Optimum Signals and the Uncertainty Principle

John Price

Feifei Qu

Abstract The classical uncertainty principle works for smooth signal functions. In our work, we apply the Fourier transform derivatives for the study of uncertainty principle, so that the smoothness condition for the signal functions is not required. At first, the amplitude and phase derivatives of vector-valued signal functions based on the Fourier transform are defined. Then we obtain a strong form of the uncertainty principle.

P Marquardt, G. Galeczki

Jianping Li

The uncertainty principle of signal is proved according to the uncertainty principle of quantum mechanics,and is further interpreted by quantum mechanics.The signal can be considered as a kind of the quasi quantum system with wave-particle dualism.Because of the wave-particle dualism of the signal,time and frequency can not be measured at the same time.The frequency operator(FO) of signal is defined in this study,and it can be widely used in signal theory and,especially in wavelet research.The quantum interpretation for the signal explains the essence of the signal and provides the theory support for applying the framework of quantum mechanics to the signal processing.

Steeve Zozor, M. Portesi, Christophe Vignat

Patrick J. Loughlin, Leon W. Cohen

We address the issue of the relation between local quantities and the uncertainty principle. We approach the problem by defining local quantities as conditional standard deviations, and we relate these to the uncertainty product appearing in the standard uncertainty principle. We show that the uncertainty product for the average local standard deviations is always less than or equal to the standard uncertainty product and that it can be arbitrarily small. We apply these results to the short-time Fourier transform/spectrogram to explore the commonly held notion that the uncertainty principle somehow limits local quantities. We show that, indeed, for the spectrogram, there is a lower bound on the local uncertainty product of the spectrogram due to the windowing operation of this method. This limitation is an inherent property of the spectrogram and is not a property of the signal or a fundamental limit. We also examine the local uncertainty product for a large class of time-frequency distributions that satisfy the usual uncertainty principle, including the Wigner distribution, the Choi-Williams distribution, and many other commonly used distributions. We obtain an expression for the local uncertainty product in terms of the signal and show that for these distributions, the local uncertainty product is less than that of the spectrogram and can be arbitrarily small. Extension of our approach to an entropy formulation of the uncertainty principle is also considered.