The uncertainty principle: global, local, or both?

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.