Expected number of maxima in the envelope of a spherically invariant random process
In many engineering applications, specially in communication engineering, one 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.
envelope characteristic function gaussian processes level crossing problems maxima of the envelope spherically invariant processes