On Translation Invariant Subspaces and Critically Sampled Wavelet Transforms

The discrete wavelet transform (DWT) is attractive for many reasons. Its sparse sampling grid eliminates redundancy and is very efficient. Its localized basis functions are well suited for processing non–stationary signals such as transients. On the other hand, its lack of translation invariance is a major pitfall for applications such as radar and sonar, particularly in a multipath environment where numerous signal components arrive with arbitrary delays. The paper proposes the use of robust representations as a solution to the translation invariance problem. We measure robustness in terms of a mean square error for which we derive an expression that describes this translation error in the Zak domain. We develop an iterative algorithm in the Zak domain for designing increasingly robust representations. The result is an approach for generating multiresolution subspaces that retain most of their coefficient energy as the input signal is shifted. A typical robust subspace retains 98% of its energy, a significant improvement over more traditional wavelet representations.