A Signal Dependent Time Frequency Representation: Optimal Kernel Design
Baraniuk, Richard G.
Jones, Douglas L.
Time-frequency distributions (TFDs), 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 TFDs, 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 <i>signal-dependent</i> 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 withwhich the formulation can be customized to incorporate application-specific knowledge into the design process.