Frames for Signal Processing, Wireless Communication and Transmission Losses

0102686 Casazza这个项目将集中在几个新的应用框架理论有关的信息传输。 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，并且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，并且需要从最初接收的内容进行最佳可能的重建。 第二，已经表明，采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，建设性的方法来实现这一能力依赖于假设接收机知道复值瑞利衰落系数。 这种假设在实际中往往是不切实际的，最近已经提出了新的酉空时信号的发送方和接收方都不知道衰落系数。 研究人员将促进这两个项目的实施，沿着信号/图像处理中的一系列相关问题。 这将需要：（1）构造具有特定优化性质的一致紧框架族;（2）发展实现它们的算法;（3）新的发展（和计算效率）的方法，用于反演的Gabor帧运算符，以允许更广泛的功能类被用于信号/图像处理。有没有可能开发出一种几乎在任何情况下都不会褪色的手机？ 有没有可能让您的计算机在发送后立即收到几乎完美的消息？ 有没有可能开发出一种助听器，不仅可以让佩戴者听到他/她想要的声音，而且还可以让一个人过滤掉他们不想听到的声音？有没有可能做一个乳房X光片，它是如此清晰，癌症诊断是99%的准确率？ 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个称为"统一紧框架“的数学分支。研究人员正在与全国各地的十几个研究小组合作，开发这些应用所需的数学，并开发在上述特定情况下实施这些结果的方法以及大量其他应用领域。

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.