Polynomial decomposition algorithms in signal processing

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2013.Cataloged from PDF version of thesis.Includes bibliographical references (p. 107-109).Polynomial decomposition has attracted considerable attention in computational mathematics. In general, the field identifies polynomials f(x) and g(x) such that their composition f(g(x)) equals or approximates a given polynomial h(x). Despite potentially promising applications, polynomial decomposition has not been significantly utilized in signal processing. This thesis studies the sensitivities of polynomial composition and decomposition to explore their robustness in potential signal processing applications and develops effective polynomial decomposition algorithms to be applied in a signal processing context. First, we state the problems of sensitivity, exact decomposition, and approximate decomposition. After that, the sensitivities of the composition and decomposition operations are theoretically derived from the perspective of robustness. In particular, we present and validate an approach to decrease certain sensitivities by using equivalent compositions, and a practical rule for parameter selection is proposed to get to a point that is near the minimum of these sensitivities. Then, new algorithms are proposed for the exact decomposition problems, and simulations are performed to make comparison with existing approaches. Finally, existing and new algorithms for the approximate decomposition problems are presented and evaluated using numerical simulations.by Guolong Su.S.M

Composition structures for system representation

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Electrical Engineering and Computer Science, 2017.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Cataloged from student-submitted PDF version of thesis.Includes bibliographical references (pages 171-183).This thesis discusses parameter estimation algorithms for a number of structures for system representation that can be interpreted as different types of composition. We refer to the term composition as the systematic replacement of elements in an object by other object modules, where the objects can be functions that have a single or multiple input variables as well as operators that work on a set of signals of interest. In general, composition structures can be regarded as an important class of constrained parametric representations, which are widely used in signal processing. Different types of composition are considered in this thesis, including multivariate function composition, operator composition that naturally corresponds to cascade systems, and modular composition that we refer to as the replacement of each delay element in a system block diagram with an identical copy of another system module. There are a number of potential advantages of the use of composition structures in signal processing, such as reduction of the total number of independent parameters that achieves representational and computational efficiency, modular structures that benefit hardware implementation, and the ability to form more sophisticated models that can represent significantly larger classes of systems or functions. The first part of this thesis considers operator composition, which is an alternative interpretation of the class of cascade systems that has been widely studied in signal processing. As an important class of linear time-invariant (LTI) systems, we develop new algorithms to approximate a two-dimensional (2D) finite impulse response (FIR) filter as a cascade of a pair of 2D FIR filters with lower orders, which can gain computational efficiency. For nonlinear systems with a cascade structure, we generalize a two-step parameter estimation algorithm for the Hammerstein model, and propose a generalized all-pole modeling technique with the cascade of multiple nonlinear memoryless functions and LTI subsystems. The second part of this thesis discusses modular composition, which replaces each delay element in a FIR filter with another subsystem. As an example, we propose the modular Volterra system where the subsystem has the form of the Volterra series. Given statistical information between input and output signals, an algorithm is proposed to estimate the coefficients of the FIR filter and the kernels of the Volterra subsystem, under the assumption that the coefficients of the nonlinear kernels have sufficiently small magnitude. The third part of this thesis focuses on composition of multivariate functions. In particular, we consider two-level Boolean functions in the conjunctive or disjunctive normal forms, which can be considered as the composition of one-level multivariate Boolean functions that take the logical conjunction (or disjunction) over a subset of binary input variables. We propose new optimization-based approaches for learning a two-level Boolean function from a training dataset for classification purposes, with the joint criteria of accuracy and simplicity of the learned function.by Guolong Su.Ph. D