Abstract
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 language | English (US) |
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Pages (from-to) | 1226-1231 |
Number of pages | 6 |
Journal | IEEE Transactions on Information Theory |
Volume | 48 |
Issue number | 5 |
DOIs | |
State | Published - May 2002 |
All Science Journal Classification (ASJC) codes
- Information Systems
- Computer Science Applications
- Library and Information Sciences
Keywords
- Almost everywhere differentiability
- Differentiability with probability one
- Envelope
- Envelope second derivative
- Fading channels
- Gaussian process
- Mean square differentiability
- Rayleigh process