On the density of phase-space expansions

Phase-space decompositions for signals of finite energy have been used to formulate the intuitive but elusive idea that frequency content of a signal can vary with time as the signal evolves. The decompositions consist of subdividing the time and frequency axes into certain ranges and expanding signals stably in a two-parameter basis of fixed functions of time, in which the (k,m)th coefficient is viewed as describing the part of the signal that is concentrated in the kth time and mth frequency range. It is shown that in any such expansion, the time-frequency space must be subdivided, or, equivalently, the coefficients computed, at least at the Nyquist rate of two per unit of time and cycle of bandwidth