, rather than the unitary transformations required in the H^2 case, further investigation is needed to determine the numerical behavior of such algorithms

Superfast Inference for Stationary Gaussian Processes in Particle Tracking Microrheology

Particle tracking of passive microscopic species has become the experimental measurement of choice in diverse applications, where either the material volumes are limited, or the materials themselves are so soft that they deform uncontrollably under the stresses and strains of traditional instruments. As such, the results of countless biological and rheological analyses hinge pivotally on extracting reliable dynamical information from large datasets of particle trajectory recordings. However, to do this in a statistically and computationally efficient manner presents a number of important challenges. Addressing some of these challenges is the focus of the present work. In Chapter 2, we present a superfast set of tools for parametric inference in single-particle tracking. Parametric likelihoods for particle trajectory measurements typically consist of stationary Gaussian time series, for which traditional fast inference algorithms scale as N-square in the number of observations. We present a superfast algorithm for parametric inference for stationary Gaussian processes and propose novel superfast algorithms for score and Hessian calculations. This effectively enables superfast inference for stationary Gaussian process via a wide array of frequentist and Bayesian methods. In Chapters 3 and 4, we use the superfast toolkit to address two outstanding problems prevalent in many particle tracking analyses. The first is that particle position measurements are generally contaminated by various forms of high-frequency errors. Failure to account for these errors leads to considerable bias in estimation results. In Chapter 3 we propose a novel strategy to filter high-frequency noise from measurements of particle positions. Our filters are shown theoretically to cover a vast range of high-frequency noise regimes and lead to an efficient computational estimator of model coefficients. Analyses of numerous experimental and simulated datasets suggest that our filtering approach performs remarkably well. The second problem we address is the considerable heterogeneity of typical biological fluids in which particle tracking experiments are conducted. In Chapter 4, we propose a simple metric by which to quantify the degree of heterogeneity of a fluid, along with a computationally efficient estimator and statistical test against the hypothesis that the fluid is homogeneous. The thesis is concluded by outlining several directions for future research