A function of time, frequency, lag, and Doppler

In signal processing, four functions of one variable are commonly used. The y are the signal in time, the spectrum, the auto-correlation function of th e 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 ex tended to quadratic functions of two variables. In this paper, we investiga te a method for creating quartic functions of three of these variables as w ell as a quartic function of all four variables. These quartic functions pr ovide a meaningful representation of the signal that goes beyond the well-k nown quadratic functions. The quartic functions are applied to the design o f signal-adaptive kernels for the Cohen class and shown to provide improvem ents over previous methods.