Current Research in the Mathematics of Communications Department

Our work in Communication Theory and Signal Processing spans a spectrum of goals, encompassing both fundamentals and practice. But in either case our motivations are to expand knowledge base and, with occasional exceptions, are not project specific. We are fortunate that some of our people have, in the past, been members of development organizations, and have kept ties to these organizations as they evolved into business units. This continuing interaction provides valuable input regarding the direction of our investigations.

One major research goal is the search for fundamental limits, usually disregarding complexity and other practical issues, and is broadly described by Information Theory. Present efforts include difficult generalizations of point-to-point Shannon Theory to multi-access problems involving a sharing of a communication resource, with particular emphasis on wireless applications. Basic capacity studies of multiple antenna communication systems are included here.

In addition to this fundamental work we also engage in studies which propose or evaluate techniques, systems, or devices that are much closer to the practical art. Such investigations can be related to signal design, equalization, synchronization, performance evaluation, or communication theory for wireless systems. They can relate to new system proposals or existing infrastructure. The study of tradeoffs involved in choosing between equalization or coding to correct certain channel impairments on communication channels, or on recording channels modeled as communication channels, is part of this work. The search for constrained block codes that provide coding gain is of particular relevance to magnetic recording.

A very long standing project has been spectral estimation, the estimation of the power spectrum (perhaps short term) of time series. The theory includes non-stationary time series and non-uniform sampling; the applications are often to unusual and difficult situations. Spectral estimation techniques have been applied to global warming, black-holes, and so-called g-modes of oscillation of the sun. The latter modulates the solar wind, a flux of energetic charged particles capable of inducing damagingly high voltages on long ocean cables our in expensive communication satellites. Prediction of this phenomenon would enable advance action to prevent damage.

Wavelet research is another active area. One of the pioneers of the theory of wavelets, Ingrid Daubechies, was a member of our department for a number of years. While she was here, she worked on orthonormal bases of compactly supported wavelets, the work that spurred a wide use of wavelets in signal compression. More recently, we have been involved in the work on local trigonometric bases and their use in signal processing as well as in the lifting scheme, a new technique for constructing second generation wavelets. Other work in this area includes transform methods for multiple-description source coding, and geometry processing, a general attack on description, compression and filtering of surfaces in three dimensions. More details on any of these topics can be found on the Bell Labs Wavelet Home Page.

In preparing this brief report, it is natural to reflect on earlier history. It would be too lengthy, and probably arbitrary and incomplete, to start to name outstanding individuals who have worked, or indeed spent their careers in this department during its many years of existence. Instead, allow me just to chauvinistically mention that the department was founded and first headed by S.O. Rice, and was subsequently headed by D. Slepian and A.D. Wyner.

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