Processing and Transmission of Information

Contains reports on four research projects.Lincoln Laboratory, Purchase Order DDL-B222U.S. Air Force under Air Force Contract AF19(604)-520

Processing and Transmission of Information

Contains reports on five research projects

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

Some comments on developments in exact solutions in statistical mechanics since 1944

Lars Onsager and Bruria Kaufman calculated the partition function of the Ising model exactly in 1944 and 1949. Since then there have been many developments in the exact solution of similar, but usually more complicated, models. Here I shall mention a few, and show how some of the latest work seems to be returning once again to the properties observed by Onsager and Kaufman.Comment: 28 pages, 5 figures, section on six-vertex model revise

Calcretes in the Thar desert: genesis, chronology and palaeoenvironment

The calcretes in the Thar desert occur in a variety of settings, including the piedmonts, sheetwash aggraded plains; and this study adds calcretes in regolith and colluvio-alluvial plains to the group of settings in which calcretes occur in the region. Field logs, morphological details and analytical data such as petrographic, cathodoluminescence and geochemical characteristics are described along with a discussion on their implications. Sand dunes and sandy plains dating to <20 ka have weakly developed calcretes. The better-developed calcrete horizons occur in piedmonts, interdunes or in areas that have sufficient groundwater. Deep sections in the region show phases of calcrete development in aeolian sand aggradation at ~150, ~100, ~60 and 27-14 ka. The extensive sheetwash plains have mature calcretes and date to mid-Pleistocene. Our studies indicate that these calcretes represent a hybrid process, where carbonate enrichment of the originally calcareous host occurred due to periodically raised groundwaters, and its differentiation into nodules occurred under subaerial environment i.e., after recession of groundwater. Deep sections also show a stack of discrete calcretes that developed in individual aggradation episodes with hiatuses as indicated by ESR dating results. Nodules display a multiplicity of carbonate precipi tation events and internal reorganization of calcitic groundmass. The process is accompanied by degradation and transformation of unstable minerals, particularly clays and with a neosynthesis of palygorskite. The ancient calcretes are dated from the beginning of the Quaternary to ~600 ka and show more evolved morphologies marked by brecciation, dissolution, laminar growth on brecciated surfaces, pisolites and several generations of re-cementation. Mica/chlorite schists and such other rocks are particularly vulnerable to replacement by carbonate. In an extreme case, replacement of quartzose sandstone was observed also. The presence of stretches of alluvio-colluvial plains in an area presently devoid of drainage bespeaks of occasional high-energy fluvial regime, under a semi-arid climate. The mid-Pleistocene period saw a shift towards more arid climate and this facilitated sheetwash aggradation. Finally, during the late Pleistocene, aggradation of aeolian sands indicated a progressively drier climate. However, this does not find its reflection in stable isotope data. The amount of carbonate in the form of calcretes is substantial. The present studies indicate that aeolian dust or rainwater are minor contributors to the carbonate budget. A more important source was provided by the pre-existing calcretes in the sheetwash aggraded plains and detrital carbonate in the aeolian sediments. The original source of carbonate in the region, however, remains unresolved and will need further investigations. Electron spin resonance protocols for the dating of calcretes were developed as a part of this study and the results accorded well with geological reasoning

A weakly stable algorithm for general Toeplitz systems

We show that a fast algorithm for the QR factorization of a Toeplitz or Hankel matrix A is weakly stable in the sense that R^T.R is close to A^T.A. Thus, when the algorithm is used to solve the semi-normal equations R^T.Rx = A^Tb, we obtain a weakly stable method for the solution of a nonsingular Toeplitz or Hankel linear system Ax = b. The algorithm also applies to the solution of the full-rank Toeplitz or Hankel least squares problem.Comment: 17 pages. An old Technical Report with postscript added. For further details, see http://wwwmaths.anu.edu.au/~brent/pub/pub143.htm

The quantum capacity is properly defined without encodings

We show that no source encoding is needed in the definition of the capacity of a quantum channel for carrying quantum information. This allows us to use the coherent information maximized over all sources and and block sizes, but not encodings, to bound the quantum capacity. We perform an explicit calculation of this maximum coherent information for the quantum erasure channel and apply the bound in order find the erasure channel's capacity without relying on an unproven assumption as in an earlier paper.Comment: 19 pages revtex with two eps figures. Submitted to Phys. Rev. A. Replaced with revised and simplified version, and improved references, etc. Why can't the last line of the comments field end with a period using this web submission form

Coarse Bifurcation Diagrams via Microscopic Simulators: A State-Feedback Control-Based Approach

The arc-length continuation framework is used for the design of state feedback control laws that enable a microscopic simulator trace its own open-loop coarse bifurcation diagram. The steering of the system along solution branches is achieved through the manipulation of the bifurcation parameter, which becomes our actuator. The design approach is based on the assumption that the eigenvalues of the linearized system can be decomposed into two well separated clusters: one containing eigenvalues with large negative real parts and one containing (possibly unstable) eigenvalues close to the origin

Online Convex Optimization Using Predictions

Making use of predictions is a crucial, but under-explored, area of online algorithms. This paper studies a class of online optimization problems where we have external noisy predictions available. We propose a stochastic prediction error model that generalizes prior models in the learning and stochastic control communities, incorporates correlation among prediction errors, and captures the fact that predictions improve as time passes. We prove that achieving sublinear regret and constant competitive ratio for online algorithms requires the use of an unbounded prediction window in adversarial settings, but that under more realistic stochastic prediction error models it is possible to use Averaging Fixed Horizon Control (AFHC) to simultaneously achieve sublinear regret and constant competitive ratio in expectation using only a constant-sized prediction window. Furthermore, we show that the performance of AFHC is tightly concentrated around its mean

The filtering equations revisited

The problem of nonlinear filtering has engendered a surprising number of mathematical techniques for its treatment. A notable example is the change-of--probability-measure method originally introduced by Kallianpur and Striebel to derive the filtering equations and the Bayes-like formula that bears their names. More recent work, however, has generally preferred other methods. In this paper, we reconsider the change-of-measure approach to the derivation of the filtering equations and show that many of the technical conditions present in previous work can be relaxed. The filtering equations are established for general Markov signal processes that can be described by a martingale-problem formulation. Two specific applications are treated