A function of time, frequency, lag, and Doppler

In signal processing, four functions of one variable are commonly used. They are the signal in time, the spectrum, the auto-correlation function of the signal, and the auto-correlation function of the spectrum. The variables of these functions are denoted, respectively, as time, frequency, lag, and Doppler. In time-frequency analysis, these functions of one variable are extended to quadratic functions of two variables. In this paper, we investigate a method for creating quartic functions of three of these variables as well as a quartic function of all four variables. These quartic functions provide a meaningful representation of the signal that goes beyond the well-known quadratic functions. The quartic functions are applied to the design of signal-adaptive kernels for the Cohen class and shown to provide improvements over previous methods.