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Expected Number of Maxima in the Envelope of a
Spherically Invariant Random Process
Ali Abdi and Said Nader-Esfahani
Abstract--In many engineering applications, specially in communication engineering, one
usually encounters a bandpass non-Gaussian random process, with a slowly varying envelope.
Among the available models for non-Gaussian random processes, spherically invariant random
processes (SIRP's) play an important role. These processes are of interest mainly due to the fact
that they allow one to relax the assumption of Gaussianity, while keeping many of its useful
characteristics. In this paper, we have derived a simple and closed-form formula for the expected
number of maxima of a SIRP envelope. Since Gaussian random processes are special cases of
SIRP's, this formula holds for Gaussian random processes as well. In contrast with the available
complicated expression for the expected number of maxima in the envelope of a Gaussian
random process, our simple result holds for an arbitrary power spectrum. The key idea in
deriving this result is the application of the characteristic function, rather than the probability
density function, for calculating the expected level crossing rate of a random process.
Index Terms--Characteristic function, Envelope, Gaussian processes, Level crossing problems,
Maxima of the envelope, Spherically invariant processes.
· This work has been supported in part by Grant No. 612-1-204 of the University of Tehran.
· Parts of this work have been presented in IEEE Int. Symp. Inform. Theory, Whistler, B.C., Canada, 1995,