 
Summary: Page 1 of 19
On the Second Derivative of a Gaussian Process Envelope
Ali Abdi
AbstractIn this paper 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
integralform for the probability density function of the second derivative of the envelope,
assuming an arbitrary power spectrum.
Index TermsEnvelope, Envelope second derivative, Gaussian process, Rayleigh process, Mean
square differentiability, Almost everywhere differentiability, Differentiability
with probability one, Fading channels.
List of figure captions:
Fig. 1. The joint probability density function of R and for two different power spectra:
Upper: Exponential nonsymmetric spectrum ( ) exp( ), 2 2I m mw f f f f f=   <  < ,
Lower: Gaussian symmetric spectrum 2
( ) exp[ ( ) ], 1 1I m mw f f f f f=    <  < .
Fig. 2. The probability density function of R for two different power spectra:
