Adaptive step-size blind multi-user detectors for DS-CDMA systems under deterministically and Markovian time-varying multipath environments

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Adaptive Stochastic Systems: Estimation, Filtering, And Noise Attenuation

This dissertation investigates problems arising in identification and control of stochastic systems. When the parameters determining the underlying systems are unknown and/or time varying, estimation and adaptive filter- ing are invoked to to identify parameters or to track time-varying systems. We begin by considering linear systems whose coefficients evolve as a slowly- varying Markov Chain. We propose three families of constant step-size (or gain size) algorithms for estimating and tracking the coefficient parameter: Least-Mean Squares (LMS), Sign-Regressor (SR), and Sign-Error (SE) algorithms. The analysis is carried out in a multi-scale framework considering the relative size of the gain (rate of adaptation) to the transition rate of the Markovian system parameter. Mean-square error bounds are established, and weak convergence methods are employed to show the convergence of suitably interpolated sequences of estimates to solutions of systems of ordinary and stochastic differential equations with regime switching. Next we consider problems in noise attenuation in systems with unmodeled dynamics and stochastic signal errors. A robust two-phase design procedure is developed which first estimates the signal in a simplified form, and then applies a control to tune out the noise. Worst-case error bounds are derived in terms of the unmodeled dynamics and variances of the disturbance and measurement errors

On The Dynamics and Control Strategy of Time-Delayed Vibro-Impact Oscillators

Being able to control nonlinear oscillators, which are ubiquitous, has significant engineering implications in process development and product sustainability design. The fundamental characteristics of a vibro-impact oscillator, a non-autonomous time-delayed feedback oscillator, and a time-delayed vibro-impact oscillator are studied. Their being stochastic, nonstationary, non-smooth, and dynamically complex render the mitigation of their behaviors in response to linear and stationary inputs very difficult if not entirely impossible. A novel nonlinear control concept featuring simultaneous control of vibration amplitude in the time-domain and spectral response in the frequency-domain is developed and subsequently incorporated to maintain dynamic stability in these nonlinear oscillators by denying bifurcation and route-to-chaos from coming to pass. Convergence of the controller is formulated to be inherently unconditional with the optimization step size being self-adaptive to system identification and control force input. Optimal initial filter weights are also derived to warrant fast convergence rate and short response time. These novel features impart adaptivity, intelligence, and universal applicability to the wavelet based nonlinear time-frequency control methodology. The validity of the controller design is demonstrated by evaluating its performance against PID and fuzzy logic controllers in controlling the aperiodic, broad bandwidth, discontinuous responses characteristic of the time-delayed, vibro-impact oscillator

Tracking Rhythmicity in Biomedical Signals using Sequential Monte Carlo methods

Cyclical patterns are common in signals that originate from natural systems such as the human body and man-made machinery. Often these cyclical patterns are not perfectly periodic. In that case, the signals are called pseudo-periodic or quasi-periodic and can be modeled as a sum of time-varying sinusoids, whose frequencies, phases, and amplitudes change slowly over time. Each time-varying sinusoid represents an individual rhythmical component, called a partial, that can be characterized by three parameters: frequency, phase, and amplitude. Quasi-periodic signals often contain multiple partials that are harmonically related. In that case, the frequencies of other partials become exact integer multiples of that of the slowest partial. These signals are referred to as multi-harmonic signals. Examples of such signals are electrocardiogram (ECG), arterial blood pressure (ABP), and human voice. A Markov process is a mathematical model for a random system whose future and past states are independent conditional on the present state. Multi-harmonic signals can be modeled as a stochastic process with the Markov property. The Markovian representation of multi-harmonic signals enables us to use state-space tracking methods to continuously estimate the frequencies, phases, and amplitudes of the partials. Several research groups have proposed various signal analysis methods such as hidden Markov Models (HMM), short time Fourier transform (STFT), and Wigner-Ville distribution to solve this problem. Recently, a few groups of researchers have proposed Monte Carlo methods which estimate the posterior distribution of the fundamental frequency in multi-harmonic signals sequentially. However, multi-harmonic tracking is more challenging than single-frequency tracking, though the reason for this has not been well understood. The main objectives of this dissertation are to elucidate the fundamental obstacles to multi-harmonic tracking and to develop a reliable multi-harmonic tracker that can track cyclical patterns in multi-harmonic signals

Error Bounds and Applications for Stochastic Approximation with Non-Decaying Gain

This work analyzes the stochastic approximation algorithm with non-decaying gains as applied in time-varying problems. The setting is to minimize a sequence of scalar-valued loss functions $f_k(\cdot)$ at sampling times $\tau_k$ or to locate the root of a sequence of vector-valued functions $g_k(\cdot)$ at $\tau_k$ with respect to a parameter $\theta\in R^p$. The available information is the noise-corrupted observation(s) of either $f_k(\cdot)$ or $g_k(\cdot)$ evaluated at one or two design points only. Given the time-varying stochastic approximation setup, we apply stochastic approximation algorithms with non-decaying gains, so that the recursive estimate denoted as $\hat{\theta}_k$ can maintain its momentum in tracking the time-varying optimum denoted as $\theta_k^*$. Chapter 3 provides a bound for the root-mean-squared error $\sqrt{E(\|\hat{\theta}_k-\theta_k^*\|^2})$. Overall, the bounds are applicable under a mild assumption on the time-varying drift and a modest restriction on the observation noise and the bias term. After establishing the tracking capability in Chapter 3, we also discuss the concentration behavior of $\hat{\theta}_k$ in Chapter 4. The weak convergence limit of the continuous interpolation of $\hat{\theta}_k$ is shown to follow the trajectory of a non-autonomous ordinary differential equation. Both Chapter 3 and Chapter 4 are probabilistic arguments and may not provide much guidance on the gain-tuning strategies useful for one single experiment run. Therefore, Chapter 5 discusses a data-dependent gain-tuning strategy based on estimating the Hessian information and the noise level. Overall, this work answers the questions "what is the estimate for the dynamical system $\theta_k^*$" and "how much we can trust $\hat{\theta}_k$ as an estimate for $\theta_k^*$."Comment: arXiv admin note: text overlap with arXiv:1906.0953