Time-varying orthonormal tilings of the time-frequency plane

Expansions that give arbitrary orthonormal tilings of the time-frequency plane are considered. These differ from the short-time Fourier transform, wavelet transform, and wavelet packet tilings in that they change over time. It is shown how this can be achieved using time-varying orthogonal tree structures, which preserve orthogonality, even across transitions. One method is based on lapped orthogonal transforms, which makes it possible to change the number of channels in the transform. A second method is based on the construction of orthogonal boundary filters to construct essentially arbitrary tilings. A double-tree algorithm is presented that, for a given signal, decides on the best binary segmentation and on which tree split to use for each segment. That is, it is a joint optimization of time and frequency splitting. The algorithm is optimal for additive cost functions (e.g., rate distortion), which gives the best time-varying bases. Results of experiments on test signals are shown.