An operator theory approach to discrete time-frequency distributions

The theoretical link between a discrete-time sequence and its discrete-time/discrete-frequency representation has heretofore been established via a uniform sampling of their continuous-time counterparts. We provide a direct link between the two which we establish using the concepts of operator theory. We see that many similarities, but also some important differences, exist between the results of the continuous-time operator approach and our discrete one. The differences between the continuous distributions and discrete ones may not be the simple sampling relationship which has so often been assumed. Through basic matrix operations, discrete-time/discrete-frequency distributions can be generated using our operators, and we show that: (a) key properties like positivity are much easier to formulate and solve in the discrete case, and (b) while proper quadratic distributions are not possible using the Fourier transform, they do indeed exist for other transforms.