Ten lectures on wavelets
An M-band, 2-dimensional translation-invariant wavelet transform and applications
ICASSP '95 Proceedings of the Acoustics, Speech, and Signal Processing, 1995. on International Conference - Volume 02
Shift invariant wavelet packet bases
ICASSP '95 Proceedings of the Acoustics, Speech, and Signal Processing, 1995. on International Conference - Volume 02
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The discrete wavelet transform(DWT) is attractive for many reasons. Its sparse sampling grideliminates redundancy and is very efficient. Its localized basisfunctions are well suited for processing non–stationarysignals such as transients. On the other hand, its lack of translationinvariance is a major pitfall for applications such as radarand sonar, particularly in a multipath environment where numeroussignal components arrive with arbitrary delays. The paper proposesthe use of robust representations as a solution to the translationinvariance problem. We measure robustness in terms of a meansquare error for which we derive an expression that describesthis translation error in the Zak domain. We develop an iterativealgorithm in the Zak domain for designing increasingly robustrepresentations. The result is an approach for generating multiresolutionsubspaces that retain most of their coefficient energy as theinput signal is shifted. A typical robust subspace retains 98\%of its energy, a significant improvement over more traditionalwavelet representations.