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 choi ce of the Heisenberg rectangles tiling the time-frequency domain over which the energy of the signal is distributed. It is also shown how op timizing a concave objective function is equivalent to concentrating m aximal energy into a few basis elements. Signal-adaptive basis selecti on algorithms currently in use do not generally find the absolute best basis, and moreover have an asymmetric time-frequency adaptivity-alth ough a complete wavepacket decomposition comprises a symmetric set of tilings with respect to time and frequency. The higher adaptivity in f requency than in time can lead to ignoring frequencies that exist over short time intervals (short as compared to the length of the whole si gnal, not to the period corresponding to these frequencies). Revealing short-lived frequencies to the investigator can bring up important fe atures of the studied process, such as the presence of coherent ('pers istent') structures in a time series.