Absolute optimal time-frequency basis - a research tool

The paper presents a method for finding the absolute best basis out of the library of bases offered by the wavelet packet decomposition of a discrete signal. Data-adaptive optimality is achieved with respect to an objective function, e.g. minimizing entropy, and concerns the choice of the Heisenberg rectangles tiling the time-frequency domain over which the energy of the signal is distributed. It is also shown how optimizing a concave objective function is equivalent to concentrating maximal energy into a few basis elements. Signal-adaptive basis selection algorithms currently in use do not generally find the absolute best basis, and moreover have an asymmetric time-frequency adaptivity-although a complete wavepacket decomposition comprises a symmetric set of tilings with respect to time and frequency. The higher adaptivity in frequency than in time can lead to ignoring frequencies that exist over short time intervals (short as compared to the length of the whole signal, not to the period corresponding to these frequencies). Revealing short-lived frequencies to the investigator can bring up important features of the studied process, such as the presence of coherent ('persistent') structures in a time series.