Hirschman uncertainty using Rényi, instead of shannon, entropy is invariant to the Rényi entropy order

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]. In this paper, we develop a new uncertainty measure that incorporates the Rényi entropy instead of the Shannon entropy, and we show that this new uncertainty, which may be viewed as a generalized form of the Hirschman Uncertainty, is in fact invariant to the Rényi entropy order, typically denoted by the term α. This powerful characteristic strongly suggests that Hirschman Uncertainty is a fundamental characteristic of digital signals. Note, too, that the picket fence functions found in [2] minimize the generalized Hirschman Uncertainty for all α, not just for the Shannon entropy of α = 1.