Frames for Signal Processing, Wireless Communication and Transmission Losses

这个项目将集中于框架理论在信息传递方面的几个新的应用。首先，对于诸如因特网的通信网络，数据分组经常在传输中丢失，并根据用户不可见的协议重新传输。在许多应用中，随后的延迟是不可接受的，需要从最初接收到的数据进行尽可能好的重建。其次，已经证明使用多个天线的无线通信系统可以具有非常高的信道容量。到目前为止，实现这一容量的建设性方法依赖于假设接收器知道复值瑞利衰落系数。这种假设在实践中往往是不现实的，最近提出了一类新的酉空时信号，其中发送器和接收器都不知道衰落系数。调查人员将协助实施这两个方案以及信号/图像处理中的一系列相关问题。这将需要：(1)构建具有特定优化特性的全新的一致紧帧系列；(2)为其实现开发算法；(3)开发新的(并且在计算上有效的)方法来求逆Gabor帧操作符，以允许更广泛的函数类别用于信号/图像处理。有可能开发出在几乎任何情况下都不会褪色的手机吗？有没有可能让您的计算机在发送后立即接收近乎完美的消息？有没有可能开发出一种助听器，不仅可以让佩戴者听到他/她想听的东西，还可以让人过滤掉他们不想听的声音？有没有可能做一张如此清晰的乳房x光片，从而使癌症诊断的准确率达到99%？答案是，所有这些(以及许多其他重要问题)都是“理论上可能的”。实现这些重要应用所需要的是发展一门数学分支，称为“一致紧框架”。与全国各地的十几个研究小组合作，调查人员正在开发这些应用所需的数学知识，并开发在上述具体案例和大量其他应用领域实施这些结果的方法。

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.