Time-frequency projection filters and time-frequency signal expansions

We consider the problems of designing a linear, time-varying filter with a specified “time-frequency (TF) pass region” and of constructing an orthonormal basis for the parsimonious expansion of signals located in a given TF support region. These problems of TF filtering and TF signal expansion are reduced to the problem of designing a “TF subspace”, i.e., a linear signal space comprising all signals located in a given TF legion. Specifically, the TF filter is taken to be the orthogonal projection operator on the TF subspace. We present an optimum design of TF subspaces that is based on the Wigner distribution of a linear signal space and is an extension of the well-known signal synthesis problem. The optimum TF subspace is shown to be an “eigenspace” of the TF region, and some properties of eigenspaces are discussed. The performance of TF projection filters and TF signal expansions is studied both analytically and via computer simulation