Uncertainty Relations and Time-Frequency Distributions for Unsharp Observables

This paper deals with a new framework in analyzing the formal mathematical correspondence between quantum mechanics and time-frequency representations of a signal. It is also shown that joint time-frequency distributions have a close link with Heisenberg uncertainty relations if the observables are taken as fuzzy entities. This result contradicts the arguments of Cohen [IEEE Proc. 77(7):941 (1989)] regarding the time-frequency distributions and the uncertainty relation. It is postulated that these mechanisms will be of crucial importance in highly fragmented computation structures, such as neural networks, as they may exhibit a strong mutual interaction between data and operator. 1. I N T R O D U C T I O N Gabor [1] published a pioneer ing paper in 1946 on t ime-frequency representat ions of signals in the context of communica t ion theory. It was Wigner [2] who first investigated the possibility of construct ing joint distribution functions in phase-space and their impor tance in the domain of quan tum mechanics. Later on, Ville [3] studied the joint distribution functions in signal analysis in the spirit of Wigner. In 1965, Cohen [4] suggested a general ized approach for construct ing the joint distributions in INFORMATION SCIENCES 89, 193-209 (1996) © Elsevier Science Inc. 1996 0020-0255/96/$15.00 655 Avenue of the Americas, New York, NY 10010 SSDI 0020-0255(95)00232-4