A Signal-Dependent Time-Frequency Representation: Optimal Kernel Design

Time-frequency distributions (TFD‘s) which indicate the energy content of a signal as a function of both time and frequency, are powerful tools for time-varying signal analysis. The lack of a single distribution that is “best” for all applications has resulted in a proliferation of TFD's, each corresponding to a different, fixed mapping from signals to the time-frequency plane. A major drawback of all fixed mappings is that, for each mapping, the resulting time-frequency representation is satisfactory only for a limited class of signals. In this paper, we introduce a new TFD that adapts to each signal and so offers good performance for a large class of signals. The design of the signal-dependent TFD is formulated in Cohen's class as an optimization problem and results in a special linear program. Given a signal to be analyzed, the solution to the linear program yields the optimal kernel and, hence, the optimal time-frequency mapping for that signal. A fast algorithm has been developed for solving the linear program, allowing the computation of the signal-dependent TFD with a time complexity on the same order as a fixed-kernel distribution. Besides this computational efficiency, an attractive feature of the optimization-based approach is the ease with which the formulation can be customized to incorporate application-specific knowledge into the design process.