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IEEE-SP International Symposium on Time-Frequency and Time-Scale Analysis

The uncertainty principle in signal analysis

Leon W. Cohen

Published Dec 17, 2002• 31 citations
Publisher

Abstract

We give a simple derivation for the covariance of time and frequency. The simplicity remains when applied to arbitrary variables. We show that the covariance enters into the uncertainty principle in a fundamental way. Also we present a seeming paradox: the uncertainty principle depends only on the marginals, but the marginals carry no information about covariance. Then how is it possible that the uncertainty principle involves the covariance? The resolution of the paradox clarifies a number of issues regarding the existence of well behaved manifestly positive joint distributions of time and frequency.>

DOI
10.1109/tfsa.1994.467263

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We give a simple derivation for the covariance of time and frequency. The simplicity remains when applied to arbitrary variables. We show that the covariance enters into the uncertainty principle in a fundamental way. Also we present a seeming paradox: the uncertainty principle depends only on the marginals, but the marginals carry no information about covariance. Then how is it possible that the uncertainty principle involves the covariance? The resolution of the paradox clarifies a number of issues regarding the existence of well behaved manifestly positive joint distributions of time and frequency.>
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.
We examine the construction of joint probabilities for non-commuting observables. We show that there are indications in standard quantum mechanics that imply the existence of conditional expectation values, which in turn implies the existence of a joint distribution. We also argue that the uncertainty principle has no bearing on the existence of joint distributions but only constrains the marginal distributions. In addition, we show that within classical probability theory there are mathematical quantities that are similar to quantum mechanical wave functions. This is shown by generalising a theorem of Khinchin on the necessary and sufficient conditions for a function to be a characteristic function.
Does the Heisenberg uncertainty principle refer to errors of measurement, or to the spread of values of the physical variables intrinsic to a particle’s state, or to some combination of these? The most commonly quoted form of the uncertainty principle relates the spread of an ensemble of separate measurements of some variable to the analogous spread of its conjugate variable. In contrast, Heisenberg’s original argument for the uncertainty principle involved the perturbation to a particle’s state by a measurement of one variable, which affects one’s ability to predict the outcome of a subsequent measurement of the conjugate variable. The relation between these two views of the uncertainty principle is discussed in this paper. A familiar example is considered: an ensemble of identically prepared particles passing through a slit, and after further propagation being detected. From this arrangement it is possible to infer joint (although necessarily imprecise) information on both transverse position and conjugate momentum for each member of the ensemble. It is shown that in this case of joint measurement the product of standard deviations for the measurement outcomes is at least twice as large as the lower bound implied by the usual uncertainty principle. The discussion is meant to help clarify the different roles played in the various statements of the uncertainty principle by the initial state and by measurement error.