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Orthonormal shift-invariant adaptive local trigonometric decomposition. (English) Zbl 1007.94514
Summary: An extended library of smooth local trigonometric bases is defined, and an appropriate fast “best-basis” search algorithm is introduced. When compared with the standard local cosine decomposition (LCD), the proposed algorithm is advantageous in three respects. First, it leads to a best-basis expansion that is shift-invariant. Second, the resulting representation is characterized by a lower information cost. Third, the polarity of the folding operator is adapted to the parity properties of the segmented signal at the end-points. The shift invariance stems from an adaptive relative shift of expansions in distinct resolution levels. We show that at any resolution level $$\ell$$ it suffices to examine and select one of two relative shift options – a zero shift or a $$2^{-\ell -1}$$ shift. A variable folding operator, whose polarity is locally adapted to the parity properties of the signal, further enhances the representation. The computational complexity is manageable and comparable to that of the LCD.

##### MSC:
 94A12 Signal theory (characterization, reconstruction, filtering, etc.) 65T40 Numerical methods for trigonometric approximation and interpolation
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