Information Theory And Geometric Methods Of Radar Signal Processing

Information Geometry is the fundamental and cutting-edge discipline which studiesstatistical problems on Riemannian manifolds of probability distributions using the methodsof Differential Geometry. It is identified as the second generation of modern InformationTheory pioneered by Shannon and exhibits great potential for development in the field ofinformation science and systems theory. The underlying thesis is to explore the applications ofinformation geometry to signal processing, especially to radar signal processing, and studiesthe fundamental and scientific problems in radar signal processing from a bran-new viewpoint.In particular, the scientific problems such as information resolution of radar systems, signaldetection, parameter estimation, the information gathering capacity and accumulativeinformation of sensor networks are explored in depth, with the development of a new set ofanalysis methods as well as a new set of methods to deal with the existing problems.The first chapter refines the scientific content of information geometry and elaborates thehistory and applications of information geometry as well as its basic ideas and basic methods.The second chapter introduces the principles and mathematical foundations of infor-mation geometry and provides a basis for the following chapters.The third chapter explores the definition, significance and measure of radar resolution aswell as its influence on the performance of target detection and tracking. A new concept calledinformation resolution for a sensor measurement system, which is defined in the frameworkof information geometry, is proposed. In particular, the information resolution of radarsystems is generalized from the work on existing radar resolution pioneered by Woodwardand defined on statistical manifolds where the intrinsic geometrical structure of waveform,measurement and noise models of the underlying sensing devices are convenientlycharacterized in terms of the Fisher information metric. Information resolution provides ametric to measure the practical resolution capacity of radar systems as well as the resolutionof joint detection-tracking systems. A set of important conclusions of information resolutionenriches the conventional understanding of radar resolution. Based on resolution cells fortarget detection, the key problems of radar measurement, detection and resolution, as well asthe measurement extraction schemes are studied. A definition of detection-based radarresolution, called differential resolution, is developed to describe the system’s capacity todistinguish two closely spaced targets. The SNR effects on resolution are analyzed and themeasurement-extraction scheme based on the differential resolution cell is discussed andsimulations show that an enhanced tracking performance can be obtained by such adevelopment.The fourth chapter studies the information geometric methods of signal detection. Aconcise geometric interpretation of deterministic and random signal detection in the theory ofinformation geometry is established. In such a framework, both hypotheses and detector canbe treated as geometrical objects on the statistical manifold of a parameterized family ofprobability distributions. Both the detector and detection performance are elucidated geometrically in terms of the Kullback-Leibler divergence. Then, the generalized likelihoodratio test (GLRT) for composite hypothesis testing problems is considered from a geometricviewpoint. Two pictures of the GLRT for curved exponential families are presented, based onwhich the performance deterioration when performing the GLRT under finite number ofsamples is discussed. Further more, a concrete geometric interpretation of the locally mostpowerful (LMP) test for weak signal detection is presented. In particular, the LMP test isidentified as the norm of natural whitened gradient on the statistical manifold, which indicatesthat the LMP test pursues the steepest learning directions from the null hypothesis to theempirical distribution of the observed data on the manifold. Due to this nicety, an immediateextension of the LMP test, called the ELMP test which removes the scalar and the one-sidedrestrictions in the LMP test, is proposed. The above analyses reveal equivalence between thedetection problems and the discrimination problems and provide a unified geometricframework for the analysis of detection problems, which extends the existing analyses to anew level.The fifth chapter investigates information geometric methods of parameter estimation.Firstly, the basic theory of information geometry on information loss of parameter estimationis summarized and examples of the inherent information loss caused by the nonlinearity ofmeasurement model are presented, while a tighter lower bound of the error variance withrespect to the well-known Cramér-Rao Lower Bound (CRLB) is calculated. Secondly, anatural gradient-based maximum likelihood estimator (NG-MLE) on statistical manifolds isproposed to deal with the nonlinear parameter estimation problem of curved exponentialfamilies. We demonstrate that the nonlinear estimation problem can be simply viewed as adeterministic optimization problem with respect to the structure of a statistical manifold. Inthis way, information geometry offers an elegant geometric interpretation for the definitionand convergence of the estimator. The theory is interpreted via the analysis of a distributedmote network localization problem where the Radio Interferometric Positioning System(RIPS) measurements are used for free mote location estimation. The analysis resultspresented demonstrate the advanced computational philosophy of the proposed methodology.Moreover, a noisy measurement model that takes the location uncertainties of anchor nodesinto account in the node localization process has been derived, which effectively deals withthe problem of progressive localization when the localization error is nonlinearly propagatedover the sensor network.The sixth chapter studies the information geometry of target tracking sensor networks.The connections between information geometry and performance of sensor networks fortarget tracking are explored to pursue a better understanding of placement, planning andscheduling issues. Firstly, the integrated Fisher information distance (IFID) between the statesof two targets is analyzed by solving the geodesic equations and is adopted as a measure oftarget resolvability by the sensor. The differences between the IFID and the well knownKullback-Leibler divergence (KLD) are highlighted. We also explain how the energyfunctional, which is the “integrated, differential” KLD, relates to the other distance measures.Secondly, the structures of statistical manifolds are elucidated by computing the canonical Levi-Civita affine connection as well as Riemannian and scalar curvatures. We show therelationship between the Ricci curvature tensor field and the amount of information that canbe obtained by the network sensors. Thirdly, an analytical presentation of statistical manifoldsas an immersion in the Euclidean space for distributions of exponential type is given. Thesignificance and potential to address system definition and planning issues using informationgeometry, such as the sensing capability to distinguish closely spaced targets, calculation ofthe amount of information collected by sensors and the problem of optimal scheduling ofnetwork sensor and resources, etc., are demonstrated. Finally, the cumulative effect ofinformation when there is relative motion between the target and sensor is discussed. Theaccumulative information is used as a criterion for the sensor trajectory scheduling problem inbearings-only tracking.The seventh chapter explores the target tracking and localization problem in the presenceof phase measurement ambiguities. The main focus of this chapter is to deal with theambiguities caused by phase measurements and to elucidate how to identify and remove theseambiguities in tracking and localization context. Specifically, we combine the aboveapproaches in terms of creating mappings between target location and phase measurementspaces so that the nonlinear and indeterminate Diophantine problem reduces to the acquisitionof a finite set of possible target locations over a region of interest. Then firstly, we show thatwhen the target motion is significant between data sampling intervals the location ambiguitycan be resolved over time via known target-in-cluster tracking techniques. Secondly, when thetarget is undergoing micromotions which results in the same collection of candidate locationsfrom phase measurements over time, the location ambiguity can be resolved using a novelphase distribution discrimination method. In this method a probability density function of theambiguous phase-only measurement is derived that takes both sensor noise and target motiondistributions into account based on directional statistics. Optimal locations are inferred fromsuch distributions. The inference algorithm is interpreted from the viewpoint of manifoldlearning, which provides a reference for solving identification problems based on manifold.The eighth chapter makes a summary of the thesis, while several open problems ofinformation geometry in applications of signal processing are proposed.In conclusion, the studies and results in this paper not only enrich the basic theory ofstatistical signal processing and radar signal processing, but also establish comprehensiveconnections between statistics and information geometry, and provides an exemplification ofadvantages of the geometrical perspective on studying statistical problems.