Some theory and an experiment on the fundamentals of Hirschman uncertainty

The Heisenberg Uncertainty principle is a fundamental concept from Quantum Mechanics that also describes the Fourier Transform. Unfortunately, it does not directly apply to the digital signals. However, it can be generalized if we use entropy rather than energy to form an uncertainty relation. This form of uncertainty, called the Hirschman Uncertainty, uses the Shannon Entropy. The Hirschman Uncertainty is defined as the average of the Shannon entropies of the discrete-time signal and its Fourier Transform. The functions that minimize this uncertainty are not the wellknown Gaussians from the Heisenberg theory, but are the picket fence functions first noticed in wavelet denoising. This connection suggests that the Hirschman Uncertainty is fundamental, but not conclusively. Here in this research, we develop two new uncertainty measures that are derived from the Hirschman Uncertainty. We want to use these measures to explore the fundamental nature of the Hirschman Uncertainty. In the first case, we replace the Shannon entropy with the R´enyi entropy and study the impact of varying the R´enyi order on the uncertainty of various digital signals. We call this new uncertainty measure, the Hirschman-R´enyi uncertainty denoted by U.;alpha1/2(x). We find that the derived uncertaintymeasure is invariant to the R´enyi order in case of the picket fence signals and varies in case of other the digital signals like rectagular, cosine, square wave signals to name a few. This new uncertainty measure that utilizes the R´enyi entropy decays with the increase in R´enyi order value. Considering the invariance in uncertainty in case of picket fence signal, we can use either Shannon or R´enyi entropy with any value of R´enyi order to calculate Hirschman Uncertainty. In the second case, we derive an uncertainty measure that replaces the Fourier Transform with the Fractional Fourier Transform. The Hirschman Uncertainty using dFRT denoted by U.;a1/ 2(x)is explored with the help of the minimizers of the Hirschman Uncertainty (the picket fence signals) along with the other digital signals. In this case, we find that the degree of rotation in the Fractional Fourier Transform does impact the uncertainty at the integer values of the transfer order. But for the non-interger values of the transfer order, the uncertainty variations are greatly reduced or are minimal. Finally to help verify our theory, we perform a classical texture recognition experiment. We find that the recognition performance follows directly as our developed Hirschman R´enyi Uncertainty and the Hirschman Uncertainty using dFRT theory suggests. Additionally, it appears that a predictive solution for the proper selection of the R´enyi order and the rotation angle can be developed that could significantly aid in image analysis. Our recognition results are consistent with entropic invariance theory in case of the two uncertainty measures. These results suggests that the Hirschman Uncertainty may be a fundamental characteristic of the digital signals.