Multiresolution sequence detection in rapidly fading channels based on focused wavelet decompositions

The problem of detecting a sequence using the maximum-likelihood strategy, when the time-varying multipath channel is unknown, can be in principle solved using the generalized likelihood detector (GLD). The GLD metric involves finding the orthogonal projection of the received signal onto the subspace of the transmitted signal distorted by the multipath channel: the faded signal corresponding to one of the transmitted sequences is in fact known to lie in a subspace but its exact location is not known, because the channel parameters are unknown. This detector (which in the case of a static channel collapses to the per-survivor processing method), is also called a matched subspace detector because its statistic is "matched" to the a priori known signal subspace. Unfortunately, the computation of the (perfectly matched) orthogonal projection of the received signal onto the multipath faded signal subspace is, in the general time-varying case, impossible. We introduce in this work the idea of using new wavelet-based subspaces that approximate the original signal subspace. The nested sequence of linear vector spaces, defined by a wavelet-based multiresolution decomposition of the fading channel time variations, provides a set of subspaces that, at an increasingly high level of detail, are "efficient" representations of the original signal subspace. For each of these representations sequence detection at different levels of resolution can be performed.