A study on the impact of the fourier transform on Hirschman Uncertainty

The Hirschman Uncertainty [1] is defined by the average of the Shannon entropies of a discrete-time signal and its Fourier transform. The picket fence functions has been found to be the optimal basis for the Hirschman Uncertainty, 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. In [3] and [4], we showed that the minimizers, and thus the uncertainty of these minimizers, are invariant to the Rényi entropy order. 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 (dFRT) instead of the DFT in the predefined Hirschman Unceratinty and develop a new uncertainty measure Hirschman uncertainty using the dFRT denoted by Ua1/2(x). We explore this new uncertainty measure using the discrete signals and study how the transfer order variations affects the uncertainty of different discrete signals. To help verify our theory, we study the effect of transfer order value variations on the classification rate in an image classification experiment.