Time-Frequency Spatial-Spatial Frequency Represent a tions" LEON COHEN Hunter College and Graduate Center zyxw of CUNY New York, New York 10021 INTRODUCTION zyxw For many signals in nature the frequency content, the spectrum, changes in time. The basic reason is that the physical processes producing the signal are time depen- dent. The attempt to understand how the spectrum of signals change in time is called time-frequency analysis. In recent years there has been significant progress made in developing the mathematical and physical ideas for describing a time- varying spectrum.' Analogous to time-frequency analysis is spatial-spatial frequency analysis. It generally deals with images and how their spatial frequency content changes with l ~ c a t i o n . ~ - ~ It appears, though, that a combined time-frequency spatial-spatial frequency analysis has not been developed and it is the purpose of this chapter to do so. Possible applications are situations where there is considerable variation in both space and time, such as on the surface of the sun. Also, for propagation of pulses we expect there to be a strong relation between spatial frequencies and ordinary fre- quencies and hence we need a tool that would establish these correlations or lack of them. Before developing these representations we give a brief outline for the time- frequency case so that we can see how to generalize to this situation. CHARACTERISTIC FUNCTION OPERATOR METHOD For a density of two variables P(a, b) the characteristic function is defined byb zy M(6, T) zyxwvutsrqp = {lP(t, w)eiet+i'o dt do = (eier+irw ) and, inversely, the density can be obtained from the characteristic function, "Work supported in part zyxwvu by the NSA HBCU/MI Program and the PSC-CUNY Research Award Program. Also, this work was done zyxwv as part of a NATO Collaborative Research Grants Pro am. ! L integrals go from - zyx co to co. 97