Cornerstones of Sampling of Operator Theory

This paper reviews some results on the identifiability of classes of operators whose Kohn-Nirenberg symbols are band-limited (called band-limited operators), which we refer to as sampling of operators. We trace the motivation and history of the subject back to the original work of the third-named author in the late 1950s and early 1960s, and to the innovations in spread-spectrum communications that preceded that work. We give a brief overview of the NOMAC (Noise Modulation and Correlation) and Rake receivers, which were early implementations of spread-spectrum multi-path wireless communication systems. We examine in detail the original proof of the third-named author characterizing identifiability of channels in terms of the maximum time and Doppler spread of the channel, and do the same for the subsequent generalization of that work by Bello. The mathematical limitations inherent in the proofs of Bello and the third author are removed by using mathematical tools unavailable at the time. We survey more recent advances in sampling of operators and discuss the implications of the use of periodically-weighted delta-trains as identifiers for operator classes that satisfy Bello's criterion for identifiability, leading to new insights into the theory of finite-dimensional Gabor systems. We present novel results on operator sampling in higher dimensions, and review implications and generalizations of the results to stochastic operators, MIMO systems, and operators with unknown spreading domains

Efficient algorithms and implementations for signal processing

A scheme is presented to regain a finite number of lost samples from a Nyquist-rate-sampled band-limited signal f of finite energy by replenishing new sample values of the same number. The result can also be viewed as the solution to a special non-uniform sampling problem. A scheme is also presented to recover a band-limited function f of finite energy from its sampling values on real sequences with an accumulation point. The result given here can also be viewed as an approach to the extrapolation problem of determination a band-limited function in terms of its given values on a finite interval. An error estimate is also obtained. The existence of two kinds of frames, Weyl-Heisenberg frames and affine frames, is studied. The conditions given in this dissertation improve the known conditions and, in addition, are easy to verify. A parallel algorithm for the two-dimensional forward fast wavelet transform is developed and implemented on the AP1000 multiprocessor system. The algorithm is carefully analyzed before implementation. Experiments are performed on different input sizes on different numbers of processors. The results from the experiments coincide with the theoretical analysis. The parallel algorithm gains expected speedup on the mesh architecture. Further work is suggested

Quantum phases of interacting bosons in optical lattices

This thesis presents a theoretical analysis of the phase diagram of ultracold bosons in a lattice and interacting with long-range interac- tions. The theoretical model is an extended Bose-Hubbard model and describes the dynamics of ultracold atoms in optical lattices realised in present experimental platforms. We consider here two situations, where either the long-range forces are global and emerge from the coupling with a high-finesse cavity, or they decay with the interparti- cle distance and can be due to Rydberg interactions or to the atoms permanent dipoles. We determine the ground state in one and two dimensions using mean-field treatments. In one dimension we comple- ment our studies using numerical programs based on tensor networks. We focus in particular on parameters for which the hopping induced by the kinetic energy competes with the interaction-induced corre- lated hopping between lattice sites. We analyse the superfluid phases emerging from the competition of these two mechanisms, and identify the parameters, where the two processes destructively interfere. For power-law interactions this quantum interference leads to insulating phases at relatively large kinetic energies, where one would otherwise expect superfluidity. When correlated tunnelling is due to the global potential of a resonator, the ground state is a self-organised topological insulator.Diese Arbeit präsentiert eine theoretische Analyse des Phasendiagramms von ultrakalten Bosonen in einem Gitter, die langreichweitige Wechselwirkungen erfahren. Das theoretische Modell ist ein erweitertes Bose-Hubbard Modell und beschreibt die Dynamik von ultrakalten Atomen in einem optischen Gitter, wie sie in heutigen Experimenten realisiert werden kann. Wir betrachten hier zwei Situationen: Zum einen sind die langreichweitigen Kräfte global und entstehen aus der Kopplung mit einem Resonator. Zum anderen zerfällt das Wechselwirkungspotential mit dem Abstand zwischen den Teilchen, wie es zwischen Rydbergatomen oder Atomen mit einem permanenten Dipolmoment auftritt. Wir bestimmen den Grundzustand in einer und zwei Dimensionen durch Mean-Field Analysen. In einer Dimension benutzen wir zudem ein auf Tensornetzwerken basierendes numerisches Programm. Wir betrachten insbesondere Parameter, für die das durch die kinetische Energie induzierte Tunneln mit dem von der Wechselwirkung induzierten Tunneln konkurriert. Wir analysieren die superfluiden Phasen, die sich aus dieser Kompetition ergeben, und identifizieren die Parameter, bei denen die beiden Prozesse destruktiv interferieren. Für die mit dem Abstand zerfallenden Wechselwirkungen führt diese Quanteninterferenz zu Isolatoren in Parameterbereichen, in denen man sonst Superfluidität erwarten würde. Wenn das Tunneln vom globalen Potential herrührt, ist der Grundzustand ein selbstorganisierter topologischer Isolator

Sensor array signal processing : two decades later

Caption title.Includes bibliographical references (p. 55-65).Supported by Army Research Office. DAAL03-92-G-115 Supported by the Air Force Office of Scientific Research. F49620-92-J-2002 Supported by the National Science Foundation. MIP-9015281 Supported by the ONR. N00014-91-J-1967 Supported by the AFOSR. F49620-93-1-0102Hamid Krim, Mats Viberg

Sampling theory in shift-invariant spaces: generalizations

Roughly speaking sampling theory deals with determining whether we can or can not recover a continuous function from some discrete set of its values. The most important result and main pillar of this theory is the well-known Shannon’s sampling theorem which states that: If a signal f(t) contains no frequencies higher than 1/2 cycles per second, it is completely determined by giving its ordinates at a sequence of points spaced one second apart….A grandes rasgos la teoría de muestreo estudia el problema de recuperar una función continua a partir de un conjunto discreto de sus valores. El resultado más importante y pilar fundamental de esta teoría es el conocido teorema de muestreo de Shannon que afirma que: Si una señal f(t) no contiene frecuencias mayores que 1/2 ciclos por segundo entonces está completamente determinada por sus ordenadas en una sucesión de puntos espaciados en un segundo….Proyecto de investigación MTM2009–08345 del Ministerio de Ciencia e Innovación de España.Programa Oficial de Doctorado en Ingeniería MatemáticaPresidente: Luis Alberto Ibort Latre.- Secretario: Eugenio Hernández Rodríguez.- Vocal: Ole Christense

Iterative algorithms for optimal signal reconstruction and parameter identification given noisy and incomplete data

Originally published as thesis (Dept. of Electrical Engineering and Computer Science, Ph.D., 1982).Includes bibliographies.Supported in part by the Advanced Research Projects Agency monitored by ONR under Contract N00014-81-K-0742 NR-049-506 Supported in part by the National Science Foundation under Grant ECS80-07102Bruce Ronald Musicus