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Electrical engineering and applied signal processing series/The electrical engineering and applied signal processing series

Positive Time–Frequency Distributions

Leon W. Cohen, Patrick J. Loughlin

Published Oct 29, 2002• 1 citation
Publisher
DOI
10.1201/9781420042467.ch3

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The position-momentum quasidistribution obtained from an Arthurs and Kelly joint measurement model is used to obtain indirectly an ``operational'' time-of-arrival (TOA) distribution following a quantization procedure proposed by Kocha\ifmmode \acute{n}\else \'{n}\fi{}ski and W\'odkiewicz [Phys. Rev. A 60, 2689 (1999)]. This TOA distribution is not time covariant. The procedure is generalized by using other phase-space quasidistributions, and sufficient conditions are provided for time covariance that limit the possible phase-space quasidistributions essentially to the Wigner function, which, however, provides a nonpositive TOA quasidistribution. These problems are remedied with a different quantization approach which, on the other hand, does not guarantee normalization. Finally, an Arthurs and Kelly measurement model for TOA and energy is worked out, valid also for arbitrary conjugate variables when one of the variables is bounded from below. The marginal TOA distribution so obtained, a distorted version of Kijowski's distribution, is time covariant, positive, and normalized.
Let a “complex probability’’ be a normalizable complex distribution P(x) defined on RD. A real and positive probability distribution p(z), defined on the complex plane CD, is said to be a positive representation of P(x) if 〈Q(x)〉P=〈Q(z)〉p, where Q(x) is any polynomial in RD and Q(z) its analytical extension on CD. In this paper it is shown that every complex probability admits a real representation and a constructive method is given. Among other results, explicit positive representations, in any number of dimensions, are given for any complex distribution of the form Gaussian times polynomial, for any complex distributions with support at one point and for any periodic Gaussian times polynomial.