Algorithms for time -frequency distributions: Suppressing cross terms and enhancing resolution.

Abstract

The importance of Time-Frequency Distributions (TFDs) has been widely recognized in many fields after years of intensive research. However, its usage is shadowed by the Uncertainty Principle and the conviction that ideal TFDs are impossible to obtain in general cases. It is realized that for different situations, specific time-frequency analysis tools can be devised to be superbly effective. The key to successful design of such a tool is to understand and match the construction mechanism of TFDs and the nature of the signal. In this dissertation, several approaches of constructing TFDs are studied. The major contributions include three new TFD algorithms with emphasis on different aspects of TFDs. The first (in Chapter 3), an adaptive kernel design procedure, aims at minimizing the Renyi entropy on the TF plane. The second algorithm (in Chapter 4), an algorithm for positive TFDs, forces a positive TFD to satisfy the time and frequency marginals and to be closest to the Wigner distribution at the same time. For the positive TFD algorithm, the concept of collective marginal error is also introduced. The last algorithm (in Chapter 5) is to construct the high-concentration TFDs introduced in this thesis for the first time, in the framework of considering the concentration issue explicitly with the construction of TFDs. Typical test signals are provided in each case to demonstrate the effectiveness and, sometimes, the limitations of the three new algorithms. A quantitative indication of performance, the Renyi entropy measure is also provided to validate the claims. Besides the major contributions, basic issues of the mechanism of constructing TFDs and the limitation on energy concentration also obtain proper attention in Chapter 2. In addition, a sampling theory is developed for the Wigner distribution, out of the consideration of applying TFDs to real-world problems. Finally, I conclude the dissertation by pointing out that a comprehensive way to match a suitable TF analysis tool with the signal to be analyzed will be an important research direction in the future.