The optimal solutions to the continuous and discrete-time versions of the Hirschman uncertainty principle

We have previously developed an uncertainty measure that is suitable for finitely-supported (N samples) discrete-time signals. A specific instance of our measure has been termed the “discrete Hirschman (1957) uncertainty principal” in the literature, and we have adopted this terminology for our more general measure. We compare the optimal signals of this discrete version to the already determined optimal signals of the (continuous-time) Hirschman uncertainty principal. From our comparison, we conclude that a basic premise in signal processing, that if we sample densely enough, the discrete-time case directly corresponds to the continuous-time case, is not correct in this instance. The arithmetic of N, which seems to have no analog in continuous time, is crucial to the construction of the Hirschman optimal discrete representation. We suggest that more work in this important area be performed to determine what impact this has, and to find out how widespread this problem may be