Applied Time-frequency Analysis

但由于其自身的性质，只有极其微弱的足迹深深隐藏在绝大多数正常数据之下。在许多情况下，他们揭示自己作为不寻常的局部结构沿着联合时间和频率域的一个大的非平稳数据集。设计时频分析（TFA）旨在揭示具有时变频率成分的非平稳信号的局部特征。如果原始数据能够被智能地转换并沿时域和频域沿着表示，则它提供了发现隐藏的异常的能力。例如，不规则的心跳，异常的大脑活动，或突然发作并发生在不同频率范围内的异常飞机振动都可以通过TFA识别。早期发现严重异常使我们能够采取适当的行动，以防止昂贵的和致命的损害。 尽管它的有用性，两个主要问题严重阻碍了基于TFA的诊断的有效性。首先，由于TFA将一维信号表示为时间和频率两个变量的函数，因此计算工作量通常很大。这对于处理超大数据集和/或在实时设置中使用尤其成问题。然而，在许多情况下，预期的应用程序具有独特的功能，可以利用这些功能来大大降低计算复杂性。其次，许多标准TFA技术需要简化假设或标准特性，例如信号是确定性的、均匀采样的或很好地拟合正弦波形的线性模型。然而，在实践中，大多数重要的应用程序将严重违反这些假设。然而，目前的数学理论可以扩展，使具体的TFA技术可以开发，而不依赖于一些简化的假设。此外，如果实际数据具有某些可以有效利用的特征，则可以将这些“非标准特征”纳入技术的理论开发中，以帮助提高有效性。 该研究计划的目标是从根本上克服现有TFA技术的关键限制，以释放其在实际应用中的全部潜力。更具体地说，我们的目标是概括时频分析背后的基本原理：1）首先处理超大数据量，然后进行基于TFA的实时信号处理; 2）研究实际数据中经常出现的一些“非标准特征”; 3）随着1）和2）中建立的有效计算方案和更精细的理论框架的可用性，可以探索更广泛的计算机辅助诊断应用。 本研究具有跨学科性和实用性。它集成了数学，统计，计算和应用程序开发。我希望该计划的学员将与医学科学和工业领域的工程师和科学家密切合作，并为现实世界的问题做出重要的原创性贡献。该计划的成功将推动时频分析领域的发展，创建针对特定类型信号的复杂时频分析技术，并导致用于各种目的的计算机辅助监测和诊断软件的研发开发。

Abnormalities by their own nature have only extremely weak footprints hidden deeply underneath the vast majority of normal data. In many cases, they reveal themselves as unusual localized structures along a joint time and frequency domain of a large set of non-stationary data. Time-frequency analysis (TFA) by design aims to reveal local features of non-stationary signals with time-varying frequency content. It provides the ability to uncover hidden abnormalities if the original data can be intelligently transformed and represented along the time and frequency domains. For example, irregular heartbeats, abnormal brain activities, or abnormal airplane vibrations that have a sudden onset and occur at a distinct frequency range can all be identified by TFA. Early detection of critical abnormalities allows us to take appropriate action to prevent expensive and fatal damages. Despite of its usefulness, two main issues severely impede the effectiveness of the TFA-based diagnoses. First, as TFA represents a one-dimensional signal as a function of two variables, time and frequency, the computational efforts are often substantial. This is particularly problematic for processing ultra large data set and/or used in a real-time setting. However, in many cases, the intended application has unique features that can be exploited to considerably reduce the computational complexity. Second, many standard TFA techniques require simplifying assumptions or standard characteristics such as that the signals are deterministic, sampled evenly, or fit well to linear models of sinusoidal waveforms. However, in practice, the majority of important applications will violate these assumptions significantly. Nevertheless, current mathematical theories can be extended so that specific TFA techniques can be developed without the reliance of some of the simplifying assumptions. Furthermore, if the actual data has certain characteristics that can be usefully exploit, these “non-standard characteristics” can be incorporated in the theoretical development of the techniques to help increase the effectiveness. The objective of this research program is to fundamentally overcome the critical limitations of the existing TFA techniques, in order to release its full potential for practical applications. More specifically, we aim to generalize the rationale behind the time-frequency analysis 1) to first tackle ultra large data size and then real-time TFA-based signal processing; 2) to investigate a number of “non-standard characteristics” that are regularly presented in the actual data; 3) with the availability of efficient computational schemes and more refined theoretical framework established in 1) and 2), a much broader range of computer-aided diagnostic applications can be explored. This research proposal is interdisciplinary and practical. It integrates mathematics, statistics, computing, and applications development. I expect that the trainees in the program will work closely with engineers and scientists in medical science and industries and make significant original contributions relevant to real-world problems. The success of this program will advance time-frequency analysis field, create sophisticated time-frequency analysis techniques tailored for specific types of signals and lead to the R&D development of computer-assisted monitoring and diagnostic software for various purposes.