Characterization of Information Channels for Asymptotic Mean Stationarity and Stochastic Stability of Non-stationary/Unstable Linear Systems

Stabilization of non-stationary linear systems over noisy communication channels is considered. Stochastically stable sources, and unstable but noise-free or bounded-noise systems have been extensively studied in information theory and control theory literature since 1970s, with a renewed interest in the past decade. There have also been studies on non-causal and causal coding of unstable/non-stationary linear Gaussian sources. In this paper, tight necessary and sufficient conditions for stochastic stabilizability of unstable (non-stationary) possibly multi-dimensional linear systems driven by Gaussian noise over discrete channels (possibly with memory and feedback) are presented. Stochastic stability notions include recurrence, asymptotic mean stationarity and sample path ergodicity, and the existence of finite second moments. Our constructive proof uses random-time state-dependent stochastic drift criteria for stabilization of Markov chains. For asymptotic mean stationarity (and thus sample path ergodicity), it is sufficient that the capacity of a channel is (strictly) greater than the sum of the logarithms of the unstable pole magnitudes for memoryless channels and a class of channels with memory. This condition is also necessary under a mild technical condition. Sufficient conditions for the existence of finite average second moments for such systems driven by unbounded noise are provided.Comment: To appear in IEEE Transactions on Information Theor

Some new developments in image compression

This study is divided into two parts. The first part involves an investigation of near-lossless compression of digitized images using the entropy-coded DPCM method with a large number of quantization levels. Through the investigation, a new scheme that combines both lossy and lossless DPCM methods into a common framework is developed. This new scheme uses known results on the design of predictors and quantizers that incorporate properties of human visual perception. In order to enhance the compression performance of the scheme, an adaptively generated source model with multiple contexts is employed for the coding of the quantized prediction errors, rather than a memoryless model as in the conventional DPCM method. Experiments show that the scheme can provide compression in the range from 4 to 11 with a peak SNR of about 50 dB for 8-bit medical images. Also, the use of multiple contexts is found to improve compression performance by about 25% to 35%;The second part of the study is devoted to the problem of lossy image compression using tree-structured vector quantization. As a result of the study, a new design method for codebook generation is developed together with four different implementation algorithms. In the new method, an unbalanced tree-structured vector codebook is designed in a greedy fashion under the constraint of rate-distortion trade-off which can then be used to implement a variable-rate compression system. From experiments, it is found that the new method can achieve a very good rate-distortion performance while being computationally efficient. Also, due to the tree-structure of the codebook, the new method is amenable to progressive transmission applications

Geometric modeling and analysis of dynamic resource allocation mechanisms

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2001.Includes bibliographical references (p. 159-163).The major contribution of this thesis is the investigation of a specific resource allocation optimization problem whose solution has both practical application as well as theoretical interest. It is presented as a specific case of a more general modeling framework we put forth. The underlying question asks how to partition a given resource into a fixed number of parts such that the elements of the resulting partition can be scheduled among a set of user requests to minimize the worst case difference between the schedule and the requests. This particular allocation problem has not been studied before. The general problem is difficult in part because the evaluation of the objective problem is a difficult task by itself. We present a novel algorithm for its exact solution in a constrained setting and discussion of the unconstrained setting in, followed by a number of practical applications of these solutions. The solution to the constrained optimization problem is shown to provide sizable benefits in allocation efficiency in a number of contexts at a minimal implementation cost. The specific contexts we look at include communication over a shared channel, allocation of many small channels to a few users and package delivery from a central office to a number of satellite offices. We also present a set of new fairness results for auction-based allocation mechanisms and show how these mechanisms also fall within our modeling framework. Specifically, we look at using auctions as mechanisms to allocate an indivisible shared resource fairly among a number of users. We establish that a straightforward approach as has been tried in the literature does not guarantee an fair allocation over a long time scale and provide a modified approach that does guarantee a fair allocation. We also show that by allowing users to strategize when bidding on the resource we can avoid the problem of unfairness, for some simple cases. This analysis has not been seen in existing literature. Finally, an analysis of the deterministic and stochastic stability of our class of models is presented that applies to a large subset of the models within our framework. The deterministic stability results presented establish the ultimate boundedness of the lag of deterministically stabilizable models in our framework under a wide variety of quantizer-based scheduling rules. This variety of available rules can be used to further control the behavior of the lag of a stable mechanism. We also discuss the application of existing stochastic stability theory to a large subset of the stochastic models in our framework. This is a straightforward usage of existing stability results based on verifying the satisfaction of a stochastic drift condition.by Matthew Secor.Ph.D