Unitary equivalence: a new twist on signal processing

Unitary similarity transformations furnish a powerful vehicle for generating innnite generic classes of signal analysis and processing tools based on concepts diierent from time, frequency, and scale. Implementation of these new tools involves simply preprocessing the signal by a unitary transformation , performing standard processing techniques on the transformed signal, and then (in some cases) transforming the resulting output. The resulting unitarily equivalent systems focus on the critical signal characteristics in large classes of signals and, hence, prove useful for representing and processing signals that are not well matched by current techniques. As speciic examples of this procedure, we generalize linear time-invariant systems, orthonormal basis and frame decomposi-tions, and joint time-frequency and timescale distributions, illustrating the utility of the unitary equivalence concept for uniting seemingly disparate approaches proposed in the literature.