The principal objectives of the present work is to obtain a formulation of the distribution for the intensity fluctuation of a sampled random signal having an arbitrary correlation and distribution function without simplifying the problem by assuming a simplified input model such as the Gaussian distribution, white noise characteristics etc., and to confirm experimentally the validity of the formulation for few special cases of interest by the method of digital simulation. Letting
xi, denote the
i-th sampled data of the random signal at time
ti(
i=1, 2, …,
K), an explicit expression of p. d. f.
P(
E) of
E=_??_
xi2 is found to be
P(
E)=
Em-1/
Γ(m)Sme-E/S{1+_??_
BnLn(m-1)(
E/S)}(
m=
K/2),
Bn=(-1)
mΓ(m)n!/
Γ(
m+
n)_??_‹_??_1/
ni!
Hni(
xi/σ
i)›(1/π)
m2
nΓ(
n1/2)
Γ(
n2+1/2)…
Γ(
nK+1/2)
as a solution of an integral equation.
Introducing a dimensionless variable η=
E/
S into the above equation, one obtains
P(η)=
PΓ(η,
m)+_??_(-1)
nDnPΓ(n)(η,
n+
m),
Dn=(-1)
nΓ(
m+
n)/
Γ(m)n!
Bnwhere
PΓ(η,
m)_??_η
m-1exp(-η)/
Γ(m) is the well-known gamma distribution density. It should be noticed that the expansion coefficients
Dn express respectively the effect of linear and nonlinear correlations among
xi's.
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