Hirschman uncertainty with the discrete fractional fourier transform

The Hirschman Uncertainty [1] is defined by the average of the Shannon entropies of a discrete-time signal and its Fourier transform. The optimal basis for the Hirschman Uncertainty has been shown to be the picket fence function, as given in a previous paper of ours [2]. We have seen that a basis can be constructed from signals with minimum Hirschman Uncertainty in that paper, leading to a transform we called the Hirschman Optimal Transform. Recently, we showed that the minimizers, and thus the uncertainty, are invariant to the Rényi entropy order [3]. This characteristic strongly suggests that Hirschman Uncertainty is a fundamental characteristic of digital signals. In this paper, we study the effect of incorporating the discrete fractional fourier transfom (discrete FRT) instead of the DFT and develop a new uncertainty measure denoted by U^a_1/2(x).