Applied Time-frequency Analysis

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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.