Lattice theory, circular statistics and polynomial phase signals
Robert Mckilliam · 2010
This thesis studies connections between two fields, lattice theory and circular statistics. We focus on the estimation and theoretical analysis of polynomial phase signals. These signals have a vast array of applications in science, in particular in astronomy, optics, biology, geology, geography and meteorology and also in engineering, particularly in communications and radar. Despite this, we find that the theoretical tools for analysing these signals are lacking. We discover a special family of lattices, called Vn/m*, that are particularly useful for studying polynomial phase signals. Using these lattices we are able to close a number of the theoretical gaps that exist in the literature, and also produce some remarkably accurate estimators. We firstly describe some new results in the field of lattice theory. The most significant result is the discovery of a fast nearest point algorithm for the lattice An* and also a related family of lattices called the Coxeter lattices. The new algorithms all require a linear number of operations in the dimension of the lattice. This is significantly faster than previous algorithms that require, in the worst case, a quadratic number of operations. We then study the lattices Vn/m*. We describe a number of their properties and devise a nearest point algorithm that requires at most a polynomial number of operations in the dimension of the lattice. This is an improvement over the fastest nearest point algorithms for random lattices that require an exponential number of operations. We then consider polynomial phase signals and their estimation. For polynomial phase signals of order zero the estimation problem is equivalent to a fundamental problem in circular statistics, that of estimating the mean direction of a set of circular data. A standard approach to mean direction estimation is to compute the sample circular mean. In this thesis we consider an alternative estimator called the angular least squares estimator, and we discover that it can be computed rapidly by finding a nearest lattice point in the lattice An*, a problem we have solved. In some scenarios the angular least squares estimator is statistically more accurate and also computationally simpler than the sample circular mean. Therefore the results of this thesis potentially have implications for the wide variety of fields in science, engineering and statistics that currently use the sample circular mean. For higher order polynomial phase signals the estimation problem is equivalent to single frequency estimation (when the order is equal to one) and polynomial phase estimation (when the order is greater than one). These problems are common to radar, sonar, astronomy and telecommunications. We find that a very accurate estimator results from computing a nearest point in the lattice Vn/m* and derive the asymptotic properties of this estimator. We show that the estimator is strongly consistent and describe its central limit theorem. For polynomial phase signals of order greater than one these theoretical results are the first of their kind. While deriving these statistical results, we produce a number of new theorems that describe the aliasing properties of polynomial phase signals. These results can be viewed as higher order versions of the Nyquist sampling theorem. Lattice theory is crucial in the description of these aliasing properties. These results will be of great value to engineers, scientists and statisticians studying polynomial phase signals.