On the second derivative of a Gaussian process envelope

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In this correspondence, we explore some dynamic characteristics of the envelope of a bandpass Gaussian process, which are of interest in wireless fading channels. Specifically, we show that unlike the first derivative, the second derivative of the envelope, which appears in a number of applications, does not exist in the traditional mean square sense. However, we prove that the envelope is twice differentiable almost everywhere (with probability one) if the power spectrum of the bandpass Gaussian process satisfies a certain condition. We also derive an integral form for the probability density function (pdf) of the second derivative of the envelope, assuming an arbitrary power spectrum.

Original languageEnglish (US)
Pages (from-to)1226-1231
Number of pages6
JournalIEEE Transactions on Information Theory
Issue number5
StatePublished - May 2002

All Science Journal Classification (ASJC) codes

  • Information Systems
  • Computer Science Applications
  • Library and Information Sciences


  • Almost everywhere differentiability
  • Differentiability with probability one
  • Envelope
  • Envelope second derivative
  • Fading channels
  • Gaussian process
  • Mean square differentiability
  • Rayleigh process


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