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Zero-Crossings of Random Processes with Application to Estimation and Detection

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Nonuniform Sampling

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Abstract

The origin of Rice’s formula for the average level-crossing rate of a general class of random processes can be traced back to his 1936 notes on “Singing Transmission Lines,” [35]. For a stationary, ergodic, random process {X(t)} , -∞ < t < ∞, with sufficiently smooth sample paths, the average number of crossings about the level u, per unit time, is given by Rice’s formula [36]

$$ E\left[ {D_u } \right] = \int_{ - \infty }^\infty {\left| {x'} \right|p\left( {u,x'} \right)dx'} $$
((1))

where p(x, x’) is the joint probability density of the X(t) and its mean-square derivative X’(t), and Du is the number of u level-crossings of {X(t)} for t in the unit interval [0,1]. Rice’s formula is quite general in nature, and as we shall see, has a simplified form when the process is Gaussian.

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Authors and Affiliations

  1. Intelligence, Surveillance, Reconnaissance Department, Space and Naval Warfare Systems Center, San Diego, California, 92152, USA

    J. T. Barnett

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Editors and Affiliations

  1. King’s College, London, UK

    Farokh Marvasti

  2. Sharif College of Technology, Tehran, Iran

    Farokh Marvasti

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Barnett, J.T. (2001). Zero-Crossings of Random Processes with Application to Estimation and Detection. In: Marvasti, F. (eds) Nonuniform Sampling. Information Technology: Transmission, Processing, and Storage. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-1229-5_9

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  • DOI: https://doi.org/10.1007/978-1-4615-1229-5_9

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4613-5451-2

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