Adaptive Least Mean Square (LMS) equalizers are widely used in digital communication systems primarily for their ease of implementation and lack of dependence on a priori knowledge of input signal statistics. LMS equalizers exhibit non-Wiener characteristics in the presence of a strong narrowband interference and can outperform the optimal Wiener equalizer in terms of both mean square error (MSE) and bit error rate (BER). There has been significant work in the past related to the analysis of the non-Wiener characteristics of the LMS equalizer, which includes the discovery of the shift in the mean of the LMS weights from the corresponding Wiener weights and the modeling of steady state MSE performance. BER performance is ultimately a more practically relevant metric than MSE for characterizing system performance. The present work focuses on modeling the steady state BER performance of the normalized LMS (NLMS) equalizer operating in the presence of a strong narrowband interference. Initial observations showed that a 2 dB improvement in MSE may result in two orders of magnitude improvement in BER. However, some differences in the MSE and BER behavior of the NLMS equalizer were also seen, most notably the significant dependence (one order of magnitude variation) of the BER behavior on the interference frequency, a dependence not seen in MSE. Thus, MSE cannot be used as a predictor for the BER performance; the latter further motivates the pursuit of a separate BER model. The primary contribution of this work is the derivation of the probability density of the output of the NLMS equalizer conditioned on a particular symbol having been transmitted, which can then be leveraged to predict its BER performance. The analysis of the NLMS equalizer, operating in a strong narrowband interference environment, resulted in a conditional probability density function in the form of a Gaussian Sum Mixture (GSM). Simulation results verify the efficacy of the GSM expression for a wide range of system parameters, such as signal-to-noise ratio (SNR), interference-to-signal (ISR) ratio, interference frequency, and step-sizes over the range of mean-square stable operation of NLMS. Additionally, a low complexity approximate version of the GSM model is also derived and can be used to give a conservative lower bound on BER performance. A thorough analysis of the MSE and BER behavior of the Bi-scale NLMS equalizer (BNLMS), a variant of the NLMS equalizer, constitutes another important contribution of this work. Prior results indicated a 2 dB MSE improvement of BNLMS over NLMS in the presence of a strong narrowband interference. A closed form MSE model is derived for the BLMS algorithm. Additionally, BNLMS BER behavior was studied and showed the potential of two orders of magnitude improvement over NLMS. Analysis led to a BER model in the form of a GSM similar to the NLMS case but with different parameters. Simulation results verified that both models for MSE and BER provided accurate prediction of system performance for different combinations of SNR, ISR, interference frequency, and step-size. An enhanced GSM (EGSM) model to predict the BER performance for the NLMS equalizer is also introduced, specifically to address certain cases (low ISR cases) where the original GSM expression (derived for high ISR) was less accurate. Simulation results show that the EGSM model is more accurate in the low ISR region than the GSM expression. For the situations where the derived GSM expression was accurate, the BER estimates provided by the heuristic EGSM model coincided with those computed from the GSM expression. Finally, the two-interferer problem is introduced, where NLMS equalizer performance is studied in the presence of two narrowband interferers. Initial results show the presence of non-Wiener characteristics for the two-interferer case. Additionally, experimental results indicate that the BER performance of the NLMS equalizer operating in the presence of a single narrowband interferer may be improved by purposeful injection of a second narrowband interferer. / PHD

Let Fqt be the finite field with qt elements and let F*qt be its multiplicative group. We study the diagonal equation axq−1 + byq−1 = c, where a,b and c ∈ F*qt. This equation can be written as xq−1+αyq−1 = β, where α, β ∈ F ∗ q t . Let Nt(α, β) denote the number of solutions (x,y) ∈ F*qt × F*qt of xq−1 + αyq−1 = β and I(r; a, b) be the number of monic irreducible polynomials f ∈ Fq[x] of degree r with f(0) = a and f(1) = b. We show that Nt(α, β) can be expressed in terms of I(r; a, b), where r | t and a, b ∈ F*q are related to α and β. A recursive formula for I(r; a, b) will be given and we illustrate this by computing I(r; a, b) for 2 ≤ r ≤ 4. We also show that N3(α, β) can be expressed in terms of the number of monic irreducible cubic polynomials over Fq with prescribed trace and norm. Consequently, N3(α, β) can be expressed in terms of the number of rational points on a certain elliptic curve. We give a proof that given any a, b ∈ F*q and integer r ≥ 3, there always exists a monic irreducible polynomial f ∈ Fq[x] of degree r such that f(0) = a and f(1) = b. We also use the result on N2(α, β) to construct a new family of planar functions.

Practical control problems are subject to dealing with instrumentation noise and inaccurate models. These can be modelled as measurement and state noise, respectively. Nonlinear state estimators, for example a particle filter, can be used to mitigate these effects. However, they are usually computationally expensive which makes them impractical for industrial use. This text investigates using General Purpose Graphics Processing Units (GPGPU) to improve the performance particle and Gaussian sum filters by parallelizing their prediction, update and resampling steps. GPGPU accelerated filters are found to outperform non-accelerated filters as the number of particle increases. GPGPU acceleration also allows particle filters with 2^19.5 particles to be used on systems with dynamic time constants on the order of 0.1 second and for Gaussian sum filters with 2^18.5 particles to be used with time constants on the order of 1 second. The filters are applied to a bioreactor system containing R. Oryzae, where MPC control is applied to the production phase fumaric acid and glucose concentrations. The bioreactor is modelled using results from Iplik (2017) and Swart (2019). It is found that the GPGPU filters improved run times allow for more particles to be used which provides increased filter accuracy and thus better performance. This improved performance comes at the cost of consuming more energy. Thus, it is believed that the GPGPU implementations should be used for applications with complex dynamics/noise that require large numbers of particles and/or high sampling rates. / Dissertation (MEng (Control Engineering))--University of Pretoria, 2021. / Chemical Engineering / MEng (Control Engineering) / Unrestricted

This thesis essentially deals with the development and numerical explorations of a few improved Monte Carlo filters for nonlinear dynamical systems with a view to estimating the associated states and parameters (i.e. the hidden states appearing in the system or process model) based on the available noisy partial observations. The hidden states are characterized, subject to modelling errors, by the weak solutions of the process model, which is typically in the form of a system of stochastic ordinary differential equations (SDEs). The unknown system parameters, when included as pseudo-states within the process model, are made to evolve as Wiener processes. The observations may also be modelled by a set of measurement SDEs or, when collected at discrete time instants, their temporally discretized maps. The proposed Monte Carlo filters aim at achieving robustness (i.e. insensitivity to variations in the noise parameters) and higher accuracy in the estimates whilst retaining the important feature of applicability to large dimensional nonlinear filtering problems. The thesis begins with a brief review of the literature in Chapter 1. The first development, reported in Chapter 2, is that of a nearly exact, semi-analytical, weak and explicit linearization scheme called Girsanov Corrected Linearization Method (GCLM) for nonlinear mechanical oscillators under additive stochastic excitations. At the heart of the linearization is a temporally localized rejection sampling strategy that, combined with a resampling scheme, enables selecting from and appropriately modifying an ensemble of locally linearized trajectories whilst weakly applying the Girsanov correction (the Radon- Nikodym derivative) for the linearization errors. Through their numeric implementations for a few workhorse nonlinear oscillators, the proposed variants of the scheme are shown to exhibit significantly higher numerical accuracy over a much larger range of the time step size than is possible with the local drift-linearization schemes on their own. The above scheme for linearization correction is exploited and extended in Chapter 3, wherein novel variations within a particle filtering algorithm are proposed to weakly correct for the linearization or integration errors that occur while numerically propagating the process dynamics. Specifically, the correction for linearization, provided by the likelihood or the Radon-Nikodym derivative, is incorporated in two steps. Once the likelihood, an exponential martingale, is split into a product of two factors, correction owing to the first factor is implemented via rejection sampling in the first step. The second factor, being directly computable, is accounted for via two schemes, one employing resampling and the other, a gain-weighted innovation term added to the drift field of the process SDE thereby overcoming excessive sample dispersion by resampling. The proposed strategies, employed as add-ons to existing particle filters, the bootstrap and auxiliary SIR filters in this work, are found to non-trivially improve the convergence and accuracy of the estimates and also yield reduced mean square errors of such estimates visà-vis those obtained through the parent filtering schemes. In Chapter 4, we explore the possibility of unscented transformation on Gaussian random variables, as employed within a scaled Gaussian sum stochastic filter, as a means of applying the nonlinear stochastic filtering theory to higher dimensional system identification problems. As an additional strategy to reconcile the evolving process dynamics with the observation history, the proposed filtering scheme also modifies the process model via the incorporation of gain-weighted innovation terms. The reported numerical work on the identification of dynamic models of dimension up to 100 is indicative of the potential of the proposed filter in realizing the stated aim of successfully treating relatively larger dimensional filtering problems. We propose in Chapter 5 an iterated gain-based particle filter that is consistent with the form of the nonlinear filtering (Kushner-Stratonovich) equation in our attempt to treat larger dimensional filtering problems with enhanced estimation accuracy. A crucial aspect of the proposed filtering set-up is that it retains the simplicity of implementation of the ensemble Kalman filter (EnKF). The numerical results obtained via EnKF-like simulations with or without a reduced-rank unscented transformation also indicate substantively improved filter convergence. The final contribution, reported in Chapter 6, is an iterative, gain-based filter bank incorporating an artificial diffusion parameter and may be viewed as an extension of the iterative filter in Chapter 5. While the filter bank helps in exploring the phase space of the state variables better, the iterative strategy based on the artificial diffusion parameter, which is lowered to zero over successive iterations, helps improve the mixing property of the associated iterative update kernels and these are aspects that gather importance for highly nonlinear filtering problems, including those involving significant initial mismatch of the process states and the measured ones. Numerical evidence of remarkably enhanced filter performance is exemplified by target tracking and structural health assessment applications. The thesis is finally wound up in Chapter 7 by summarizing these developments and briefly outlining the future research directions