New time-frequency symbol classification

We propose new time-frequency (TF) symbols as the narrowband Weyl symbol (WS) smoothed by an appropriate kernel. These new symbols preserve time and frequency shifts on a random process. Choosing specific smoothing kernels, we can obtain various new symbols (e.g. Levin symbol and Page symbol). We link a quadratic form of the signal to the new symbols and Cohen’s class of quadratic time-frequency representations, and we derive a simple kernel constraint for unitary symbols. We also propose an affine class of symbols in terms of the wideband Weyl symbol (P0WS). These symbols preserve scale changes and time shifts. Furthermore, we generalize the smoothed versions of the WS and P0WS to analyze random processes undergoing generalized frequency shifts or generalized time shifts.