Time/scale-adjusted dyadic wavelet packet bases
Stephen P. Del Marco · Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE · 1996
This paper generalizes the dyadic wavelet packet bases (DWP), developed by Coifman and Wickerhauser, to time/scale-adjusted DWP bases. These generalized DWP bases provide more flexibility in matching the time-scale characteristics of the input signal. Development of these generalized bases is achieved by combining the previously defined time-invariant DWP bases of Pesquet, Krim, Carfantan, and Proakis with a generalized scale sampling. The generalized scale sampling extends the usual dyadic sampling by adding a real-valued offset parameter to the integer power of two in the scale parameter. This offset parameter value is taken between zero and one. By combining both scale and translation generalizations, signal components existing between consecutive dyadic scales, or consecutive time translations, may be captured. It is shown how these DWP coefficients may be generated from a two step process; first projecting the input signal onto an appropriate space. Then, performing the usual wavelet low and highpass filtering operations, followed by downsampling. The projection operation is shown to be equivalent to a filtering operation. An expression for the filter taps is derived, and basic properties are proven. A translation-invariant transform defined on these scale-adjusted wavelet packets, is developed. An application to transient detection is presented, by developing a transient detector based on this transform. ROC curves, generated by Monte- Carlo simulation, are presented demonstrating detector performance. Detector performance is shown to be independent of the signal translation. It is further shown how matching the basis functions to the time-scale-frequency characteristics of the transient can provide improved detection performance.