Frames for Signal Processing, Wireless Communication and Transmission Losses

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0102686CasazzaThis project will concentrate on several new applications of frame theory concerning the transmission of information. First, for communication networks such as the Internet, packets of data are often lost in transmission and are retransmitted according to a protocol invisible to the user. In many applications, the subsequent delay is unacceptable and best possible reconstruction from what is initially received is necessary. Second, it has been shown that wireless communication systems which employ multiple antennas can have very high channel capacities. Until now, constructive approaches to achieving this capacity have relied on the assumption that the receiver knows the complex-valued Rayleigh fading coefficients. This assumption is often unrealistic in practice and recently new classes of unitary space-time signals have been proposed where neither the sender nor the receiver knows the fading coefficients. The investigators will facilitate the implementation of both of these programs along with a host of related problems in signal/image processing. This will require: (1) The construction of totally new families of uniform tight frames with specific optimization properties; (2) The development of algorithms for their implementation; (3) The development of new (and computationally efficient) methods for inverting the Gabor frame operator to allow a much broader class of functions to be used for signal/image processing.Is it possible to develop a cell phone which will not fade under almost any circumstance? Is it possible to have your computer receive an almost perfect message immediately after it is sent? Is it possible to develop a hearing aid which not only allows the wearer to hear what he/she wants, but also allows a person to filter out sounds they do not want to hear? Is it possible to have a breast x-ray which is so clear that cancer diagnoses are 99% accurate? The answer is that all of these (and a host of other important questions) are ``theoretically possible''. What is needed for the implementation of these important applications is the development of a branch of mathematics called ``uniform tight frames''. Working with a dozen research groups around the country, the investigators are developing the mathematics needed for these applications as well as developing the means to implement these results in the specific cases listed above and a large number of other areas of application.