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Area of Science:

  • Stochastic processes
  • Signal analysis
  • Probability theory

Background:

  • Calculating signal level crossings is crucial in fields like image analysis, speech recognition, and material science.
  • The Rice formula provides the mean number of zero crossings for Gaussian stationary processes, but higher-order statistics remain challenging.
  • Existing methods struggle with complex, non-Gaussian, or non-Markovian processes.

Purpose of the Study:

  • To derive analytic expressions for higher-order zero-crossing cumulants and moments.
  • To extend the capabilities beyond the traditional Rice formula for stochastic processes.
  • To provide a more comprehensive understanding of signal crossing behavior.

Main Methods:

  • Utilized the independent interval approximation.
  • Derived analytic expressions for all higher-order zero-crossing cumulants and moments.
  • Validated results against simulations for non-Markovian autoregressive models.

Main Results:

  • Successfully derived analytic expressions for higher-order zero-crossing statistics.
  • The derived formulas provide a more detailed characterization of signal crossings.
  • Results demonstrated good agreement with simulation data for complex models.

Conclusions:

  • The independent interval approximation offers a powerful tool to extend Rice's formula.
  • The new analytic expressions enable more accurate analysis of signal level crossings.
  • This work advances the theoretical understanding and practical application of stochastic process analysis.