Wideband Weyl symbols for dispersive time-varying processing of systems and random signals

We extend the narrowband Weyl symbol (WS) and the wideband P_O -Weyl symbol (PoWS) for dispersive time-frequency (TF) analysis of nonstationary random processes and time-varying systems. We obtain the new TF symbols using unitary transformations on the WS and the P_O WS. For example, whereas the WS is matched to systems with constant or linear TF characteristics, the new symbols are better matched to systems with dispersive (nonlinear) TF structures. This results from matching the geometry of the unitary transformation to the specific TF characteristics of a system. We also develop new classes of smoothed Weyl symbols that are covariant to TF shifts or time shift and scaling system transformations. These classes of symbols are also extended via unitary warpings to obtain classes of TF symbols covariant to dispersive shifts. We provide examples of the new symbols and symbol classes, and we list some of their desirable properties. Using simulation examples, we demonstrate the advantage of using TF symbols that are matched to the changes in the TF characteristics of a system or random process. We also provide new TF formulations for matched detection applications