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 (1992) 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