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.