Time-frequency duality

This paper develops a concept of duality, called time-frequency duality, which is applicable to a class of networks called communication-signal-processing networks. Such networks consist of an interconnection of basic elements such as filters, mixers, delay lines, etc. The usefulness of time-frequency duality stems from the fact that two situations which may be quite distinct physically can have identical behavior patterns except for an interchange of the roles played by time and frequency. As a result, the solution to a detection or estimation problem may be found directly from the solution of the dual problem, if known, merely by replacing variables and quantities by their duals. This application of time-frequency duality is illustrated in the paper by the problem of measuring the transfer function of a scatter medium by means of an optimal gating operation prior to spectrum analysis. Another benefit to be obtained from time-frequency duality is the generation of new ideas for communication signal processing techniques. We illustrate this type of benefit by constructing the dual of the Kineplex communication system. A third benefit is the additional insight gained into a communication problem by the ability to look at it in another way. This benefit is illustrated by the characterization of time-variant linear channels. It is demonstrated that such channels may be characterized in an interesting symmetrical manner in time and frequency variables by defining dual system functions.