Scaling functions robust to translations

The discrete wavelet transform (DWT) is popular in a wide variety of applications. Its sparse sampling eliminates redundancy in the representation of signals and leads to efficient processing. However, the DWT lacks translation invariance. This makes it ill suited for many problems where the received signal is the superposition of arbitrarily shifted replicas of a transmitted signal as when multipath occurs, for example. The paper develops algorithms for the design of orthogonal and biorthogonal compact support scaling functions that are robust to translations. Our approach is to maintain the critical sampling of the DWT while designing multiresolution representations for which the coefficient energy redistributes itself mostly within each subband and not across the entire time-scale plane. We obtain expedite algorithms by decoupling the optimization from the constraints on the scaling function. Examples illustrate that the designed scaling function significantly improves the robustness of the representation.